==== TABLE OF CONTENTS ==== Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//01-introduction.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//02-inductive-reduction.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03-programs.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03a-program-foundations.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03b-projective-good-model.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-base-presentations.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-restricted-induction.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//04-boundary.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-projective-program-input.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-rank-one.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//06-simple-nonvanishing.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//07-signed-rigidity.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//main.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//preamble.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//two-diagonals.tex Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//references.bib =========================== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//01-introduction.tex ==== \section{Introduction}\label{sec:introduction} The abundance problem asks when the numerical positivity of a log canonical bundle produces a holomorphic map. More precisely, a nef adjoint should have a positive multiple generated by global sections. On a compact Kähler space, nefness is an analytic condition on a cohomology class, whereas generation is a statement about an actual holomorphic line bundle. The distinction is essential in the nonprojective setting. We prove log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity for smooth projective varieties with reduced boundary. The argument passes through a stronger birational statement: every pseudo-effective smooth log adjoint has a semiample positive part and an effective rational fixed divisor equal to its analytic divisorial negative part. This form both supports the induction and retains the line-bundle information needed for generation on the original space. \subsection{The assumption and the theorem} A compact Kähler space is a compact complex analytic space with a Kähler form in the sense of local strictly plurisubharmonic potentials on local embeddings. For a rational line bundle \(L\), analytic nefness means that \(c_1(L)\) belongs to the closure of the Kähler cone. Equivalently, fix a Kähler form \(\omega\) and an integer \(r>0\) for which \(rL\) is a line bundle. For every \(\varepsilon>0\), that line has a smooth Hermitian metric \(h_\varepsilon\) satisfying \[ \frac{\sqrt{-1}}{2\pi r}\Theta_{h_\varepsilon} \geq-\varepsilon\omega. \] Smooth metrics and forms on a singular space are understood through local embeddings. We call \(L\) semiample if an actual positive integral multiple is a holomorphic line bundle generated by its global sections. \begin{assumption}[Logarithmic Iitaka subadditivity] \label{asm:log-iitaka} Let \(f:X\to Y\) be a surjective morphism with connected fibers between smooth connected projective complex varieties. Let \(D_X\) and \(D_Y\) be reduced effective simple normal crossing divisors, allowing zero divisors, such that \[ \Supp(f^*D_Y)\subseteq\Supp(D_X). \] For a very general smooth fiber \(F\), put \(D_F=D_X|_F\). Then \[ \kappa(X,K_X+D_X)\geq \kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y). \] Here \(D_F\) is reduced SNC and \(K_F+D_F\sim (K_X+D_X)|_F\). The Iitaka dimension is \(-\infty\) when all positive integral systems are empty, with \((-\infty)+b=-\infty\) also for \(b=-\infty\); the zero divisor on a point has Iitaka dimension zero. \end{assumption} \begin{theorem}[Log abundance for compact Kähler spaces]\label{thm:main} Assume Assumption~\ref{asm:log-iitaka}. Let \(X\) be a normal irreducible compact Kähler complex analytic space, and let \(\Delta\geq0\) be a rational divisor such that \((X,\Delta)\) is log canonical and \(K_X+\Delta\) is rational Cartier. If \(K_X+\Delta\) is analytically nef, then it is semiample. Explicitly, there is an integer \(m>0\) such that \(m(K_X+\Delta)\) is Cartier and \[ H^0\bigl(X,\OO_X(m(K_X+\Delta))\bigr)\otimes_{\C}\OO_X \longrightarrow \OO_X(m(K_X+\Delta)) \] is surjective at every point of \(X\). \end{theorem} The theorem includes every finite dimension and the zero-boundary case. Assumption~\ref{asm:log-iitaka} is used through the projective good-model theorem stated in Proposition~\ref{proj:good-models}. The cited proof of that projective theorem uses only its zero-boundary case, on a resolved projective Albanese fibration. \subsection{Context and relation to earlier work} In the minimal model program, abundance complements the construction of minimal models: the nef adjoint on a minimal model should determine a canonical fibration. For minimal projective threefolds, Miyaoka proved the case of numerical dimension one and Kawamata completed canonical abundance \cite{Miyaoka1988,Kawamata1992}. Keel, Matsuki, and McKernan established log abundance for threefolds \cite{KMM1994,KMM2004}. In arbitrary dimension, Birkar, Cascini, Hacon, and McKernan established log terminal models for projective klt pairs with big boundary and pseudo-effective adjoint, and finite generation for big adjoints \cite{BCHM2010}. The compact Kähler history already points to the importance of simple spaces. Peternell's threefold theorem isolated the possible case of a simple space not bimeromorphic to a finite quotient of a torus \cite{Peternell2001}; Demailly and Peternell subsequently proved canonical nonvanishing for nef terminal Kähler threefolds, including that case \cite{DemaillyPeternell2003}. Höring and Peternell constructed minimal models for normal \(\Q\)-factorial terminal compact Kähler threefolds with pseudo-effective canonical class \cite{HoeringPeternell2016}. Campana, Höring, and Peternell treated canonical abundance \cite{CHP2016,CHP2023}, and Das and Ou established log abundance for compact Kähler log canonical threefolds \cite[Corollary~1.3]{DasOu2026}. With \(D_X=D_Y=0\), Assumption~\ref{asm:log-iitaka} is Iitaka's classical \(C_{n,m}\) inequality \(\kappa(X)\geq\kappa(F)+\kappa(Y)\) for algebraic fiber spaces \cite[Equation~(1.0.1)]{CaoPaun2017}. Cao and Păun proved the analogous logarithmic inequality over an abelian base when the pair on the total space has an effective rational boundary and is klt \cite[Theorem~1.1]{CaoPaun2017}. Fujino derives logarithmic subadditivity from the conjectured equality \(\kappa_\sigma(X,K_X+D)=\kappa(X,K_X+D)\) for smooth projective \(X\) with reduced SNC \(D\) \cite{Fujino2017Subadditivity,Fujino2020Corrigendum}. Hashizume proves the reduced-SNC subadditivity statement when the log canonical divisor of the very general fiber is abundant \cite[Theorem~1.2]{Hashizume2020}. Here the implication is used in the other direction: the projective companion \cite{LI} derives good models for pseudo-effective projective lc adjoints from Assumption~\ref{asm:log-iitaka}, and those good models anchor the Kähler induction. Hacon and Xie's cone theorem, adjunction, relative positivity, and modified-big preparation \cite{HX} provide analytic inputs for our program constructions. Section~\ref{sec:programs} constructs effective generalized base data on a higher carrier, with an exact adjoint pullback identity and a multiplier-ideal inclusion. Fibrewise $L^2$ positivity and regularization give this presentation; its Stein-factor consequence supplies the non-klt induction. The section then proves the restricted semiampleness, proper relative good-model, and finite-model statements needed here, following the dimension-induction strategy of \cite{HX}. The nef data remain globally nef on a fixed carrier, and the total generalized boundary is globally modified big. Actual rational lines make the selected ray contractions projective; ordinary projective analytic results \cite{FujinoAnalyticBCHM} then construct their flips and the relative models. This produces a Kähler pullback description of the relevant Bott--Chern class. Generation of the prescribed holomorphic line requires the additional divisorial decomposition and boundary arguments. The fibration comparison and period positivity come from \cite{CI}; the boundary and lifting constructions adapt inputs in \cite{CB,BL}. The precise imported results are stated at their points of use. \subsection{The proof and its main constructions} For a pseudo-effective \((1,1)\)-class \(\alpha\), \(N(\alpha)\) records the least generic vanishing forced along prime divisors. Given a smooth compact Kähler manifold \(X\) and a rational SNC boundary \(B\) with \(J=K_X+B\) pseudo-effective, the induction constructs a modification \(\mu:Y\to X\) with \(Y\) smooth and \[ \mu^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad R=N(c_1(\mu^*J)), \] as an identity of rational holomorphic line bundles, with \(R\) an effective rational divisor. Section~\ref{sec:inductive-reduction} uses transfer rules, algebraic reduction, and generically finite covers to reduce the induction to projective manifolds, nontrivial fibrations, and simple spaces of algebraic dimension zero. Here simple means that no positive-dimensional proper compact subvariety passes through a very general point. Section~\ref{sec:programs} supplies two program results. For the fibration case it contracts a known negative part while preserving the prescribed nef line. For the simple case it proves special termination for a chosen scaling and reaches a nef model when the pseudo-effective adjoint is rationally equivalent to a divisor supported on the reduced boundary, allowing negative coefficients. The termination argument uses lower-dimensional induction to construct one model on which an interval of perturbed adjoints is nef. On a common resolution their negative multiplicities are affine, whereas every nontrivial wall of the restricted program changes a slope. Hence only finitely many such walls occur. Section~\ref{sec:boundary} turns generation on separate strata into generation on the entire reduced dlt boundary, adapting Fujino's method of admissible sections \cite[Section~4]{Fujino2000}. When a comparison is needed, a Mori contraction realizes Kollár's residue comparison at the two coefficient-one points of a general \(\PP^1\) fiber \cite[Definition~13 and Proposition~14]{Kollar2013Sources}. On a common resolution, the pulled-back restrictions of the ambient Kähler class differ by a real linear combination of Chern classes of line bundles. On a stratum whose boundary dominates the image of the map defined by its semiample adjoint, restriction determines sections. For the remaining strata, we prove finite image for self-comparisons on sufficiently divisible pluricanonical systems using an invariant integral, period and lattice arguments, and a uniform cohomological bound for cyclic covers. Products over these finite images give compatible generating sections, used in both closing arguments. Section~\ref{sec:rank-one} first reduces to fibrations whose very general fiber has log Kodaira dimension zero. On a prepared fibration \(g:X\to W\), a relative generating form gives \[ K_X+B\sim_{\Q}g^*H+A_*,\qquad H=K_W+T+M, \] where \(T\) is a rational SNC boundary and \(M\) is pulled back from a nef rational line on a projective quotient of \(W\). Fiber induction allows us to subtract the part of \(A_*\) dominating \(W\) from a positive current for \(K_X+B\). The resulting metric descends along connected smooth fibers; a zero among the nonnegative coefficients at primes dominating each base prime allows extension, proving \(H\) pseudo-effective. If \(a(W)=0\), the projective quotient is a point, so \(M\sim_{\Q}0\) and lower-dimensional induction makes \(H\) rationally equivalent to its negative divisor. For projective \(W\), a reduction via a chosen generalized program derived from Assumption~\ref{asm:log-iitaka} either lowers the positive base dimension, handled by secondary induction, or reaches a nef line that is big or torsion. At a stopping case, an intersection argument identifies the full negative divisor upstairs. A torsion positive part completes the decomposition directly. In the big nef case, Section~\ref{sec:programs} contracts that divisor while preserving the nef line; boundary generation, extension, and a new log canonical place at a hypothetical base locus prove semiampleness. Section~\ref{sec:nonvanishing} proves, independently of Assumption~\ref{asm:log-iitaka}, that a positive canonical power has a nonzero meromorphic section on every smooth connected simple compact Kähler manifold of algebraic dimension zero. Ou's uniruledness and foliation results supply the cotangent slope control \cite{Ou2025}. A point-threshold bound for big classes of volume one is contradicted using an auxiliary projective bundle over \(X\times X\). The diagonal in its square produces a subsheaf occupying a fixed positive fraction of a symmetric cotangent power of that bundle. A second construction over the diagonal of \(X\times X\) forces determinant vanishing which, after restriction to a blowup of \(X\), violates the point bound. Finally, Section~\ref{sec:signed} uses the meromorphic section to obtain, after resolving and enlarging to a reduced SNC boundary, a pseudo-effective adjoint with a signed representative supported there. Section~\ref{sec:programs} gives a nef dlt model, and Section~\ref{sec:boundary} gives generation on its reduced boundary. Pseudo-effectivity of the canonical pullback and an intersection argument isolate positive-dimensional boundary fibers away from components whose signed coefficients are nonpositive. A local root construction adapted from \cite[Proposition~3.3]{LI} separates the positive and negative divisor supports. An SNC Hodge-module calculation extends the lifting method of \cite[Sections~10--12]{BL} to residual poles and kills every finite-order obstruction to lifting such a fiber. Douady space and Artin approximation give compact deformations leaving the boundary; compactness of the cycle-space components and relative compactness of bounded-volume cycles allow a Baire argument to produce a covering family. Simplicity forces the actual nef line to be torsion. The program comparison and Lemma~\ref{bir:unique-current} then permit subtraction of the added boundary, yielding the inductive decomposition. ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//01-introduction.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//02-inductive-reduction.tex ==== \section{Birational decompositions and the geometric reduction} \label{sec:inductive-reduction} The proof will produce a semiample line bundle on a smooth model, together with its entire fixed divisorial part. This section makes that statement precise and reduces the induction to two cases: spaces admitting a nontrivial fibration, and simple spaces of algebraic dimension zero. \subsection{The inductive decomposition and its negative part} We use additive notation for rational holomorphic line bundles. Thus \(L\sim_{\Q}L'\) means that \(mL\) and \(mL'\) are holomorphically isomorphic for some positive integer \(m\). If a rational divisor occurs in such an identity, it denotes its associated rational line bundle. In contrast, \(c_1(L)\) and \(\{D\}=c_1(\OO_X(D))\) denote real Bott--Chern classes. Canonical comparisons are made with the usual local meromorphic canonical identifications. In particular, an identity of lines below contains more information than equality of their Chern classes. Let \(X\) be a smooth compact Kähler manifold and let \(\alpha\) be a pseudo-effective real \((1,1)\)-class. Fix a Kähler form \(\omega\). For a prime divisor \(D\), its minimal multiplicity is \[ \nu_D(\alpha)=\lim_{\varepsilon\downarrow0} \inf\bigl\{\nu_D(T):T\in\alpha,\ T\geq-\varepsilon\omega\bigr\}. \] Here \(T\) ranges over closed currents, and \(\nu_D(T)\) is its generic Lelong number; one may equivalently use currents with analytic singularities. The limit is independent of \(\omega\). Boucksom's divisorial decomposition is \[ N(\alpha)=\sum_D\nu_D(\alpha)D, \qquad Z(\alpha)=\alpha-\{N(\alpha)\}. \] The sum is a finite effective real divisor, and \(Z(\alpha)\) is modified nef: its minimal multiplicity at every prime is zero. We write \(N(L)\) and \(\nu_D(L)\) for \(N(c_1(L))\) and \(\nu_D(c_1(L))\). These are the analytic notions, with small Kähler perturbations, throughout the paper. We use the foundational results in \cite[Sections~2--3 and~5]{Boucksom2004}. The induction asks for a decomposition that retains both the actual line bundle and this analytic negative divisor. \begin{definition}\label{def:good-decomposition} For \(n\geq0\), let \(\Ggood_n\) be the following assertion. If \(X\) is a connected smooth compact Kähler manifold of dimension \(n\), \(B\) is a rational simple normal crossing boundary with coefficients in \([0,1]\), and \(J=K_X+B\) is pseudo-effective, then there is a smooth compact Kähler modification \(\mu:Y\to X\) and an actual rational line identity \begin{equation}\label{bir:good-decomposition} \mu^*J\sim_{\Q}P+R, \qquad P\text{ semiample},\qquad R=N(\mu^*J)\geq0, \end{equation} where \(R\) is a rational divisor. \end{definition} The next lemmas make this decomposition stable under further resolutions and allow exceptional resolution errors to be removed. They also provide the fixed-section statement needed to descend the case in which the positive part is torsion. \begin{lemma}[Negative-part calculus]\label{bir:calculus} Let \(\alpha\) be pseudo-effective on a smooth compact Kähler manifold. \begin{enumerate} \item Every positive current in \(\alpha\) contains the divisorial current \([N(\alpha)]\). If \(0\leq F\leq N(\alpha)\), then \[ \alpha-\{F\}\text{ is pseudo-effective},\qquad N(\alpha-\{F\})=N(\alpha)-F. \] \item Minimal multiplicities are homogeneous and subadditive. In particular, if \(\beta\) is nef, then \(N(\alpha+\beta)\leq N(\alpha)\). \item If \(\mu:Y\to X\) is a smooth modification and \(D'\) is the strict transform of a prime \(D\), then \[ \nu_{D'}(\mu^*\alpha)=\nu_D(\alpha). \] \item If \(\gamma\) is modified nef and \(j:\widetilde D\to X\) is a resolution of a prime divisor, then \(j^*\gamma\) is pseudo-effective. \end{enumerate} \end{lemma} \begin{proof} The first assertion about currents follows from the definition of minimal multiplicity and Siu decomposition. Homogeneity and subadditivity follow by scaling and adding testing currents. The subtraction identity is the corresponding property in the big cone, followed by a small Kähler perturbation. More explicitly, for \(\alpha_\varepsilon=\alpha+ \varepsilon\{\omega\}\), subtract \(F_\varepsilon=\min\{F,N(\alpha_\varepsilon)\}\) coefficientwise. In the big cone all positive currents contain this divisor, so subtracting it translates each minimal multiplicity by its coefficient. As \(\varepsilon\downarrow0\), \(F_\varepsilon\to F\). The class of \(F-F_\varepsilon\) can be absorbed in a Kähler perturbation tending to zero (there are only finitely many components). This gives the upper bound for the asserted identity; subadditivity applied after adding \(F\) gives the reverse bound. Weak compactness of positive currents gives pseudo-effectivity of the limit. This is also the subtraction property of the divisorial decomposition in \cite[Section~3]{Boucksom2004}. For the strict-transform formula, pull almost-positive testing currents to \(Y\). Near the generic point of \(D'\), the map is an isomorphism, so the generic order is unchanged. This proves one inequality. Conversely, push a testing current on \(Y\) to \(X\). To control its negative error, write the error as a small multiple of the fixed positive current \(\mu_*\omega_Y\), and add a fixed smooth Kähler representative of \(C\omega_X-\{\mu_*\omega_Y\}\) for \(C\) sufficiently large. The result is a positive test in a Kähler perturbation tending to zero. The pushforward of \(\omega_Y\) has no divisorial order at the generic point of \(D\), where \(\mu\) is an isomorphism. Thus this test has the same generic order as the original one at \(D'\). Regularization, if needed, returns to tests with analytic singularities. This proves the other inequality and the formula. It is the analytic form of the strict-transform comparison in \cite[Lemma~5.8]{Boucksom2004}. Finally, choose analytic-singularity tests for a modified-nef class whose generic order at the chosen prime tends to zero. Subtract that generic divisorial order before restricting to a resolution of the prime. The remaining singular set does not contain its generic point, so restriction gives an almost-positive current there. The subtracted multiple and the Kähler error tend to zero. Taking the limit proves the last assertion; see also \cite[Propositions~2.4 and~3.8]{Boucksom2004}. \end{proof} For a rational line \(J\) on a normal compact Kähler space \(X\) whose pullback to a smooth resolution is pseudo-effective, we also use minimal multiplicities for divisorial places. If \(Q\) appears on a smooth resolution \(p:Y\to X\), set \[ \nu_Q(J):=\nu_Q(p^*J). \] The strict-transform formula on a common smooth refinement makes this independent of the chosen resolution. When a real class is already on a fixed smooth model, \(\nu_Q\) continues to denote its coefficient in the negative part on that model. \begin{lemma}[Exceptional translation]\label{bir:exceptional} Let \(p:Y\to X\) be a proper bimeromorphic morphism from a smooth compact Kähler manifold to a normal compact Kähler space. Let \(M\) be a real \((1,1)\)-class on \(X\) represented by smooth local potentials, and let \(E\geq0\) be a real \(p\)-exceptional divisor. If \(\alpha=p^*M+\{E\}\) is pseudo-effective, then \(p^*M\) is pseudo-effective and \[ N(\alpha)=N(p^*M)+E. \] \end{lemma} \begin{proof} It suffices by Lemma~\ref{bir:calculus} to prove \(E\leq N(\alpha)\). Write \[ E-N(\alpha)=U-V, \qquad U,V\geq0, \] with no common component, and suppose \(U\neq0\). Its components are \(p\)-exceptional. Put \(n=\dim Y\) and \(\ell=\max\{\dim p(U_i):U_i\text{ a component of }U\}\). Normality gives \(\ell\leq n-2\). Let \(\eta\) be the pullback of a Kähler form on \(X\), let \(\omega\) be Kähler on \(Y\), and set \[ q(\gamma,\delta)= \int_Y\gamma\,\delta\,\eta^{\ell}\omega^{n-\ell-2}. \] The restriction property in Lemma~\ref{bir:calculus} gives \(q(Z(\alpha),U)\geq0\). The term \(q(p^*M,U)\) is zero: on each component of \(U\), it contains \(\ell+1\) factors pulled back from an image of dimension at most \(\ell\). Distinct effective divisors have nonnegative intersection against the semipositive and Kähler factors in \(q\). Hence \[ 0\leq q(Z(\alpha),U)=q(U,U)-q(V,U)\leq q(U,U). \] On the other hand, \(q(U,\eta)=0\) by the same dimension count, whereas \(q(\eta,\eta)>0\). The mixed Hodge index theorem, applied first with Kähler factors and then by a semipositive limit, says that \(q\) is negative semidefinite on \(\eta^\perp\). It follows that \(q(U,U)=0\) and that \(U\) is in the radical of \(q\): the radical assertion follows from negative semidefiniteness on \(\eta^\perp\), and then from \(q(U,\eta)=0\). But \[ q(U,\omega)=\sum_i u_i\int_{U_i} \eta^{\ell}\omega^{n-\ell-1}>0. \] Indeed, a component with image dimension \(\ell\) has strictly positive integrand on a dense open. This is a contradiction. The mixed Hodge index statement used here is the mixed Hodge--Riemann relation \cite[Theorems~A and~C]{DinhNguyen}; the limit preserves the assertion that there is at most one positive direction. \end{proof} \begin{lemma}[Pulling up a known decomposition]\label{bir:pullup} Let \(\alpha\) be a pseudo-effective real \((1,1)\)-class on a smooth compact Kähler manifold, and suppose \[ \alpha=\beta+\{R\},\qquad \beta\text{ nef},\qquad R=N(\alpha)\geq0. \] For every smooth modification \(\mu:Y\to X\), \[ N(\mu^*\alpha)=\mu^*R. \] In particular, this applies to the Chern classes of an actual rational line decomposition with a nef rational positive part. \end{lemma} \begin{proof} Subadditivity and nefness give \(N(\mu^*\alpha)\leq\mu^*R\). By the strict-transform formula, the difference \(U=\mu^*R-N(\mu^*\alpha)\) is effective and exceptional. Moreover \(Z(\mu^*\alpha)=\mu^*\beta+\{U\}\) is modified nef. Apply Lemma~\ref{bir:exceptional} to this class. It gives \(U\leq N(Z(\mu^*\alpha))=0\), proving the assertion. \end{proof} \begin{lemma}[Fixed sections]\label{bir:sections} Let \(L\) be a pseudo-effective rational line on a smooth compact Kähler manifold. Every section \(s\) of an integral multiple \(mL\) satisfies \[ \ord_D(s)\geq m\nu_D(L) \] at every prime \(D\). If \(L\sim_{\Q}P+R\), with \(P\) semiample and \(R=N(L)\) rational, then in every sufficiently divisible degree multiplication by the canonical section of \(mR\) identifies \(H^0(X,mP)\) with \(H^0(X,mL)\). \end{lemma} \begin{proof} The divisor current \([\operatorname{div}(s)]/m\) is a positive current in \(c_1(L)\), so Lemma~\ref{bir:calculus} gives the order bound. In the stated decomposition, every section therefore divides holomorphically by the canonical section of \(mR\). Conversely, multiplication gives a section of \(mL\). The actual rational line identity makes these operations inverse after clearing denominators. \end{proof} \begin{lemma}[Uniqueness when the positive part is torsion] \label{bir:unique-current} Suppose \(L\sim_{\Q}R\), where \(R=N(L)\) is an effective rational divisor on a smooth compact Kähler manifold. Then the only positive current in \(c_1(L)\) is \([R]\). If \(F\geq0\) is a rational divisor and \(L-F\) is pseudo-effective, then \[ F\leq R,\qquad L-F\sim_{\Q}R-F=N(L-F). \] The same conclusions hold if \(L\sim_{\Q}P+R\) with \(P\) torsion. \end{lemma} \begin{proof} Write \(X\) for the manifold. The assertion is immediate if \(\dim X=0\), so assume \(\dim X>0\). Every positive current \(T\) in \(c_1(L)\) contains \([R]\). The residual \(T-[R]\) is positive with zero cohomology class, so its mass against a Kähler form to the power \(\dim X-1\) is zero. Thus it vanishes. For the second assertion, add \([F]\) to any positive current in \(c_1(L-F)\). Uniqueness gives \(F\leq R\), and subtraction in Lemma~\ref{bir:calculus} gives the asserted negative part. A torsion line has zero Chern class and becomes trivial after taking a positive multiple, so the last statement is the same argument. \end{proof} \subsection{Birational transfer of the decomposition} All modifications resolving spaces, maps, or ideals can be chosen projective. A projective modification of a compact Kähler space is Kähler; graphs between compact Kähler models can likewise be resolved by smooth compact Kähler manifolds. We use these standard analytic resolution and flattening results without further mention. For log discrepancies our convention is \[ a(E;X,\Delta)=1+\ord_E\bigl(K_Y-p^*(K_X+\Delta)\bigr) \] on a smooth model \(p:Y\to X\). In particular, on a log resolution of an lc pair, giving each new exceptional divisor coefficient one produces an effective exceptional error. The next proposition explains why this convention is harmless. \begin{proposition}[Resolution and descent]\label{bir:resolution-transfer} Let \((X,\Delta)\) be a normal compact Kähler lc pair with rational boundary and rational Cartier adjoint \(J\). Let \(p:Y\to X\) be a log resolution, and put \[ B_Y=p_*^{-1}\Delta+\sum_{E\text{ exceptional}}E. \] If \(p^*J\) is pseudo-effective and the conclusion of Definition~\ref{def:good-decomposition} holds for \((Y,B_Y)\), then there is a smooth compact Kähler modification \(r:V\to X\) with an actual rational line identity \[ r^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad R=N(r^*J)\geq0, \] where \(R\) is a rational divisor. If \(J\) is analytically nef, it is semiample on \(X\). \end{proposition} \begin{proof} With compatible canonical choices there is an actual rational divisor identity \[ K_Y+B_Y=p^*J+E_p,\qquad E_p=\sum_{E\text{ exceptional}}a(E;X,\Delta)E\geq0. \] Suppose \(q:V\to Y\) realizes the decomposition for the left side. Lemma~\ref{bir:exceptional}, applied over \(X\), gives \[ N\bigl(q^*(K_Y+B_Y)\bigr) =N(q^*p^*J)+q^*E_p. \] Cancelling \(q^*E_p\) in the actual rational line identities proves the first assertion. This argument also shows that further resolutions using the reduced-exceptional convention do not change the assertion \(\Ggood_n\). If \(J\) is nef, its pullback is nef, so its negative part is zero. The semiample line in the decomposition is therefore \(q^*p^*J\) itself, up to rational line isomorphism. Choose an actual Cartier multiple generated on \(V\). Normality gives \((p q)_*\OO_V=\OO_X\), and projection formula identifies its sections with the sections downstairs. A base point downstairs would make every pullback section vanish on its nonempty fiber. There is no such point, so that multiple of \(J\) is globally generated. \end{proof} The second transfer concerns a torsion positive part. Its proof uses the following local construction for finite maps, which will also be used for meromorphic sections and for covers of spaces of algebraic dimension zero. \begin{lemma}[Finite analytic norms and characteristic polynomials] \label{bir:finite-norm} Let \(\nu:Z\to X\) be a finite surjective morphism of normal irreducible complex spaces, of degree \(d>0\), and let \(L\) be a holomorphic line bundle on \(X\). A nonzero meromorphic section \(s\) of \(\nu^*L\) has a nonzero meromorphic norm \(\operatorname{Nm}_\nu(s)\) in \(L^{\otimes d}\), with \[ \operatorname{div}(\operatorname{Nm}_\nu(s)) =\nu_*\operatorname{div}(s). \] If \(s\) is holomorphic, its norm is holomorphic. If, in addition, \(s\) is nonzero at every point of \(\nu^{-1}(x)\), then its norm is nonzero at \(x\). For a global meromorphic function \(f\) on \(Z\), there is a monic polynomial \(\chi_f(T)\) of degree \(d\) with global meromorphic coefficient functions on \(X\) such that \(\chi_f(f)=0\) after pulling the coefficients to \(Z\). \end{lemma} \begin{proof} At \(x\in X\), choose a holomorphic frame of \(L\) and put \[ R=\OO_{X,x},\qquad K=\operatorname{Frac}(R),\qquad A=(\nu_*\OO_Z)_x. \] The generic algebra \(A\otimes_RK\) is a product of finite field extensions of \(K\), of total dimension \(d\). The local meromorphic coefficient \(a\) of \(s\) belongs to this algebra. Since \(Z\) is irreducible and \(s\) is not identically zero, its restriction is nonzero on every generic local component. Thus \(a\) is nonzero in every field factor, and the determinant of multiplication by \(a\) is nonzero. Under a change of frame by a unit \(g\), this determinant changes by \(g^{-d}\). The determinants therefore glue to the asserted meromorphic section. The valuation formula for a norm over a discrete valuation ring gives the displayed divisor identity at every prime of the normal space \(X\). If \(s\) is holomorphic, then \(a\in A\) is integral over \(R\). Its norm is integral over \(R\) and belongs to \(K\), hence belongs to \(R\) by normality. This does not require \(A\) to be locally free. If \(s\) is nonzero at every point over \(x\), then \(a\) is a unit in the finite semilocal algebra \(A\); applying the same argument to \(a^{-1}\) shows that its norm is a unit in \(R\). For a meromorphic function \(f\), use instead the determinant of \(T\) minus multiplication by its local coefficient. These monic polynomials agree on overlaps, and Cayley--Hamilton shows that they annihilate \(f\). All these constructions use fractions of local analytic germs, so they remain available when the global meromorphic functions on \(X\) are constant. \end{proof} We now descend a decomposition whose positive part is torsion, the form needed for a covering family on a space of algebraic dimension zero. \begin{proposition}[Descent of a purely negative decomposition] \label{bir:finite-descent} Let \(e:Y\to X\) be a proper generically finite surjective morphism between connected smooth compact Kähler manifolds. Let \(L\) be a pseudo-effective rational line on \(X\), and let \(F\geq0\) be a rational divisor on \(Y\). Suppose that on a smooth model over \(Y\), the line \(e^*L+F\) is rationally linearly equivalent to its rational negative part. Then there is an effective rational divisor \(D\) on \(X\) such that \[ L\sim_{\Q}D=N(L). \] \end{proposition} \begin{proof} Replace \(Y\) by the smooth model in the hypothesis and pull \(F\) back. Write \(e^*L+F\sim_{\Q}R=N(e^*L+F)\). Pull back a positive current in \(c_1(L)\) and add \([F]\). Lemma~\ref{bir:unique-current} gives \(F\leq R\). Put \(D_Y=R-F\). Subtraction in Lemma~\ref{bir:calculus} gives \[ e^*L\sim_{\Q}D_Y=N(e^*L)\geq0. \] In particular \([D_Y]\) is the unique positive current in \(c_1(e^*L)\). Choose \(m\) so that \(mL\) and \(mD_Y\) are integral and the latter is the divisor of a holomorphic section \(s\) of \(e^*(mL)\). Factor \(e\) through its normal Stein space: \[ Y\xrightarrow{h}Z\xrightarrow{\nu}X, \qquad \nu\text{ finite of degree }d=\deg e. \] The connected birational map \(h\) descends \(s\) to a holomorphic section \(s_Z\) of \(\nu^*(mL)\), by normality and projection formula. Lemma~\ref{bir:finite-norm} gives a holomorphic section of \(mdL\) whose divisor is \[ \operatorname{div}(\operatorname{Nm}_\nu(s_Z)) =\nu_*\operatorname{div}(s_Z)=m e_*D_Y. \] Thus \(D=d^{-1}e_*D_Y\) is effective and satisfies \(L\sim_{\Q}D\). Both \(e^*D\) and \(D_Y\) are positive divisor currents in \(c_1(e^*L)\), so uniqueness gives \(e^*D=D_Y\). We already know \(N(L)\leq D\). If this inequality were strict at a prime \(A\subset X\), choose analytic-singularity tests for \(L\) whose generic orders at \(A\) tend to \(\nu_A(L)\), and pull them to \(Y\). For any prime \(A'\) dominating \(A\), the generic orders are multiplied by \(\operatorname{mult}_{A'}e^*A\). The pulled-back Kähler errors can be enlarged to Kähler errors upstairs tending to zero. Consequently \[ \nu_{A'}(e^*L) \leq \operatorname{mult}_{A'}(e^*A)\,\nu_A(L) <\operatorname{mult}_{A'}(e^*A)\,\operatorname{coeff}_A(D), \] contrary to \(N(e^*L)=D_Y=e^*D\). This proves \(D=N(L)\). \end{proof} We will also use Proposition~\ref{bir:finite-descent} when \(e\) is a modification and \(F=0\). Once the positive part of a decomposition is known to be torsion, it gives \(L\sim_{\Q}N(L)\) on the original smooth space; Lemma~\ref{bir:pullup} then identifies the negative divisor upstairs with the pullback of \(N(L)\). \subsection{The projective anchor and the two geometric cases} \begin{proposition}[Projective good models]\label{proj:good-models} Under Assumption~\ref{asm:log-iitaka}, every projective lc pair over \(\C\) with rational boundary and pseudo-effective rational Cartier adjoint has a good log minimal model. In particular, \(\Ggood_n\) holds for projective manifolds in every dimension, and every nef rational lc adjoint on a projective variety is semiample as an actual rational line. \end{proposition} \begin{proof} The good-model assertion is the rational-boundary consequence of the full argument in \cite[Proposition~2.5, Theorem~9.6, and Section~10]{LI}. That argument establishes the additional inductive and nonvanishing premises of its Proposition~2.5 before applying it, and proves a stronger real-boundary induction. Its only use of Assumption~\ref{asm:log-iitaka} is in its Lemma~6.1, on a resolved projective Albanese fibration with both boundaries zero. Thus its premise is precisely available here. For clarity, its good-model conclusion gives on a common smooth projective resolution \[ u^*(K_X+B)=v^*(K_{X_m}+B_m)+F, \qquad F\geq0\text{ and }v\text{-exceptional}, \] as an actual rational divisor identity; the line on \(X_m\) is semiample. This is the comparison in \cite[Lemma~2.2]{LI}. Algebraic nefness of a rational line on a projective variety implies analytic nefness, by adding arbitrarily small ample rational classes. Lemma~\ref{bir:exceptional} therefore identifies \(F\) with the analytic negative part of the left side. On a smooth projective variety, analytic pseudo-effectivity of a divisor class agrees with algebraic pseudo-effectivity \cite[Proposition~1.2]{BDPP}; hence the good-model assertion applies to every projective instance of \(\Ggood_n\). This proves the stated form of \(\Ggood_n\). If the original adjoint is nef, the same comparison and normal descent give its semiampleness. \end{proof} The remainder of the paper proves the following two propositions. They are stated here so that the global induction can be completed before its technical components are developed. A compact space is \emph{simple} if no positive-dimensional proper compact analytic subvariety passes through a very general point. We write \(a(X)\) for its algebraic dimension. \begin{proposition}[The fibration case]\label{rank:fibration-case} Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume \(\Ggood_j\) for \(j0\) and assume \(\Ggood_j\) for \(j0\). If \(a(X)=n\), the Kähler Moishezon theorem makes \(X\) projective, and Proposition~\ref{proj:good-models} applies. If \(00\) and the evaluation ideal \[ \operatorname{im}\bigl(f^*f_*\OO_T(rJ)\longrightarrow\OO_T(rJ)\bigr) =\mathcal I\otimes\OO_T(rJ). \] Outside the exceptional locus of \(f\), evaluation is an isomorphism. On a positive-dimensional projective fiber, every section vanishes: \(rJ\) has negative degree on every fiber curve, and curves cover every positive-dimensional fiber component. Connectedness gives the same vanishing on the whole nontrivial fiber. Hence the zero set of \(\mathcal I\) is exactly the exceptional locus. On a common resolution principalizing \(\mathcal I\), relative generation on the positive side identifies \(rF\) in Equation~\eqref{prog:comparison} with the divisor of \(\mathcal I\OO_W\). Thus \(\Supp F\) is the full inverse image of that locus. On a common resolution of a longer negative program, discrepancy monotonicity makes the cumulative comparison dominate this first-step divisor. These are support statements for the actual comparison, stronger than effectiveness and exceptionality alone. We next state the restricted model inputs used in this section. A nef b-class is a Bott--Chern class that is globally nef on one fixed smooth compact Kähler carrier, with its linear traces on the other models. The generalized adjoint on \(T\) is \(\alpha=\{K_T+B\}+\beta_T\). Generalized klt, abbreviated gklt, means that all generalized log discrepancies, computed on that carrier, are positive. The modified-bigness hypothesis below concerns the whole boundary-plus-nef trace \(\{B\}+\beta_T\). We use the convention of \cite[Definition~2.8]{HX}: a modified-big trace is the pushforward of a big Bott--Chern class on a modification. The trace itself may be a current; the generalized adjoint is required to define a Bott--Chern class. For a proper map $f:T\to S$, let $q_f$ be the quotient map to $H^{1,1}_{\BC}(T,\R)/f^*H^{1,1}_{\BC}(S,\R)$. Relative pseudo-effectivity means $q_f(\alpha)\in\overline{q_f(\operatorname{Psef}(T))}$, where $\operatorname{Psef}(T)$ is the absolute pseudo-effective cone. Equivalently, for Kähler classes $\omega_T,\omega_S$, the class $\alpha$ is pseudo-effective over $S$ if for every $\epsilon>0$ some $c_\epsilon\geq0$ makes $\alpha+\epsilon\omega_T+c_\epsilon f^*\omega_S$ big. This imposes no projectivity hypothesis on $f$. For the actual relative line classes of an already projective map, this is the usual relative pseudo-effective cone. \begin{proposition}[Restricted Kähler model inputs]\label{prog:mori} Let \((T,B+\boldsymbol\beta)\) be an effective compact Kähler gklt pair of dimension \(d\), with \(T\) globally strongly \(\Q\)-factorial. Suppose that its nef b-data are globally nef on a fixed smooth carrier and that \(\{B\}+\beta_T\) is globally modified big. Write \(\alpha=\{K_T+B\}+\beta_T\). \begin{enumerate} \item If \(\alpha\) is nef, there is a proper morphism with connected fibers \(g:T\to Z\) to a normal compact Kähler space and a Kähler class \(\omega_Z\) such that \(\alpha=g^*\omega_Z\). \item For a proper morphism \(\pi:T\to V\) to a normal compact Kähler space, if \(\alpha\) is pseudo-effective over \(V\), there is a chosen good log terminal model over \(V\). It is nonextracting, compact Kähler and globally strongly \(\Q\)-factorial. Its adjoint is nef over \(V\), and on a common projective resolution its comparison with \(\alpha\) is effective and exceptional over the new model, with strictly positive coefficient at every prime contracted from \(T\). The construction retains compatible forward Bott--Chern traces of the specified data on a common smooth carrier. \item A compact polytope of these data on one fixed carrier has finitely many marked relative canonical models and relative weak log canonical models with normal compact Kähler targets. \end{enumerate} The relative assertion applies to proper morphisms, without assuming that \(\pi\) is projective. It is an assertion about a chosen model, not termination of every generalized flip sequence. \end{proposition} \begin{proof} These are the semiampleness, proper relative model and finite-model assertions of Theorem~\ref{prog:restricted-package}, proved below by dimension induction independently of \(\Ggood\). Finite marked weak-model geography is recorded in Lemma~\ref{prog:finite-geography}. Projective analytic model constructions used in that induction are supplied by Proposition~\ref{prog:projective-good-model}. None of these statements promotes an undetected generalized-ray contraction to a projective map. \end{proof} We use separately the analytic cone theorem \cite[Theorem~1.3]{HX}: an \(\alpha\)-negative analytic extremal ray of an effective gklt adjoint has a rational curve generator \(C\) with \(0<-\alpha\cdot C\leq2d\). Existence of the supporting contraction in the applications below follows from the nef assertion above. When an ordinary rational adjoint is negative on that ray, Lemma~\ref{prog:detected-projectivity} and Proposition~\ref{prog:detected-flip} give its projective contraction and ordinary flip. We call these \emph{detected ordinary steps}. The Bott--Chern transform through such a birational step has a concrete description which does not require the source to be smooth. Write the step as \(T\xrightarrow{f}Z\xleftarrow{f^+}T^+\), with \(f^+=\id\) for a divisorial contraction, and let \(J\) be its negative rational adjoint. For \(\gamma\in H^{1,1}_{\BC}(T,\R)\), choose the unique real number \(c\) for which \(\gamma-c\{J\}\) annihilates the contracted analytic ray. Every contracted curve is on that ray. Lemma~\ref{prog:bc-descent} therefore gives a unique class \(\delta\) on \(Z\) with \(\gamma-c\{J\}=f^*\delta\), and we put \[ \gamma^+=(f^+)^*\delta+c\{J^+\}. \] The rational-singularity and vanishing hypotheses of that lemma hold: slightly lowering the rational Cartier floor gives a klt adjoint still antiample over \(Z\), and relative vanishing gives rational singularities on the base. On a common resolution, the pullback difference is \(c\) times the adjoint comparison in Equation~\eqref{prog:comparison}; it is exceptional over the new model, and over both models in a flip. This defines a linear transform. For the Chern class of a rational line it agrees with its reflexive trace, by the coherent line comparison \cite[Lemma~7.2]{CB} and exceptional negativity. The already-projective relative program below only needs a ray for degrees of global rational lines; this Bott--Chern construction is used for its subsequent detected analytic-ray programs. When such a class difference lies in an exceptional divisor span, we write it as \(\{F\}\) for the representing real exceptional divisor \(F\). This divisor is unique: an exceptional divisor with zero class has zero degree on every contracted curve, and exceptional negativity applied to both signs makes it zero. We also use local projective analytic results. For a projective analytic morphism over a Stein neighborhood of a compactum with Fujino's property (P), relative klt base point freeness says that, if \(L\) is a relatively nef Cartier line and \(aL-(K+\Gamma)\) is relatively ample for some positive integer \(a\), then all sufficiently high powers of \(L\) are relatively generated after shrinking \cite[Theorems~6.2 and 6.5]{FujinoAnalyticBCHM}. We use the finite negative-ray truncation in the relative cone theorem in the same setting. Finally, the multigraded adjoint finite-generation theorem applies to a projective morphism from a smooth space, for simultaneous effective SNC klt boundaries with a common relatively ample rational summand \cite[Theorem~3.1]{DHP}. Finitely many smaller Stein neighborhoods suffice over a compact base. \begin{lemma}[Descent without changing a Cartier exponent] \label{prog:integral-descent} Let \(f:T\to Z\) be a projective bimeromorphic contraction with normal target. Suppose an ordinary rational klt adjoint is \(f\)-antiample. If a Cartier line \(L\) has degree zero on every contracted curve, then \(f_*L\) is an invertible sheaf and evaluation is an isomorphism \[ f^*(f_*L)\simeq L. \] The same conclusion holds for an ordinary dlt adjoint whose floor is rational Cartier and which is \(f\)-antiample. In a flip, a line with zero contracted degree therefore has the same Cartier exponent on both sides, and the two lines are pullbacks of a line on the common base. \end{lemma} \begin{proof} For dlt input, decrease the floor by a sufficiently small positive rational multiple. The resulting pair is klt and remains \(f\)-antiample. For this klt pair the base point free hypothesis holds for \(L\), because \(L\) has zero fiber degrees and the negative adjoint is relatively ample. Over a smaller base neighborhood every sufficiently high power of \(L\) is therefore generated. The associated morphism is constant on each connected fiber: its tautological line has degree zero on every fiber curve, and every positive-dimensional projective image contains a curve. Its graph then factors through \(Z\); the graph projection is finite bimeromorphic and \(Z\) is normal. Descend two consecutive generated powers \(L^k,L^{k+1}\) to lines \(M_k,M_{k+1}\). The line \(M_{k+1}\otimes M_k^{-1}\) pulls back to \(L\). Projection formula identifies this quotient with \(f_*L\), so these local descents and their evaluation maps agree on overlaps. Pull the descended line to the positive side of a flip. This is the argument of \cite[Lemma~7.4]{CB}. \end{proof} \subsection{Relative klt programs and small models} The local finite-generation theorem also supplies an ordinary klt program over a compact base. We use it to obtain globally strongly \(\Q\)-factorial small models of strata. The construction keeps track of degrees of global rational lines, which are the degrees needed for exact Cartier descent. \begin{proposition}[Relative klt programs over a compact base] \label{prog:relative-klt} Let \(\pi:T\to V\) be projective bimeromorphic, where \(V\) is a normal compact Kähler space and \(T\) is normal, compact Kähler, and globally strongly \(\Q\)-factorial. Let \((T,B)\) be an effective rational klt pair with rational-line adjoint \(J=K_T+B\). There is a finite sequence of ordinary \(J\)-negative steps over \(V\), all projective over \(V\), whose last adjoint is curve-nef over \(V\), meaning that it has nonnegative degree on every curve contracted over \(V\). Every working space is compact Kähler and globally strongly \(\Q\)-factorial. On a common resolution the comparison from the first adjoint to the last is effective and exceptional over the last space. \end{proposition} \begin{proof} We give the reduction to the local finite-generation input, including why it yields a single finite program over the compact base. Let \(\mathcal N_{\R}\) be the finite-dimensional space of degrees of global rational lines on curves over \(V\), and put \(r=\dim\mathcal N_{\R}\). If \(r=0\), then \(J\) is already curve-nef. Assume \(r>0\). Choose rational lines \(H_1,\ldots,H_r\) whose degree classes form a basis and such that both \(H_i\) and \(J+H_i\) are relatively ample. Such a basis exists because the relatively ample classes whose sum with \(J\) is also relatively ample form a nonempty open set containing sufficiently positive classes. Define the rational simplex of adjoints \[ \Pi=\operatorname{conv}\{J,J+H_1,\ldots,J+H_r\}. \] We will choose a general direction \(H=\sum_i\theta_iH_i\) with \(\theta_i>0\) and \(\sum_i\theta_i=1\). Its entire scaling segment lies in this simplex: \[ J+tH=(1-t)J+\sum_i t\theta_i(J+H_i)\in\Pi \qquad(0\leq t\leq1), \] and it lies in the relative interior for \(0t\) for which \(J+sH\) is relatively ample. On the zero face of the normalized compact curve slice, \[ J=-\frac{t}{s-t}(J+sH) \] in degrees. Thus \(J\) is uniformly negative on that face. Choose the positive ample truncation so that its compact remainder \(K\) misses the face. The finite rational polyhedral test and the general choice of \(H\) make the face a single ray \(R\). The wall class has a positive minimum on \(K\) and is positive on every other truncated curve generator. Hence a sufficiently small open neighborhood of \(w\) inside the annihilator \(R^\perp\) consists of nef classes with zero face exactly \(R\). Suppose first that \(R^\perp\) has positive dimension. It is a rational hyperplane because \(R\) is represented by an integral curve. Choose a rational simplex of support classes in that neighborhood containing the degree class of \(w\) in its relative interior, and represent each vertex by a rational line \(N\). For each such \(N\), compactness gives an integer \(a>0\) for which \(aN-J\) is relatively ample. Relative base point freeness contracts exactly \(R\) and descends \(N\) to a relatively ample rational line on the contraction base. These bases agree because the contracted curves agree. The difference between \(w\) and the corresponding convex combination of these rational lines has zero relative degree. In the finite rational span of the lines involved, the degree map has rational kernel, since rational lines have rational degrees on integral curves. The difference is therefore a real combination of rational lines of zero relative degree. Lemma~\ref{prog:integral-descent} descends these lines, and lifting curves from the base shows that their descents still have zero degree over \(V\). They do not affect relative ampleness. Convexity now shows that \(w\) descends to a relatively ample real line class over \(V\). If \(R^\perp=0\), the same zero-degree descent applies to \(w\); the contraction base has no curves over \(V\), hence is finite over \(V\), and relative ampleness is automatic. After a flip, let \(w^+\) be the pullback of this wall class to the positive side. For sufficiently small \(\delta>0\), \[ J^++(t-\delta)H^+ =\left(1-\frac{\delta}{t}\right)w^+ +\frac{\delta}{t}J^+ \] is relatively ample over \(V\): the wall is pulled back from a relatively ample class on the contraction base, and \(J^+\) is relatively ample over that base. After a divisorial contraction, openness of the ample cone on the base gives the same nonempty interval of relative ampleness. Each working model is consequently one of the marked ample models already counted. Indeed, for a fixed finite prefix and an interior parameter \(00\) for which \((S,\Delta-\epsilon D)\) is effective and klt. On a projective log resolution \(r:W\to S\), use the strict lowered boundary and give every exceptional prime a coefficient strictly between its crepant coefficient and \(1\), and at least zero. The resulting klt adjoint has the actual form \[ K_W+\Gamma\sim_{\Q}r^*(K_S+\Delta-\epsilon D)+F, \qquad F\geq0, \] where every \(r\)-exceptional prime has positive coefficient in \(F\). Apply Proposition~\ref{prog:relative-klt} over \(S\). On its endpoint the trace \(F_Y\) is effective, exceptional over \(S\), and relatively nef. Negativity makes \(F_Y=0\). Since all initially exceptional primes had positive coefficient, \(Y\to S\) is small and the lowered adjoint is crepant. Restoring \(\epsilon D\) gives the full crepant identity. A projective small morphism to a smooth germ is an isomorphism: transport a relatively ample line to the smooth germ, where it is Cartier; smallness makes the original line its pullback, contradicting relative ampleness on a nontrivial fiber. Thus \(h\) is an isomorphism over an SNC open meeting all lc centers of the dlt pair. The full crepant comparison then proves dlt on \(Y\). \end{proof} \subsection{Scaling toward zero and keeping a nef rational line} The next construction runs every positive truncation of an ordinary dlt scaling. It also explains why a sufficiently large multiple of a prescribed nef rational line forces every step to be trivial for that line. The integer that clears the line will be the same at every step. For the detected ordinary scalings constructed below, let \(\alpha_0(\lambda)=\{H\}+\lambda\omega\) on the first space \(T_0\), with initial parameter \(1\), and let \(\alpha_i(\lambda)\) be its linear transform on a working model \(T_i\). Put \(\lambda_{-1}=1\). If the next step occurs at threshold \(\lambda_i\), its ray is annihilated by \(\alpha_i(\lambda_i)\) and is negative for \(\alpha_i(0)\). The trace \(\alpha_i(\lambda)\) is nef on \([\lambda_i,\lambda_{i-1}]\), which we call its \emph{working interval}. If \(\alpha_i(0)\) is nef, we set \(\lambda_i=0\) and stop. Working intervals are allowed to be degenerate; strict decrease of positive thresholds will be arranged only in the proof of special termination. \begin{lemma}[Positive truncations and a fixed nef line] \label{prog:nef-trivial} Let \((T,B)\) be an effective rational dlt pair on a globally strongly \(\Q\)-factorial compact Kähler \(d\)-fold, and put \(J=K_T+B\). \begin{enumerate} \item For a sufficiently large Kähler class \(\omega\), there is an ordinary \(J\)-program with scaling of \(\omega\), starting with \(\{J\}+\omega\) Kähler. Every positive truncation is finite. If \(J\) is pseudo-effective, the program either reaches a nef adjoint or has positive thresholds tending to zero. If \(J\) is not pseudo-effective, the program is finite and ends with a Mori fiber contraction. \item Let \(P\) be a nef rational line with \(mP\) Cartier, and choose \(b\in\Q\) with \(b>4dm\). The same alternatives hold for \(H=J+bP\), according to pseudo-effectivity of \(H\), with \(\omega\) enlarged if necessary. Every birational step is an ordinary \(J\)-negative step on which \(P\) is trivial, and the final Mori ray, if present, is also \(P\)-trivial. The same line \(mP\) descends through every birational contraction and pulls back to its positive side. Its traces remain analytically nef; if \(P\) is semiample, its same generated multiple and associated morphism are preserved, also through the final Mori contraction. \end{enumerate} \end{lemma} \begin{proof} Write \(H=J\) in the first case and \(H=J+bP\) in the second. Fix \(\omega\) with \(\{H\}+\omega\) Kähler. For each \(0<\eta<1\), choose a positive rational \(\epsilon\) so small that, with \(F=\lfloor B\rfloor\), \[ (T,B-\epsilon F)\ \text{is klt},\qquad \eta\omega+\epsilon\{F\}\ \text{is Kähler}. \] For \(\lambda\in[\eta,1]\) use the presentations \[ \{H\}+\lambda\omega =\{K_T+B-\epsilon F\} +\bigl(b\{P\}+\lambda\omega+\epsilon\{F\}\bigr), \] omitting \(b\{P\}\) in the first case. Their b-data are globally nef on the fixed initial carrier, and their boundary-plus-nef traces are globally modified big. These are effective gklt data. We first construct a step whenever the unscaled working trace is not nef. Choose a positive shift strictly below the current wall and use the gklt presentation of its current trace \(A\). The ray at the wall is \(A\)-negative. On a compact slice of the analytic curve cone, the cone theorem makes the \(A\)-negative part locally polyhedral after a sufficiently small positive Kähler truncation. Separating the chosen extremal ray from the other finitely many rays in that truncation gives a Kähler class \(\omega_R\) such that \(A+\omega_R\) is nef and its zero face is precisely this ray; see the supporting construction in Proposition~\ref{prog:ordinary-step}. Add the pullback of this new Kähler class to the fixed nef carrier. The resulting adjoint still has globally nef data and modified-big boundary-plus-nef trace, so the nef assertion of Proposition~\ref{prog:mori} supplies its supporting contraction. This uses a new supporting direction on the current space; it does not require the trace of the original scaling direction to be Kähler. We verify a rational ordinary detector before asserting projectivity or constructing a flip. In the second case nefness of \(P\) makes \(J\) negative on the ray; the first case has this property directly. Lower the floor by a small rational amount, retaining negativity of \(J_\epsilon\), so that \(-J\cdot R\leq2(-J_\epsilon\cdot R)\). The ordinary klt cone length bound gives a rational curve generator \(C\) with \[ 0<-J\cdot C\leq4d. \] In the second case, if \(P\cdot C>0\), integrality of \(mP\) gives \(P\cdot C\geq1/m\), contradicting \[ H\cdot C\geq-4d+b/m>0. \] Thus in that case the ray is \(P\)-trivial. The lowered ordinary adjoint is a global rational detector. The detected projectivity and flip constructions in Lemma~\ref{prog:detected-projectivity} and Proposition~\ref{prog:detected-flip} supply the projective ordinary step. The difference of \(J\) and a positive rational multiple of the lowered adjoint has zero contracted degree; exact descent in Lemma~\ref{prog:integral-descent} shows that restoring the floor preserves the positive-side sign. This is the ordinary replacement of Proposition~\ref{prog:ordinary-replacement}, and the full pair remains dlt by Equation~\eqref{prog:comparison}. In the second case, Lemma~\ref{prog:integral-descent} descends the actual Cartier line \(mP\) without changing \(m\). Analytic nefness descends and pulls back under these projective contractions \cite[Lemma~2.44]{HX}. A generated multiple descends as a generated line by projection formula and pulls back as one. Thus the same length estimate and the same \(b\) apply at the next step. The corresponding assertion on a final Mori ray follows from exactly the same estimate. For a semiample \(P\), the map of its fixed generated multiple is constant on each connected projective contracted fiber: otherwise its image contains a curve of positive degree. The map therefore factors through the normal contraction base, including in the fiber-type case. The traces used in this construction are linear by the degree-zero descent description preceding Lemma~\ref{prog:integral-descent}. At a wall the scaled class is pulled back from its base. The comparison for an earlier parameter is a nonnegative multiple of the ordinary adjoint comparison. Consequently the gklt presentations on any fixed positive segment remain gklt along its program, and every working model is a weak model for a member of that segment. The finite marked weak-model assertion in Proposition~\ref{prog:mori} applies. A marked model cannot recur: its boundary trace would be the same, whereas a nontrivial intervening ordinary step strictly increases a discrepancy. There are therefore only finitely many steps with threshold at least \(\eta\). For decreasing cutoffs, any already-constructed finite prefix remains valid. The floor perturbations cancel in the displayed classes and continue to cancel under their linear transforms; their nonpositive comparisons preserve the new gklt data. Starting at its last nef wall therefore extends the prefix to the smaller cutoff. This gives one compatible scaling program. A positive limiting threshold would lie in a finite positive truncation, so an infinite program has limit zero. Pseudo-effectivity excludes a negative Mori endpoint. If \(H\) is not pseudo-effective, its pseudo-effective threshold in the initial Kähler direction is positive. Every nef working trace, together with its effective exceptional comparison, makes the initial scaled class pseudo-effective, so the scaling thresholds are bounded below by that positive number. The program is therefore finite. Its endpoint cannot be nef, since that comparison would make \(H\) pseudo-effective; it is the stated Mori fiber contraction. \end{proof} We call these programs \emph{detected ordinary scalings}. Their steps are ordinary negative steps of one analytic extremal ray, with the linear Bott--Chern transform described above. The next lemma connects their nef working traces to the analytic negative part, including its limit at parameter zero. \begin{lemma}[Negative parts along a detected ordinary scaling] \label{prog:negative-along-scaling} In a scaling from Lemma~\ref{prog:nef-trivial}, write \(H=J\) in its first case and \(H=J+bP\) in its second, and use the notation \(\alpha_0(\lambda)=\{H\}+\lambda\omega\) above. For a parameter \(\lambda\) in the working interval of \(T_i\), a common smooth resolution \(p:W\to T_0\), \(q:W\to T_i\) gives \begin{equation} p^*\alpha_0(\lambda)=q^*\alpha_i(\lambda)+\{E_i(\lambda)\}, \qquad E_i(\lambda)\geq0,\qquad q_*E_i(\lambda)=0. \label{prog:working-negative} \end{equation} Here \(E_i(\lambda)\) is an actual real exceptional divisor, and \[ N(p^*\alpha_0(\lambda))=E_i(\lambda). \] For the limit statement, let \(H\) be any rational line on a normal compact Kähler space \(T\) whose pullback to a smooth resolution is pseudo-effective, and let \(A\) be a Kähler class on \(T\). On any fixed smooth resolution \(r:U\to T\) and for every prime \(Q\subset U\), \[ \lim_{\lambda\downarrow0}\nu_Q(r^*(\{H\}+\lambda A)) =\nu_Q(r^*\{H\}). \] Thus the same convergence holds for every divisorial place, after choosing a resolution on which it appears. \end{lemma} \begin{proof} For \(\lambda\) in the working interval of \(T_i\), every earlier threshold is at least \(\lambda\). Each earlier step is therefore negative or trivial for this parameter. Composing its comparisons gives Equation~\eqref{prog:working-negative}. Since \(\alpha_i(\lambda)\) is nef, Lemma~\ref{bir:exceptional} identifies the exceptional divisor with the negative part. On the fixed resolution \(U\), choose a Kähler class \(\Omega\) and \(c>0\) such that \(c\Omega-r^*A\) is Kähler. Monotonicity of each minimal multiplicity under addition of a nef class gives \[ \nu_Q(r^*\{H\}+\lambda c\Omega) \leq \nu_Q(r^*(\{H\}+\lambda A)) \leq \nu_Q(r^*\{H\}). \] The left side tends to the right side by the definition using small Kähler perturbations. This proves the convergence on \(U\), and the same argument applies on a resolution representing any chosen divisorial place. \end{proof} \begin{proposition}[Contracting a known negative part] \label{prog:contract-negative} Let \((T,B)\) be an effective rational dlt pair on a globally strongly \(\Q\)-factorial compact Kähler space, and put \(J=K_T+B\). Suppose there is an actual rational-line identity \[ J\sim_{\Q}P+E, \] where \(P\) is analytically nef, \(E\) is an effective rational Cartier divisor, and, on one smooth compact Kähler resolution \(\pi:W\to T\) projective over \(T\), \[ N(\pi^*\{J\})=\pi^*E. \] There is a finite ordinary \(J\)-negative program to a nef model \(T'\) which contracts precisely the prime components of \(E\) among the primes of \(T\). Every step is \(P\)-trivial, with a fixed Cartier exponent as in Lemma~\ref{prog:nef-trivial}, and \[ J_{T'}\sim_{\Q}P_{T'}. \] If \(P\) is semiample, then so is this actual last adjoint. \end{proposition} \begin{proof} Lemma~\ref{bir:pullup} makes the stated equality valid on any higher resolution. Choose \(b\) as in Lemma~\ref{prog:nef-trivial}. Adding \(bP\) leaves the same negative part. Indeed, on a smooth resolution write \(\alpha=\pi^*\{J\}\), \(p=\pi^*\{P\}\), and \(e=\pi^*E=N(\alpha)\). Subadditivity and homogeneity of negative multiplicities give \begin{align*} N(\alpha+bp)&\leq e,\\ (1+b)e=N((1+b)\alpha) &\leq N(\alpha+bp)+bN(\{e\}) \leq N(\alpha+bp)+be. \end{align*} Thus \(N(\alpha+bp)=e\). Run the scaling for \(H=J+bP\). Let \(Q\) be the strict transform on a fixed resolution of a prime component of \(E\). Its multiplicity in \(N(\pi^*\{H\})\) is positive. The limit in Lemma~\ref{prog:negative-along-scaling} makes its multiplicity positive for every sufficiently small positive parameter. If that prime still survived on the corresponding working model, Equation~\eqref{prog:working-negative} would make its multiplicity zero, because the comparison there is exceptional over that model. It must therefore be contracted after finitely many steps. The same conclusion holds if a nef endpoint is reached before taking the limit. There are only finitely many components of \(E\), so after a finite prefix all have disappeared. Throughout the program \(P\) descends and remains nef. The actual identity \(J_i\sim_{\Q}P_i+E_i\), with \(E_i\) the codimension-one pushforward of \(E\), follows from the reflexive trace convention. Once \(E_i=0\), we have \(J_i\sim_{\Q}P_i\); since \(P_i\) is nef, the program stops. Conversely, a divisorial negative step can contract only a prime of \(E_i\): its ray is \(P_i\)-trivial, so the effective divisor \(E_i\) has negative degree on every contracted curve, and those curves are contained in its support. This proves the exact assertion about contracted primes. \end{proof} The next lemma records the cohomology carried by a detected ordinary scaling that begins on a smooth space. Its surjectivity will let us choose one generic initial scaling direction for all later models. Its exceptional splitting will also carry the span of line classes to the terminal models used in Section~\ref{sec:boundary}. \begin{lemma}[Cohomology through ordinary steps] \label{prog:cohomology-transport} Let a finite prefix of one of the detected ordinary scalings constructed in Lemma~\ref{prog:nef-trivial} start on a smooth compact Kähler space, and let \(V\) be any working space. For every smooth compact Kähler resolution \(p:U\to V\) projective over \(V\), \begin{align} H^2(U,\R) &=p^*H^2(V,\R)\ \oplus\ \bigoplus_{Q\ {\rm exceptional\ for}\ p}\R\{Q\}, \label{prog:H2}\\ H^{1,1}(U,\R) &=p^*H^{1,1}_{\BC}(V,\R)\ \oplus\ \bigoplus_{Q\ {\rm exceptional\ for}\ p}\R\{Q\}. \label{prog:H11} \end{align} The linear transform from the first space is surjective onto \(H^2(V,\R)\) and \(H^{1,1}_{\BC}(V,\R)\), respectively. For a small correspondence between working spaces, the difference of the pulled-back classes and their transforms on a common resolution lies in the span of the primes exceptional over both spaces. Under the displayed decompositions, the Chern class of a global holomorphic line on \(U\) is the sum of the pullback of a rational holomorphic line class on \(V\) and an exceptional rational divisor class. In particular, passage to the exceptional quotient preserves the real span of line Chern classes. \end{lemma} \begin{proof} For a modification of a smooth compact Kähler manifold, the degree-two modification theorem gives these decompositions, and the exceptional classes have type \((1,1)\). Suppose they hold at one step. Its contraction base and both working spaces have rational singularities: slight lowering of the rational Cartier floor makes the ordinary pair klt, and the detected contraction has a base with rational singularities. These contractions are bimeromorphic, so a common resolution and Leray give \(R^if_*\OO=0\) for \(i>0\); this verifies the additional hypothesis in Lemma~\ref{prog:bc-descent}. The ray of this detected contraction is one ray for Bott--Chern degrees. This also suffices for degree-two classes. Indeed, pull a class \(\gamma\in H^2(V,\R)\) to a smooth resolution \(p:U\to V\) and take its Hodge decomposition. By the decomposition already proved for this working model in Equation~\eqref{prog:H11}, its \((1,1)\)-part is \(p^*\beta+\{E\}\) for a Bott--Chern class \(\beta\) and an exceptional real divisor \(E\). The other Hodge components have zero degree on curves. On every \(p\)-contracted curve the degree of \(E\) is therefore zero, so both signs of negativity make \(E=0\). Lifting any curve of \(V\) through the projective resolution now shows that \(\gamma\) and \(\beta\) have the same curve degrees. Thus subtracting a multiple of \(\{J\}\) from either a degree-two or a Bott--Chern class can annihilate every curve of the contracted fiber. By Lemma~\ref{prog:bc-descent}, a degree-two class, or a Bott--Chern class, with these zero degrees is pulled back from the base. Subtract a multiple of \(\{J\}\) to make any given class annihilate that ray, descend the remainder, and use the trace \(\{J^+\}\) to define its transform. On a common resolution the difference is a multiple of the exceptional adjoint comparison in Equation~\eqref{prog:comparison}. For a flip the two exceptional prime sets on that resolution are the same. For a divisorial contraction the target resolution has just the one additional exceptional prime of the step. The old decomposition therefore spans the new one and proves surjectivity of the transform. The sums are direct. If the pullback of a class from \(V\) is an exceptional divisor class, that divisor has zero degree on all curves over \(V\). Applying exceptional negativity to both signs makes the divisor zero; injectivity of pullback gives that the class is zero. The pullback injectivity in degree two and in Bott--Chern cohomology is also Lemma~\ref{prog:bc-descent}, applied to the resolution. Passing to higher smooth resolutions proves the assertion for every \(p\). For the line assertion, let \(\mathscr H\) be a holomorphic line on \(U\), and put \(\mathscr Q=(p_*\mathscr H)^{**}\). Global strong \(\Q\)-factoriality gives an invertible sheaf \(\mathscr M=\mathscr Q^{[m]}\) for some \(m>0\). The coherent evaluation comparison of \cite[Lemma~7.2]{CB} gives \[ \mathscr H^{\otimes m}\simeq p^*\mathscr M\otimes\OO_U(E), \qquad E\ \text{an integral \(p\)-exceptional divisor}. \] Indeed \((p_*\mathscr H^{\otimes m})^{**}=\mathscr M\), since the two sheaves agree at the general points of all divisors of \(V\). Local generators of \(p_*\mathscr H^{\otimes m}\) in a frame of \(\mathscr M\) compare their evaluations with the pulled-back coefficients by meromorphic quotients. Those quotients agree on the isomorphism locus and hence glue; their zeros and poles are exceptional. Taking Chern classes proves the assertion without choosing a meromorphic section of \(\mathscr H\). \end{proof} Before returning to special termination, we record the consequence used in the boundary argument: a lower-dimensional manifold of Kodaira dimension zero has a terminal torsion model carrying the preceding exceptional splitting. \begin{corollary}[Terminal models in Kodaira dimension zero] \label{prog:terminal-kappa-zero} Let \(X\) be a smooth compact Kähler \(d\)-fold with \(d0\), and transformed classes \(\{J_V\},A_V\) such that \begin{equation} \{J_V\}+tA_V\ \text{is nef for every }0\leq t\leq\delta. \label{prog:whole-nef-interval} \end{equation} On a common smooth resolution of \(S\) and \(V\), the differences of pullbacks of \(J,A\) and these transforms lie in the common exceptional span. \end{lemma} \begin{proof} Let \(P_M=J_M\) denote the semiample rational line from Lemma~\ref{prog:stratum-model}, and use its transform \(A_M\). On a common resolution \(p:W\to S,\ q:W\to M\), let \(F_A\) be the real divisor in the common exceptional span defined by \[ q^*A_M-p^*A=\{F_A\}. \] It is effective: \(-F_A\) is \(q\)-nef, because \(p^*A\) is nef and the other term is pulled back from \(M\); apply exceptional negativity. Its support is disjoint from \(p^{-1}(U)\). In particular \(A_M\) is pseudo-effective, and its negative multiplicity at every prime of \(M\) is zero. Indeed, \(q^*A_M=p^*A+\{F_A\}\), and Lemma~\ref{bir:exceptional} applied over \(S\) gives \(N(q^*A_M)=F_A\). Let \(D_M\) be the rational Cartier trace of \(D\). Choose \(\tau>0\) so small that \(A+\tau\{D\}\) is Kähler. For a fixed sufficiently small \(t>0\), consider on \(M\) the boundary \(\Delta_M-t\tau D_M\) and the nef b-class represented on \(W\) by \(t p^*(A+\tau\{D\})\). Its trace on \(M\) is \(t(A_M+\tau\{D_M\})\), so its generalized adjoint is \begin{equation} \{P_M\}+tA_M. \label{prog:interval-gadjoint} \end{equation} These are effective gklt data. Over \(U\) the b-class descends to \(M\), so its generalized discrepancy contribution is zero there, and subtracting \(t\tau D_M\) raises every zero log discrepancy of the dlt pair. Away from the floor the ordinary pair is klt; on one fixed log resolution its finitely many subunit coefficients stay subunit for small \(t\). This proves gklt. The generalized boundary is big. On a common resolution the pullback of the trace of \(A+\tau\{D\}\) minus \(p^*(A+\tau\{D\})\) is effective exceptional by the same negativity argument as for \(F_A\). The latter pullback is nef and big, so the trace is big; the remaining boundary is effective. Choose an integer \(m>0\) such that \(mP_M\) is generated, and let \(g:M\to Z\) be the Stein factor of its morphism. Then \(Z\) is normal projective and \(mP_M=g^*H\) for an ample generated Cartier line \(H\) on \(Z\). Set \(L=\{P_M\}+tA_M\). The preceding preparation makes \(L\) an effective gklt adjoint with globally nef b-data and globally modified-big boundary-plus-nef trace. It is pseudo-effective because \(P_M\) is nef and \(A_M\) is modified nef. Apply the proper relative assertion of Proposition~\ref{prog:mori} over \(Z\). This produces a chosen nonextracting good log terminal model \(\phi:M\dashrightarrow V\) and a morphism \(g_V:V\to Z\), with \(L_V\) nef over \(Z\). The hypothesis here is properness: the semiample morphism \(M\to Z\) need not be projective. Put \[ mP_V=g_V^*H,\qquad A_V=(L_V-\{P_V\})/t. \] The relative construction preserves the actual line pulled back from \(Z\), and its forward Bott--Chern transport preserves the displayed linear relation; these are the transforms of \(P_M,A_M\). The map \(\phi\) is small. On a common projective resolution \(a:W'\to M\), \(c:W'\to V\), its comparison is \[ a^*L=c^*L_V+\{E\},\qquad E\geq0, \] where \(E\) is exceptional over \(V\) and has positive coefficient at the strict transform of each prime contracted from \(M\). The forward trace \(L_V\) is pseudo-effective, so exceptional translation gives \[ N(a^*L)=N(c^*L_V)+E\geq E. \] But \(L\) is modified nef on \(M\); the coefficient of \(N(a^*L)\) at a strict prime of \(M\) is zero. Thus no prime of \(M\) is contracted. Nonextraction then proves smallness. Fix \(b>2dm\). We claim that \(L_V+b\{P_V\}\) is absolutely nef. Otherwise the cone theorem for the effective gklt adjoint \(L_V\) gives an \(L_V\)-negative rational extremal generator \(C\) such that \[ (L_V+b\{P_V\})\cdot C<0,\qquad 0<-L_V\cdot C\leq2d. \] Indeed the cone summand on which \(L_V\) is nonnegative also has nonnegative \(P_V\)-degree, so a negative ray summand must account for any failure of nefness. If \(P_V\cdot C=0\), ampleness of \(H\) makes \(C\) vertical over \(Z\), contradicting relative nefness of \(L_V\). Otherwise Cartier integrality gives \(P_V\cdot C\geq1/m\), contradicting \[ (L_V+b\{P_V\})\cdot C\geq-2d+b/m>0. \] This proves the claim. Both \(\{P_V\}\) and \((1+b)\{P_V\}+tA_V\) are nef. By convexity, \[ \{P_V\}+sA_V\ \text{is nef for }0\leq s\leq\frac{t}{1+b}. \] Thus \(\delta=t/(1+b)\) works. The actual line \(P_V\) has the same generated multiple as \(P_M\), since both are pulled back from \(H/m\). Its class is the transform of \(J\). The relative comparison and linear transport give exceptional-divisor comparisons for \(P_M,A_M\). Since \(S,M,V\) agree in codimension one, their final comparison divisors are exceptional over both \(S\) and \(V\), as required. \end{proof} \subsection{Special termination} \begin{theorem}[Special termination for a chosen dlt scaling] \label{prog:special-termination} Assume \(\Ggood_d\) for every \(d0\) on this working space, above the remaining walls, and set \[ A=\{J\}+\lambda_0H. \] It is Kähler, as the restriction of the interior ambient Kähler class. The exact change of parameter is \begin{equation} \{J\}+tA=(1+t)(\{J\}+\lambda H),\qquad \lambda=\frac{\lambda_0t}{1+t},\qquad t=\frac{\lambda}{\lambda_0-\lambda}. \label{prog:reparameterization} \end{equation} It applies for \(0\leq\lambda<\lambda_0\). The new walls \(t_i\) strictly decrease toward zero, and the traces \(\{J_i\}+tA_i\) are nef on their successive restricted working intervals. On a common resolution of a nontrivial small stratum step, let \(D_i\) be the real exceptional divisor defined by \[ p_i^*\{J_i\}-q_i^*\{J_{i+1}\}=\{D_i\}. \] The strict adjoint comparison makes \(D_i\) an effective nonzero divisor exceptional over both strata. Equality of the scaled pullbacks at its wall gives the exact Bott--Chern equality \begin{equation} p_i^*(\{J_i\}+tA_i)-q_i^*(\{J_{i+1}\}+tA_{i+1}) =\left(1-\frac{t}{t_i}\right)\{D_i\}. \label{prog:restricted-comparison} \end{equation} For \(t\) in a later working interval all preceding factors on the right are nonnegative. Telescoping on a common resolution expresses the pullback of \(\{J\}+tA\) as the pullback of its nef working trace plus an effective divisor exceptional over that working stratum. Lemma~\ref{bir:exceptional} identifies the latter divisor with the negative part. Push the resulting positive currents to one fixed smooth resolution of \(S\) and let \(t\downarrow0\). Closedness of the pseudo-effective cone proves that \(J\) is pseudo-effective. The negative multiplicities converge valuation by valuation by the limit statement in Lemma~\ref{prog:negative-along-scaling}. The comparison divisors have no strict prime of \(S\), since the maps are small. Their coefficients also vanish at every place whose center meets the floor: each finite composition is unchanged on a neighborhood of the floor by the induction on smaller centers. Passing to the limit proves Equation~\eqref{prog:away-floor}. The stratum has dimension less than \(n\), and the divisor \(D_S\) constructed above is rational Cartier with support equal to its floor. Lemmas~\ref{prog:stratum-model} and \ref{prog:nef-interval} give one small model \(V\) on which all transformed classes of \(\{J\}+tA\) are nef for \(0\leq t\leq\delta\). On a common smooth resolution \(p:W\to S,\ q:W\to V\), define the real divisor \(F(t)\) in the common exceptional span by \begin{equation} p^*(\{J\}+tA)-q^*(\{J_V\}+tA_V)=\{F(t)\}. \label{prog:affine-comparison} \end{equation} Uniqueness of the representing exceptional divisor makes \(F(t)\) coefficientwise affine in \(t\). It is effective on \([0,\delta]\): \(-F(t)\) is \(p\)-nef because the class pulled back from \(V\) is nef, so exceptional negativity applies. It is exceptional over \(V\) as well. Lemma~\ref{bir:exceptional} gives \[ N\bigl(p^*(\{J\}+tA)\bigr)=F(t) \qquad(0\leq t\leq\delta). \] Every divisorial negative multiplicity is thus affine on this whole interval. This remains true on higher resolutions by Lemma~\ref{bir:pullup}. Choose a nontrivial stratum wall \(t_i\in(0,\delta)\). On a common resolution of \(S,V\) and the adjacent strata, their nef working models compute the same negative part on their respective open intervals. Their two affine expressions agree at \(t_i\). But Equation~\eqref{prog:restricted-comparison} says that, at a prime with coefficient \(d_i>0\) in \(D_i\), their difference is \[ \left(1-\frac{t}{t_i}\right)d_i. \] Their slopes differ by \(-d_i/t_i\). A single affine function on \([0,\delta]\) cannot agree with both on two open intervals. This contradiction excludes every such wall. Infinitely many walls would tend to zero, so the nontrivial diagrams on this stratum are finite. There are finitely many surviving strata. Induction on their dimensions makes all sufficiently late ambient flipping and flipped loci disjoint from the floor. This proves the theorem. \end{proof} \begin{corollary}[A signed adjoint supported on the floor] \label{prog:signed-nef} Assume \(\Ggood_d\) for every \(d0\) \cite[Theorem~5.2]{FujinoAnalyticBCHM}. Composing a resolution of \(T\) with \(f\), the Leray spectral sequence proves that \(Z\) has rational singularities. The same reasoning works when the klt boundary is chosen separately on finitely many Stein base neighborhoods. For fiber-type contractions in case~(ii), rationality of the base follows from \cite[Lemma~8.8(i)]{DHP}; thus that application also precedes the construction of generalized data on the base. \begin{lemma}[Projectivity from a rational line detecting the ray] \label{prog:detected-projectivity} Let $f:T\to Z$ be a proper contraction between normal compact K\"ahler spaces, with $T$ having rational singularities. Suppose $\gamma$ is K\"ahler on $Z$ and \[ \alpha=f^*\gamma,\qquad \overline{\mathrm{NA}}(T)\cap\alpha^\perp=R \] is a nonzero ray. If a global rational line bundle $D$ has $D\cdot R<0$, then $-D$ is relatively ample and $f$ is projective. \end{lemma} \begin{proof} Fix a K\"ahler class $\omega$ on $T$ and normalize $\overline{\mathrm{NA}}(T)$ by $\omega\cdot z=1$. The resulting slice $S$ is compact. Its unique point on $R$ has negative $D$-degree, so $d=c_1(D)$ is negative on a neighborhood of that point in $S$. On the complementary compact set, $\alpha$ has a strictly positive minimum, while $d$ is bounded. For all sufficiently small $s>0$, the class $\alpha-sd$ is therefore strictly positive on $S$ and is K\"ahler by cone duality. Choose an integer $m>0$ making $mD$ integral. Then \[ c_1(-mD)+f^*((m/s)\gamma)=(m/s)(\alpha-sd) \] is K\"ahler. On a neighborhood in the base with a potential for $\gamma$, absorb that potential in a smooth metric on the same global line bundle $-mD$. Its curvature is positive on the fibres. The analytic positive-line-bundle criterion and the fibrewise relative-ampleness criterion make $-mD$ relatively ample. The line bundle was global throughout, so no gluing of unrelated local polarizations is required. \end{proof} An analytic contraction supplied by a supporting class therefore becomes projective as soon as one rational line has negative degree on its ray. For a small contraction, that same line will define the flip globally. The local ordinary adjoints used to prove finite generation need not glue; their section algebras will be Veronese subalgebras of one algebra. \begin{lemma}[Divisor representatives over a Stein base] \label{prog:stein-representatives} Let $\pi:Y\to U$ be a projective surjective morphism with connected fibres between normal irreducible complex spaces, where $U$ is Stein, and let $\mathcal L$ be a holomorphic line bundle on $Y$. If $\pi$ is bimeromorphic, $\mathcal L$ has a holomorphic section which is not identically zero, and hence an effective Cartier divisor representative. More generally, after shrinking $U$ around any specified point, $\mathcal L$ has a nonzero meromorphic section. \end{lemma} \begin{proof} Suppose first that $\pi$ is bimeromorphic. Grauert's proper direct-image theorem makes $\mathcal F=\pi_*\mathcal L$ coherent. On the nonempty open subset $U^\circ$ where $\pi$ is an isomorphism, it is a line bundle. Choose $u_*\in U^\circ$. Cartan's theorem~A says that global sections generate $\mathcal F_{u_*}$, so one global section has nonzero image in $\mathcal F_{u_*}/\mathfrak m_{u_*}\mathcal F_{u_*}$. Under \[ H^0(U,\pi_*\mathcal L)=H^0(Y,\mathcal L) \] it gives a section $s$ nonzero at the unique point above $u_*$. Its zero divisor is effective Cartier and represents $\mathcal L$. The section may vanish elsewhere; only its meromorphic inverse is being used to obtain a meromorphic trivialization. For the general case, fix $u\in U$ and $y\in\pi^{-1}(u)$, and choose a $\pi$-ample line bundle $\mathcal H$. Relative Serre generation, after shrinking to a Stein neighbourhood of $u$, gives an integer $n$ for which both $\mathcal L\otimes\mathcal H^n$ and $\mathcal H^n$ are relatively generated. Their proper direct images are coherent. Cartan's theorem~A and the evaluation maps supply sections $s$ and $t$ of these two bundles which are both nonzero at $y$. The quotient $s/t$ is the required nonzero meromorphic section of $\mathcal L$. At no point is $Y$ assumed Stein. \end{proof} \begin{proposition}[A global algebra for a detected flip] \label{prog:detected-flip} Let $(T,B+\mathbf M)$ be a gklt pair on a normal compact K\"ahler space which is globally strongly $\Q$-factorial, and let $A\in H^{1,1}_{\BC}(T,\R)$ be its adjoint class. Let $f:T\to Z$ be a projective small contraction onto a normal compact K\"ahler space. Suppose that the classes of all $f$-vertical curves lie on one ray $R\subset\overline{\mathrm{NA}}(T)$, that $A\cdot R<0$, and that a global line bundle $L$ satisfies $L\cdot R<0$. Then $\bigoplus_{m\geq0}f_*L^m$ is locally finitely generated. The relative Proj of a sufficiently divisible Veronese is a projective small contraction $f^+:T^+\to Z$, where $T^+$ is compact K\"ahler and globally strongly $\Q$-factorial. The reflexive transform $L^+$ is a rational line bundle, with a positive power equal to the relatively ample tautological bundle. The transformed boundary and the same b-nef datum define a gklt pair on $T^+$, with relatively K\"ahler adjoint $A^+$. This is the generalized flip, with the usual strict discrepancy increase at centres in either exceptional locus. Moreover, if $t=(A\cdot C)/(L\cdot C)>0$ for an $f$-vertical curve $C$, then there is a unique $\eta\in H^{1,1}_{\BC}(Z,\R)$ such that \[ A=t\,c_1(L)+f^*\eta,\qquad A^+=t\,c_1(L^+)+(f^+)^*\eta. \] Every class on $T$ has the corresponding forward transport with an actual exceptional-divisor correction. No assertion of surjectivity on Bott--Chern groups is required. \end{proposition} \begin{proof} Since $f$ has a global relatively ample bundle and all its vertical curves lie on $R$, numerical invariance of ampleness on the projective fibres shows that $-L$ is $f$-ample. \smallskip\noindent \emph{A local rational ordinary adjoint.} Fix $z\in Z$ and a relatively compact Stein neighbourhood $U$. Take a projective log resolution $p:V\to T_U$ carrying $\mathbf M$, and put $\rho=f p$. Write its actual structure boundary as $B_V$, so \[ [K_V+B_V]+[\mathbf M_V]=p^*A. \] Every coefficient of $B_V$ is less than one. Apply Lemma~\ref{prog:stein-representatives} to obtain a Cartier representative $D_L$ of $L$ on $T_U$. The same birational direct-image argument applies to the coherent rank-one reflexive canonical sheaf: it supplies a canonical form nonzero at a chosen point of the common smooth isomorphism locus. Use this meromorphic form on $T_U$ and its pullback to $V$ to choose compatible canonical representatives, and set \[ N=t p^*D_L-K_V-B_V. \] For every $\rho$-vertical curve, its degree equals that of $\mathbf M_V$. Thus $N$ is $\rho$-nef. The relative bigness of $N$ uses the additional fact that $\rho=f\circ p$ is \emph{bimeromorphic}: both the small contraction $f$ and the resolution $p$ are bimeromorphic. It does not follow from relative nefness alone. In fact every real Cartier divisor is relatively big for this projective bimeromorphic map over the Stein neighbourhood. Here is the required ample-plus-effective decomposition explicitly. Write $N=\sum_j r_jN_j$, where $r_j>0$, $\sum_jr_j=1$, and the $N_j$ are rational Cartier divisors in its finite Cartier span. Choose a $\rho$-ample line bundle and a Cartier representative $P$ for it using Lemma~\ref{prog:stein-representatives}. If $q_jN_j$ is integral, apply the birational case of that lemma to $\mathcal O_V(q_jN_j-P)$. Its nonzero holomorphic section has an effective Cartier zero divisor $E_j$, giving the actual relation $q_jN_j\sim P+E_j$. This use depends on $\rho$ being bimeromorphic; the section is allowed to vanish along exceptional fibres. Therefore \[ N\sim_{\R} \underbrace{\left(\sum_j\frac{r_j}{q_j}\right)P}_{H} +\underbrace{\sum_j\frac{r_j}{q_j}E_j}_{E}. \] Here $H$ is $\rho$-ample and $E\geq0$, proving $\rho$-bigness. For a map with positive-dimensional general fibres the direct image used above can be zero; no such bigness claim is made for that setting. Make the finitely many section choices on a slightly larger Stein neighbourhood, and now shrink $U$ so its closure is compact in that neighbourhood. Properness makes the inverse image of this closure compact. A log resolution of the finitely many resulting divisors therefore gives a uniform positive bound for the following choice. Choose $\epsilon>0$ small enough that $(V,B_V+\epsilon E)$ is sub-klt. The divisor $(1-\epsilon)N+\epsilon H$ is relatively ample. Express it as a positive combination of rational ample divisors and take general members of sufficiently high multiples. Relative Bertini \cite[Theorem~2.20]{DHP}, applied on a log resolution of $B_V+E$, gives an effective $\Theta\sim_{\R}N$ such that $(V,B_V+\Theta)$ is sub-klt. The high multiples make the added coefficients arbitrarily small. All choices are made near the compact fibre; shrink $U$ once to retain them. Put $\Delta=p_*(B_V+\Theta)\geq0$. Write the actual linear equivalence as $\Theta=N+\sum_j a_j\operatorname{div}_V(g_j)$. The bimeromorphic map $p$ identifies meromorphic function fields, so $g_j=p^*h_j$. Pushing down gives $K_{T_U}+\Delta=tD_L+\sum_j a_j\operatorname{div}_{T_U}(h_j)$; in particular this divisor is real Cartier. Pulling back this same identity gives \[ K_{T_U}+\Delta\sim_{\R}tD_L, \qquad p^*(K_{T_U}+\Delta)=K_V+B_V+\Theta. \] Thus $(T_U,\Delta)$ is klt. Any finitely many base line bundles in a relative linear equivalence can first be trivialized by shrinking $U$. Record the resulting equality as \begin{equation}\label{prog:detector-rationalization} K_{T_U}+\sum_i d_iD_i=tD_L+\sum_j b_j\operatorname{div}(g_j), \qquad d_i>0, \end{equation} using the positive support of $\Delta$. After a further relatively compact shrinking, the displayed divisors have finitely many prime components: their locally finite supports meet a compact inverse image. Equality of coefficients in \eqref{prog:detector-rationalization} is a finite rational affine system in $(d_i,t,b_j)$. The klt condition is open within this system, as seen on one log resolution; the pullbacks of its adjoints vary linearly by the displayed identity. A nearby rational solution therefore gives a rational effective klt boundary $\Delta_q$ and $t_q\in\Q_{>0}$ with \begin{equation}\label{prog:detector-local-adjoint} K_{T_U}+\Delta_q\sim_{\Q}t_qD_L. \end{equation} The individual local primes $D_i$ need not be $\Q$-Cartier: the identity ensures that the total adjoint is $\Q$-Cartier. \smallskip\noindent \emph{One global flip algebra.} The adjoint in \eqref{prog:detector-local-adjoint} is $f_U$-antiample. Its ordinary flip and local finite generation follow from \cite[Theorems~1.14 and~1.18]{FujinoAnalyticBCHM}. Clearing the displayed linear equivalence identifies a Veronese of its canonical algebra with a Veronese of $\mathcal R:=\bigoplus_{m\geq0}f_*L^m$ over $U$. Hence $\mathcal R$ is locally finitely generated by \cite[Lemma~2.26]{FujinoAnalyticBCHM}. The graded pieces are coherent. Over a connected normal base open the inverse image is irreducible; products of nonzero sections of line bundles are nonzero there. Thus its graded section algebra is an integral domain, without choosing any global meromorphic frame of $L$. Compactness supplies one sufficiently divisible Veronese $\mathcal R^{(r)}$ generated in degree one. Its relative Proj $T^+$ restricts to the ordinary flip over every such $U$. It is therefore normal, locally klt, and small over $Z$, and has the global relatively ample bundle $\mathcal O_{T^+}(1)$. On the common big open this bundle is $L^r$. Reflexivity gives \[ (L^+)^{[r]}\simeq\mathcal O_{T^+}(1). \] This proves that $L^+$ is a rational line bundle before any assertion of factoriality. Relative positivity over the compact K\"ahler base makes $T^+$ compact K\"ahler. For any global rank-one reflexive sheaf $\mathcal F^+$ on $T^+$, take its coherent reflexive transform $\mathcal F$ on $T$ using a common resolution. Some $M=\mathcal F^{[m]}$ is a line bundle. Choose $c\in\Q$ so that $(M+cL)\cdot C=0$, and clear its denominator to obtain an integral bundle $J=n(M+cL)$ numerically trivial over $f$. The local rational klt pairs \eqref{prog:detector-local-adjoint} have antiample adjoint, so Lemma~\ref{prog:integral-descent} yields one global line bundle $J_Z$ with $J=f^*J_Z$. On $T^+$, pull back $J_Z$, undo the rational twist $ncL^+$, and compare on the common big open. After clearing the already established index of $L^+$, reflexivity gives an invertible positive reflexive power of $\mathcal F^+$. This proves global strong factoriality. \smallskip\noindent \emph{The original generalized pair.} The local log-Fano pairs above give $R^if_*\mathcal O_T=0$ for $i>0$ by \cite[Theorem~5.2]{FujinoAnalyticBCHM}. They make $T$ locally klt, hence rational. Leray for a projective resolution then shows that $Z$ is rational. Lemma~\ref{prog:bc-descent}(i) therefore gives $f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$, with injective pullback. Thus $A=t c_1(L)+f^*\eta$ for a unique $\eta$. Define $A^+=t c_1(L^+)+(f^+)^*\eta$; it is relatively K\"ahler. To construct its actual discrepancy divisor, let $\mathcal I$ be the ideal defined by \[ \operatorname{im}(f^*f_*L^r\longrightarrow L^r) =\mathcal I\otimes L^r. \] Principalize $\mathcal I$ on a common resolution $p:W\to T$, $q:W\to T^+$ carrying $\mathbf M$. If $\mathcal I\mathcal O_W=\mathcal O_W(-F_L)$, the tautological quotient gives the actual bundle identity \[ q^*\mathcal O_{T^+}(1)=p^*L^r\otimes\mathcal O_W(-F_L). \] Hence $E_L=F_L/r$ is effective and exceptional over both sides, and $[E_L]=p^*c_1(L)-q^*c_1(L^+)$. If $B_W$ is the original structure boundary, put $B_W^+=B_W-tE_L$. Then \[ [K_W+B_W^+]+[\mathbf M_W]=q^*A^+. \] Its pushforward is $B^+$, and its discrepancies do not decrease. The local ordinary comparison in \eqref{prog:detector-local-adjoint} has discrepancy divisor $t_qE_L$, so its strictness proves the stated strictness for $tE_L$ as well. Thus the original pair remains gklt. Finally, write any $\theta\in H^{1,1}_{\BC}(T,\R)$ uniquely as $\theta=f^*\eta_\theta+s c_1(L)$ and set $\theta^+=(f^+)^*\eta_\theta+s c_1(L^+)$. Its correction is the actual divisor $sE_L$. For a rational line $M$, the number $s$ is rational. Applying Lemma~\ref{prog:integral-descent} to a Cartier multiple of $M-sL$ gives $M=sL+f^*M_Z$ as actual rational lines. Reflexive extension on the common big open gives $M^+=sL^++(f^+)^*M_Z$. Consequently the cohomological transport agrees with the reflexive transform on Chern classes. \end{proof} \subsubsection{Real boundaries and their linear transport} The rational ordinary step constructed below also determines the step for every sufficiently close real boundary. The relevant comparison is an identity of actual rational lines over the contraction base. \begin{proposition}[Ordinary real boundaries] \label{prog:ordinary-replacement} Let $T$ be normal compact K\"ahler and globally Weil $\Q$-factorial, with canonical sheaf a rational line bundle. Let $B\geq0$ be a real boundary with $(T,B)$ klt. Assume the rational ordinary-step result of Proposition~\ref{prog:ordinary-step} for $T$ and its subsequent models. Then every $(K_T+B)$-negative extremal analytic ray has an ordinary step with projective contractions, compact K\"ahler models, and the expected opposite relative ample signs. Global Weil $\Q$-factoriality is preserved; global strong $\Q$-factoriality is preserved when imposed initially. \end{proposition} \begin{proof} Put $D=K_T+B$ and fix a $D$-negative extremal ray $R$. On the finite positive support of $B$, effectiveness, the klt condition and negativity on $R$ persist under small coefficient changes. Klt openness is checked on one log resolution of this support. Choose nearby effective rational klt boundaries $B_0,\ldots,B_m$ and positive real numbers $u_j$ with \[ B=\sum_j u_jB_j,\qquad \sum_j u_j=1, \qquad D_j\cdot R<0,\quad D_j=K_T+B_j. \] For $B=0$ take just $B_0=0$. Every $D_j$ is a global rational line bundle. The rational-step theorem for $(T,B_0)$ provides a projective contraction $f:T\to Z$ onto a compact K\"ahler base, contracting exactly $R$, with $-D_0$ relatively ample. The base is rational by \cite[Lemma~8.8]{DHP}, so Lemma~\ref{prog:bc-descent} gives $f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$; use case~(ii) when $f$ is of fiber type. Let $C$ be a rational curve spanning $R$, and set $a_j=(D_j\cdot C)/(D_0\cdot C)\in\Q_{>0}$. On every projective fibre, $-D_j$ is numerically equivalent to the positive multiple $-a_jD_0$. Numerical invariance of ampleness on that fibre, followed by the relative-ampleness criterion, makes $-D_j$ relatively ample. The positive combination $-D$ is relatively ample as a real line bundle. This also handles a fibre-type contraction. Suppose henceforth that $f$ is birational. A Cartier multiple $M_j=n_j(D_j-a_jD_0)$ is numerically trivial over $Z$ and $M_j-D_0$ is relatively ample. Lemma~\ref{prog:integral-descent}, applied to the rational klt pair $(T,B_0)$ gives a rational line bundle $A_j$ on $Z$ with the actual identity \begin{equation}\label{prog:real-line-descent} D_j=a_jD_0+f^*A_j. \end{equation} Here and below an identity of rational line bundles means an isomorphism after a common integral multiple. The descent lemma is used only for birational $f$. For a small $f$, let $f^+:T^+\to Z$ be its rational $D_0$-flip. The rational theorem supplies a global relatively ample adjoint $D_0^+$ and preserves the stated factoriality and canonical-sheaf conditions. The other transformed adjoints $D_j^+$ are therefore rational line bundles. Transform \eqref{prog:real-line-descent} on the common big open and extend by reflexivity to obtain \[ D_j^+=a_jD_0^++(f^+)^*A_j. \] All $D_j^+$ are relatively ample, as is $D^+=\sum_j u_jD_j^+$. Thus this same small map is the required ordinary real-boundary flip. The real ordinary discrepancy comparison proves that $(T^+,B^+)$ is klt and gives the strict increases at exceptional centres. For a divisorial contraction, write $D_0=f^*D_0'+e_0E$ with $e_0>0$. Transforming \eqref{prog:real-line-descent} on the target and summing yields \[ D-f^*D'=\left(\sum_j u_ja_j\right)e_0E. \] This coefficient is positive. The same discrepancy comparison proves the required assertions. The ambient category was already preserved by the rational step in both cases. \end{proof} For later use, let \(D=K_T+B\) drive one of the birational steps just constructed, and write \(f':T'\to Z\) for its positive morphism (or the identity for a divisorial contraction). Lemma~\ref{prog:bc-descent} gives the unique decomposition and forward transform \begin{equation} \theta=f^*\eta+s\{D\},\qquad \theta'=(f')^*\eta+s\{D'\}. \label{prog:ordinary-forward-transport} \end{equation} On a common resolution \(p:W\to T\), \(q:W\to T'\), the ordinary comparison \(p^*D-q^*D'=F\) is an actual effective real divisor, exceptional over \(T'\) and over both sides for a flip. Hence \(p^*\theta-q^*\theta'=s\{F\}\). On Chern classes this is the reflexive transform, by the line identities in the proof above and Lemma~\ref{prog:integral-descent}. This construction gives forward transport without assuming surjectivity onto the next Bott--Chern group. \subsubsection{Actual real lines and relative numerical spaces} A second polarization argument will be used for maps whose fibers are Moishezon. It applies to a real combination of global line bundles, provided that its degrees on the fibers agree with a relatively Kähler class. \begin{lemma}[Polarization by an actual real line bundle] \label{prog:actual-line-polarization} Let $f:T\to Z$ be a proper Moishezon map, where $T$ is a compact K\"ahler space. Suppose that $L\in\operatorname{Pic}(T)\otimes\R$ and a relatively K\"ahler class $\omega$ satisfy \[ L\cdot C=\omega\cdot C \quad\text{for every compact curve $C$ contracted by $f$.} \] Then $L$ is relatively ample and $f$ is projective. \end{lemma} \begin{proof} Let $V$ be an irreducible positive-dimensional subspace of a reduced fibre. It is Moishezon. Normalize $V$ and take a smooth projective modification $\pi:\widetilde V\to V^{\mathrm n}$, chosen so that an effective exceptional divisor $E$ has $-E$ relatively ample. The pullback of the K\"ahler class from $V$ to its normalization is K\"ahler. Consequently $\pi^*\omega-\delta[E]$ is K\"ahler for sufficiently small $\delta>0$. On the projective manifold $\widetilde V$ the actual real line bundle $\pi^*L-\delta E$ has the same curve degrees as that K\"ahler class, and is therefore ample. For example, subtract a sufficiently small positive multiple of a fixed ample class from the K\"ahler class; the corresponding real line bundle is nef by the projective numerical criterion. Also $\pi^*L$ is nef, and the decomposition \[ \pi^*L=(\pi^*L-\delta E)+\delta E \] shows that it is big. Thus \[ L^{\dim V}\cdot V=(\pi^*L)^{\dim V}>0. \] The real Nakai--Moishezon criterion for proper algebraic spaces \cite[Theorem~1.6]{FujinoMiyamoto2020}, applied to the algebraizations of the Moishezon fibres, proves that $L$ is ample on every reduced fibre. The criterion is insensitive to nilpotents and reducible components. In the coefficient space of the finitely many global line bundles occurring in $L$, choose, for each fibre, a rational simplex containing $L$ in its interior whose vertices restrict to ample rational lines on that fibre. Openness of fibrewise ampleness for each of these finitely many rational lines gives a base neighbourhood on which the entire simplex is relatively ample. Choose finitely many such neighborhoods covering the base and intersect their coefficient neighborhoods of $L$. Every rational point of the resulting neighborhood is relatively ample globally. A rational simplex in this intersection also expresses $L$ as a positive combination of relatively ample rational lines. Clearing the denominator of one rational point proves projectivity. \end{proof} \begin{remark} The proof uses only positivity of the top self-intersections of $L$. Equality of curve degrees with $\omega$ does not assert equality of their higher intersection numbers or of their Bott--Chern classes. \end{remark} The projective morphism needed for the relative program can be chosen on a smooth model of the rational quotient. Here the Hodge-theoretic polarization has an elementary construction, independent of any birational projectivity-descent assertion. \begin{lemma}[Smooth Hodge-theoretic polarization] \label{prog:smooth-polarization} Let \(g:W\to S\) be a surjective morphism with connected fibers between smooth compact Kähler manifolds. If \(g^*:H^0(S,\Omega_S^2)\to H^0(W,\Omega_W^2)\) is an isomorphism, then \(g\) is projective. This applies when its general fiber is rationally connected. \end{lemma} \begin{proof} We use the smooth argument of \cite[Theorem~3.1, Step~1]{CH}. Let \(d=\dim W-\dim S\). Choose a rational class \(u\in H^2(W,\Q)\) close to a Kähler class. Write its Hodge decomposition as \(u=b+g^*v\), where \(b\) is Kähler and \(v\) has types \((2,0)\) and \((0,2)\). Hodge type and the projection formula give \[ c=g_*(u^d)=g_*(b^d)>0,\qquad g_*(u^{d+1})=g_*(b^{d+1})+(d+1)c\,v. \] Since pushforward is rational, the class \[ \ell=u-g^*\frac{g_*(u^{d+1})}{(d+1)c} =b-g^*\frac{g_*(b^{d+1})}{(d+1)c} \] is rational and of type \((1,1)\). Lefschetz's theorem makes it the Chern class of a rational line. The displayed equality supplies a metric with positive curvature locally over \(S\), by adding a potential for the base class. Thus this line is relatively ample. For rationally connected general fibers, holomorphic two-forms are pulled back from the smooth base, so the hypothesis holds. \end{proof} \begin{lemma}[A rational Hodge correction over a smooth base] \label{prog:relative-hodge} Let $f:T\to S$ be an already projective surjective morphism of normal compact K\"ahler spaces. Assume that $S$ is smooth, $T$ has globally strongly $\Q$-factorial klt singularities, and, for a projective resolution $r:W\to T$, the morphism $g=f\circ r$ satisfies \[ g^*:H^0(S,\Omega_S^2)\xrightarrow{\ \simeq\ } H^0(W,\Omega_W^2). \] Then every $\alpha\in H^{1,1}_{\BC}(T,\R)$ can be written \[ \alpha=c_1(L)+f^*\gamma, \] where $L\in\operatorname{Pic}(T)\otimes_{\mathbb Z}\mathbb R$ is a finite real combination of global line bundles and $\gamma\in H^{1,1}(S,\mathbb R)$. \end{lemma} \begin{proof} The composition $g$ is projective, since all the spaces are compact. Choose a $g$-ample line bundle on $W$, with integral Chern class $H$. Set $d=\dim W-\dim S$ and \[ c=g_*(H^d)>0. \] Thus $c$ is the positive degree of $H^d$ on a general fibre. The cohomological pushforward here is the usual pushforward between the smooth compact manifolds $W$ and $S$, defined over $\mathbb Q$ and of Hodge bidegree $(-d,-d)$. Define a rational linear operator on degree-two cohomology by \[ \Pi(\eta)=\eta-g^*\!\left(\frac{g_*(\eta\smile H^d)}{c}\right). \] Every class of type $(2,0)$ on $W$ is pulled back from $S$, and the same holds for type $(0,2)$. The projection formula therefore shows that $\Pi$ kills both types. It preserves type $(1,1)$. Consequently \[ \Pi(H^2(W,\mathbb Q))\subset H^2(W,\mathbb Q)\cap H^{1,1}(W). \] By the Lefschetz $(1,1)$ theorem the image consists of rational Chern classes of global line bundles. Applying $\Pi$ to $r^*\alpha$ gives \[ r^*\alpha=c_1(L_W)+g^*\gamma, \qquad L_W\in\operatorname{Pic}(W)\otimes\mathbb R, \qquad \gamma=\frac{g_*(r^*\alpha\smile H^d)}{c}\in H^{1,1}(S,\mathbb R). \] For example, express $r^*\alpha$ in a rational basis of $H^2(W,\mathbb Q)$ and apply $\Pi$ to each basis vector. This gives the required finite real combination $L_W$. Write $L_W=\sum_j a_jL_j$ with actual line bundles $L_j$ on $W$. For each $j$, let \[ \mathcal F_j=(r_*L_j)^{**}. \] It is a rank-one reflexive coherent sheaf. Strong $\mathbb Q$-factoriality gives an integer $m_j>0$ for which $M_j=\mathcal F_j^{[m_j]}$ is a line bundle on $T$. The canonical identification over the isomorphism locus of $r$ gives \[ L_j^{\otimes m_j}\simeq r^*M_j\otimes\mathcal O_W(E_j) \] for an integral $r$-exceptional divisor $E_j$. One may obtain this comparison directly from the evaluation map for $r_*L_j$: the two coherent rank-one sheaves agree away from the exceptional locus, so their invertible transforms differ by an exceptional divisor. No global meromorphic frame of $L_j$ or of the canonical sheaf is required. Set $L=\sum_j(a_j/m_j)M_j$ and $E=\sum_j(a_j/m_j)E_j$. The preceding identities yield \[ r^*(\alpha-c_1(L)-f^*\gamma)=[E]. \] The divisor $E$ is $r$-exceptional and is numerically trivial on every $r$-contracted curve. The exceptional negativity lemma applied in both signs gives $E=0$. Injectivity of pullback by a resolution, or pushforward of the equality of currents, now gives $\alpha=c_1(L)+f^*\gamma$. \end{proof} The hypothesis of Lemma~\ref{prog:relative-hodge} persists on each projective globally strongly \(\Q\)-factorial klt model over this fixed smooth base: on a common smooth resolution, holomorphic two-forms are invariant under modifications. Thus the lemma applies separately on every working model. It supplies real combinations of global lines; Lemma~\ref{prog:stein-representatives} supplies their Cartier-divisor representatives on sufficiently small Stein base neighborhoods. \begin{lemma}[The projective relative cone in Bott--Chern cohomology] \label{prog:projective-cone} Let \(f:T\to S\) be projective between normal compact Kähler spaces with rational singularities, and assume \[ H^{1,1}_{\BC}(T,\R) =c_1\bigl(\Pic(T)\otimes\R\bigr) +f^*H^{1,1}_{\BC}(S,\R). \] Let \(N_1^{\mathrm{line}}(T/S)\) denote the space of real combinations of \(f\)-contracted curves modulo degrees of global line bundles, and let \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) be their closed effective cone. Their analytic classes give an isomorphism onto the span of the face \[ \mathcal F_f= \overline{\mathrm{NA}}(T)\cap(f^*\omega_S)^\perp, \] where \(\omega_S\) is any Kähler class on \(S\); this isomorphism identifies \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) with \(\mathcal F_f\). In particular, relative extremal rays in the line space are extremal rays of the full analytic cone. \end{lemma} \begin{proof} The assumed decomposition makes analytic equivalence and line-degree equivalence identical on combinations of vertical curves. Their map into the analytic numerical space is therefore well-defined and injective. We must show that its closed effective cone is the entire face, including its classes represented by positive currents. Fix \(z\in\mathcal F_f\). For every \(\gamma\in H^{1,1}_{\BC}(S,\R)\), both \(a\omega_S+\gamma\) and \(a\omega_S-\gamma\) are Kähler for large \(a\). Positivity then shows that \(z\) annihilates \(f^*\gamma\). Fix a global relatively ample line \(H\). If a real line \(D\) has nonnegative degree on every vertical curve, the projective relative numerical criterion makes \(D+\epsilon H\) relatively ample for every \(\epsilon>0\). Its Chern class becomes Kähler after adding a sufficiently large multiple of \(f^*\omega_S\). Therefore \[ (c_1(D)+\epsilon c_1(H))\cdot z\geq0, \qquad c_1(D)\cdot z\geq0. \] Applying this to both signs of a relatively numerically trivial line shows that pairing with \(z\) factors through the finite-dimensional space of relative line degrees. It is nonnegative on its nef cone. By duality for this projective relative space, it is represented by an element of \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\). The decomposition in the statement makes its analytic image equal to \(z\). Conversely every vertical effective curve lies in \(\mathcal F_f\), and finite-dimensional injectivity preserves closedness. This proves the cone equality and the extremality claim. \end{proof} ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03a-program-foundations.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03b-projective-good-model.tex ==== \subsection{Good models for already projective relative adjoints} \begin{proposition}[Projective relative completion] \label{prog:projective-good-model} Let \(f:X\to S\) be an already projective surjective morphism of normal compact Kähler spaces, with \(X\) smooth. Let \((X,B+\mathbf M)\) be an effective generalized klt pair with adjoint \(A\), carried by a projective resolution \(p:W\to X\) with smooth \(W\), so that \[ p^*A=c_1(K_W)+[B_W]+\mathbf M_W. \] Suppose that there are an actual real line bundle \(J\in\Pic(X)\otimes\R\) and \(\beta\in H^{1,1}_{\BC}(S)\) such that \[ A=c_1(J)+f^*\beta, \qquad N:=p^*J-K_W-B_W \text{ is nef and big over }fp. \] Assume that \(J\) is pseudoeffective over \(S\). Then there is a finite chosen \(A\)-negative projective program over \(S\), with globally strongly \(\Q\)-factorial compact Kähler working spaces, and a nonextracting endpoint \(X_m\) such that \[ J_m\sim_{\R}g^*H, \qquad A_m=g^*\bigl(c_1(H)+\rho^*\beta\bigr). \] Here \(J_m\) is the actual real-line transform of \(J\), \(g:X_m\to Z\) is a projective connected-fibre morphism, \(\rho:Z\to S\) is projective, and \(H\) is an actual real line bundle ample over \(S\). Thus the class on \(Z\) in this formula is relatively Kähler over \(S\). The generalized pair on \(X_m\) is gklt. On a common resolution the comparison with \(A\) is effective and exceptional over \(X_m\), with strictly positive coefficient at the strict transform of every prime contracted from \(X\). \end{proposition} \begin{proof} We reduce to fixed ordinary pairs, and construct one global program. Choose finitely many Stein open sets \(U_\ell\subset S\) and semianalytic Stein compact subsets \(W_\ell\Subset U_\ell\) whose interiors cover \(S\). Such compacts are obtained by intersections with closed polydiscs in local embeddings of \(S\). In particular they satisfy condition~(P) of \cite{FujinoAnalyticBCHM}: the source is normal, the base is Stein, the compact is Stein, and its intersection with any analytic subset defined near it has finitely many connected components. This remains true on every normal projective model. Shrinking \(U_\ell\) around \(W_\ell\) preserves the finite cover. \smallskip\noindent \emph{Actual ordinary replacements.} On each chart represent the finitely many line bundles in the data by Cartier divisors. For a projective map to a Stein space this can be done by twisting a line bundle and its prospective meromorphic frame by sufficiently positive relative lines and using nonzero sections. The generic-rank-one direct-image argument is an alternative only when the map is bimeromorphic. Relative bigness gives \(N\sim_{\R}P+E\), with \(P\) relatively ample and \(E\geq0\). For sufficiently small \(\epsilon>0\), \((W,B_W+\epsilon E)\) is sub-klt near the compact inverse image. The line \((1-\epsilon)N+\epsilon P\) is relatively ample. General divided relative sections give an effective representative \(T\) such that \[ \Theta:=\epsilon E+T\sim_{\R}N, \qquad (W,B_W+\Theta)\text{ is sub-klt}. \] All supports and coefficient margins are fixed near that compact. With compatible canonical representatives, push the finite principal-divisor expression for \(\Theta-N\) down and pull it back again. For \(\Delta_\ell=p_*(B_W+\Theta)\) this gives \[ \Delta_\ell\geq0, \qquad K_X+\Delta_\ell\sim_{\R}J|_{X_{U_\ell}}, \qquad p^*(K_X+\Delta_\ell)=K_W+B_W+\Theta. \] In the last equality the chosen principal correction is included in the representative of \(K_X+\Delta_\ell\). Thus \((X_{U_\ell},\Delta_\ell)\) is ordinary klt and its boundary is relatively big. Lemma~11.15 of \cite{FujinoAnalyticBCHM}, with its big part equal to \(\Delta_\ell\) and its remaining part zero, replaces it, in its actual real linear equivalence class, by a klt boundary with an effective general relatively ample rational summand. Its condition on non-klt centres is vacuous. Choose the reserved ample summand to be \(\Q\)-linearly equivalent to a small rational multiple of a fixed global \(f\)-ample line. Its later strict transform is consequently \(\Q\)-Cartier by the transport of that global line, without a local factoriality claim. We make these choices for a finite rational polytope of actual global line data. Start with a finite rational span containing \(J\), and add relatively ample rational lines spanning the global relative degree space. The above construction is valid in a neighbourhood of \(J\): retain a small part of the ample summand and use openness of ampleness and the strict klt inequalities. It is also valid along a segment from \(J\) to a sufficiently positive adjoint, by adding divided general ample divisors. Use finitely many such representatives and one log resolution on each chart. More explicitly, represent a finite simplex inside the open ample neighbourhood used above; varying its positive coefficients represents all nearby global line parameters on the same finite support. Form the joint finite affine system consisting of the divisor identities on all charts, with common global line coordinates and separate boundary and principal-divisor coordinates. This system is rational. Allow all boundary coefficients to vary, including those originally contributed by the real boundary \(B\); fix only the reserved rational ample part and the zero-support constraints. Its projection to the global line coordinates contains the just constructed neighbourhood, so it has a rational affine section. Choose that section sufficiently close to the constructed real affine lift, by rational approximation in its affine space of sections, so that the strict klt and effectiveness margins are retained along the compact segment. In the joint solution space the strict klt and effectiveness margins permit a rational polytope containing the scaling segment. This supplies on every chart a rational polytope of ordinary klt boundaries with a fixed effective relatively ample rational summand, representing the restrictions of the same global line parameters. These boundaries, including the scaling representatives, are fixed before the program starts. Their divisors need not agree on different charts. \smallskip\noindent \emph{One global scaling construction.} Apply the global degree-space construction in the proof of Proposition~\ref{prog:relative-klt} to \(J\) and a general ample scaling direction in the chosen span. Here are the modifications that allow real line data and a nonbirational map to \(S\). The global relatively ample line still gives a compact normalized slice of the curve cone. The fixed ordinary chart pairs give finite negative-ray truncations. A general direction avoids equal wall parameters for independent ray-degree functionals, exactly as in that proof. A positive wall therefore selects a single ray in the global degree space. A rational nef support is represented by a global rational line; a large multiple minus the chartwise ordinary adjoint is relatively ample. Ordinary relative base point freeness contracts the ray using the evaluation of this one global line. If the contraction is small, choose a global rational line having negative ray degree and construct its flip by its one global graded algebra. The local ordinary proof of Proposition~\ref{prog:detected-flip} applies to this algebra: all curves of the contraction have proportional degrees for global lines, so the detector and the driving line are positively proportional over the contraction; ordinary rational replacement gives local finite generation. This part of that proof needs only the already projective contraction and these line degrees. It does not require a prior assertion about the full Bott--Chern cone. The relative Proj is the next global model. In a divisorial step the exceptional prime and Lemma~\ref{prog:integral-descent} give the same continuation as in Proposition~\ref{prog:relative-klt}. Killing the degree of any global line by a rational multiple of the detector, then applying that lemma, preserves global strong \(\Q\)-factoriality in both cases. The global lines continue to span the relative degree spaces. The rational support-simplex argument in that proposition makes the wall a descended relatively ample real line on its contraction base and gives a nonempty interval of ampleness immediately after each step. This argument uses relative line data and projectivity, not birationality of the map to \(S\). We check explicitly the persistence of the ordinary pairs. Expand the line parameters in finitely many global rational lines. Their reflexive transforms are actual rational lines by the preceding strong property. Pushing forward the fixed principal-divisor identities gives \[ K_{X_i}+\Delta_{\ell,i}\sim_{\R}J_i|_{(X_i)_{U_\ell}}. \] Hence these ordinary adjoints are \(\R\)-Cartier; no local factoriality of individual prime divisors is being assumed. On a common resolution their comparison minus the appropriate positive multiple of the detector comparison is exceptional and relatively numerically trivial. Negativity in both signs makes this difference zero. Every chosen step is consequently negative for the fixed ordinary pairs. They remain effective and klt, and their boundaries remain relatively big. The same equality of comparison divisors, after adding the fixed base pullback, preserves the generalized pair and its discrepancy inequalities. For termination, every already constructed working model is a weak log canonical model on each chart of a boundary in the fixed polytope: choose a parameter in its nonempty ample scaling interval. Indeed, actual descent at a preceding wall \(\lambda_j\) makes the comparison for \(J+tH\) equal to \((1-t/\lambda_j)\) times its \(J\)-comparison. For a parameter in the current interval all preceding \(\lambda_j\) are at least \(t\), so these comparisons are effective. Theorem~E of \cite{FujinoAnalyticBCHM} gives finitely many marked models on each chart. That theorem does not require locally \(\Q\)-factorial targets. There are therefore finitely many possible global marked working models. Indeed, isomorphisms between the restrictions of two existing marked models agree on their common dense open, and hence on overlaps. A repeated marked working model would have the same boundary transform and discrepancies, contrary to strict discrepancy increase at an intervening nontrivial step. The global program is finite. Relative pseudoeffectivity excludes a Mori fibre endpoint, so \(J_m\) is nef over \(S\). This is a finite-cover argument for a single constructed program; it does not glue independent minimal models. \smallskip\noindent \emph{Semiampleness and the global polarization.} On the endpoint the fixed ordinary pair is klt, its effective boundary is relatively big, and its adjoint is nef over \(W_\ell\). Lemma~11.16 of \cite{FujinoAnalyticBCHM} applies to this already existing weak model. More explicitly, Lemma~11.15 replaces the endpoint boundary by \(A_\ell+B_\ell\), where \(A_\ell\geq0\) is relatively ample, \(B_\ell\geq0\), and the pair is klt. Theorem~8.3 then gives semiampleness near \(W_\ell\): its log canonical, klt-existence, real Cartier, ample-summand and nef-over-compact hypotheses have all been checked. The actual equivalences identify these local semiample adjoints with restrictions of \(J_m\). Their ample models agree on overlaps by \cite[Lemma~11.3]{FujinoAnalyticBCHM}. To retain a global polarization, shrink the initial parameter polytope around \(J\) after the finite program: all its finitely many strict step signs persist, so the transformed ordinary boundaries remain klt and big. On each chart the pushforward of its initial common ample rational summand is effective and relatively big. Apply \cite[Lemma~11.13]{FujinoAnalyticBCHM} to this common big rational part and the small endpoint boundary polytope. All pairs are klt, so the condition on non-klt centres is vacuous. The lemma gives a common general relatively ample rational summand by one \(\Q\)-linearly trivial translation. In particular the rational parametrization is preserved; separate applications of Lemma~11.15 to individual parameters would not suffice here. In this finite rational span Theorem~11.17 of \cite{FujinoAnalyticBCHM} makes the endpoint nef region a rational polytope. Intersect the finitely many inverse images of these regions and express \(J_m\) as a positive combination of rational points in its minimal face. The corresponding global rational lines are semiample on every chart by the same ordinary base point free argument. After clearing denominators and taking common sufficiently divisible powers over the finite cover, each is relatively generated globally: generation on the charts is generation of its global coherent relative evaluation map. The product of these finitely many relative morphisms, followed by Stein factorization, gives a projective \(g:X_m\to Z\), a projective \(\rho:Z\to S\), and descended global lines whose positive combination \(H\) is relatively ample. It realizes the glued ample model and gives \(J_m\sim_{\R}g^*H\). Adding back \(f_m^*\beta\) proves the asserted Bott--Chern identity. Composing the effective step comparisons proves the final discrepancy statement. \end{proof} \begin{corollary}[Semiampleness on an existing projective model] \label{prog:projective-semiample} Let \(f:T\to S\) be a projective morphism of normal compact Kähler spaces with \(T\) globally strongly \(\Q\)-factorial, and let \(J\in\Pic(T)\otimes\R\) be nef over \(S\). Suppose that on a finite Stein-compact cover satisfying condition~(P), there are effective ordinary klt boundaries \(\Delta_\ell\), big over the base, such that \(K_T+\Delta_\ell\sim_{\R}J|_{T_{U_\ell}}\). Then \(J\) has the projective relative semiample contraction and actual descended ample real line of Proposition~\ref{prog:projective-good-model}. In particular this applies to any already existing weak model of the fixed ordinary chart pairs in that proof; one need not run a new program for the adjoint in question. \end{corollary} \begin{proof} Lemma~11.15 and Theorem~8.3 of \cite{FujinoAnalyticBCHM} give local semiampleness. For an existing weak model this is also Lemma~11.16. To globalize the polarization, take a finite rational span of the global lines defining \(J\). On each chart the ample part of the normalized boundary allows small perturbations in this span. The joint rational affine construction in the preceding proof gives a rational family with a common rational ample part; all boundary coefficients are allowed to vary. Theorem~11.17 then gives a rational nef region on each chart. The finite-intersection, rational-vertex and global-evaluation argument in that proof gives one projective contraction and its actual descended ample real line. Only its local ample models are identified by uniqueness. \end{proof} \begin{remark}[The two uses of projective completion] \label{prog:projective-good-model-uses} For a selective extraction over an already constructed weak model, the carrier map is projective bimeromorphic. The inherited exceptional splitting expresses the nef datum, modulo this base, by actual real lines. Relative nefness follows from the carrier datum, while relative bigness follows from birationality, using the ample-plus-effective construction in the ordinary replacement. The prepared adjoint is a base pullback plus an effective exceptional divisor supported on exactly the unwanted primes. Proposition~\ref{prog:projective-good-model} applies. On its endpoint the remaining error is effective, exceptional over the fixed base, and relatively nef; negativity makes it zero. Strict discrepancy improvement prevents contraction of a requested zero-error prime. The chosen extraction boundary must be effective and klt; in particular its prescribed primes have the usual admissible discrepancy range. For the graph construction, suppose the already projective graph resolution \(h:V\to Y_m\) has smooth source and \[ A_V=h^*A_{Y_m}+[G],\qquad G\geq0. \] Then the global relative real line is \(J=\OO_V(G)\), the base class is \(A_{Y_m}\), and its carrier line is \(N=G-K_V-B_V\). Its relative degrees equal those of the prepared nef datum. That datum is nef and big on the prepared carrier, for example a pullback of its prepared Kähler class; thus \(N\) is nef and big over \(Y_m\). This bigness is a separate hypothesis check, since \(h\) can have positive-dimensional general fibres. Effectivity of \(G\) gives relative pseudoeffectivity. Apply the proposition over \(Y_m\). The subsequent negativity argument must still remove \(G_m\); only then is the endpoint adjoint the pullback of \(A_{Y_m}\) and good over the original base. For transport of all classes, verify separately that all Bott--Chern classes on the graph carrier are real-line classes modulo \(Y_m\), using the smooth rationally connected fibration and the inherited exceptional splitting on the lower-dimensional base model. Lemma~\ref{prog:projective-cone} then identifies the relative degree rays with full Bott--Chern rays, so the detected-step transport applies. The driving identity involving \(G\) alone does not give this additional property. \end{remark} \begin{corollary}[Ordinary projective Mori output] \label{prog:projective-mori} Let \(f:X\to S\) be an already projective surjective morphism of compact Kähler spaces, with \(X\) smooth. Let \((X,B)\) be an effective ordinary real klt pair, and let \(D=K_X+B\). If \(D\) is not pseudoeffective over \(S\), then its projective program over \(S\), with scaling of a relatively ample actual real line \(N\), terminates with a Mori fibre contraction. The initial scaling parameter is chosen so that \(D+TN\) is nef over \(S\). The working spaces are globally strongly \(\Q\)-factorial and compact Kähler, and the actual line comparisons and integral descent of Proposition~\ref{prog:projective-good-model} apply. \end{corollary} \begin{proof} The line \(N\) is fixed; it need not be a general scaling direction. In particular several extremal rays may have the same wall parameter. Choose \(\epsilon>0\) such that \(D+2\epsilon N\) is still not pseudoeffective over \(S\). On the finite Stein-compact cover choose divided general effective representatives of \(N\) so that all the resulting ordinary pairs representing \(D+tN\), \(0\leq t\leq T\), are klt. For \(\epsilon\leq t\leq T\) reserve a common effective relatively ample rational summand. The joint rational-family construction in Proposition~\ref{prog:projective-good-model} puts these fixed pairs in the polytopes required by Theorem~E of \cite{FujinoAnalyticBCHM}. Suppose a finite prefix has reached \(X_i\), and put \(\mu_i=\lambda_{i-1}\), with \(\mu_0=T\). Its transformed \(D_i+\mu_iN_i\) is nef. Define \[ \lambda_i=\min\{t\in[0,\mu_i]:D_i+tN_i \text{ is nef over }S\}. \] The feasible set is a nonempty closed interval. Inductively \(\lambda_i>2\epsilon\): all previous steps have nonpositive comparison for \(D+2\epsilon N\), so its transform is still not pseudoeffective. A first threshold at most \(2\epsilon\) would make this transform nef by convexity between the current and preceding thresholds, a contradiction. There is a \(D_i\)-negative extremal ray \(R_i\) with \((D_i+\lambda_iN_i)\cdot R_i=0\). To see attainment without a generality assumption, let \(t<\lambda_i\) tend to \(\lambda_i\). A ray negative for \(D_i+tN_i\) has positive \(N_i\)-degree, because \(D_i+\mu_iN_i\) is nef. When \(t>\lambda_i/2\), it is also negative for \(D_i+(\lambda_i/2)N_i\). The latter has fixed ordinary klt big-boundary representatives on the charts, since \(\lambda_i/2>\epsilon\). Normalize each of these boundaries by \cite[Lemma~11.15]{FujinoAnalyticBCHM} as \(A_\ell+B'_\ell\), with \(A_\ell\geq0\) relatively ample and the new pair klt. The \(A_\ell\)-truncation of the cone theorem for \(K+B'_\ell\) gives finitely many rays negative for the entire adjoint in question. The finite cover therefore gives finitely many global negative rays. One of these rays attains the wall, as required. Choose one such ray, even if the wall vanishes on other rays. The rational-support argument for an individual negative ray in Proposition~\ref{prog:relative-klt} gives a global rational nef support annihilating exactly \(R_i\), and ordinary relative base point freeness contracts it. For the real ordinary boundary one may first choose a nearby rational klt boundary that is still negative on \(R_i\). The contraction therefore has a rational ordinary antiample adjoint. Its flip, when small, is constructed from one global detector algebra as above. If the contraction is of fibre type, stop. Otherwise the wall \(w_i=D_i+\lambda_iN_i\) has degree zero on \(R_i\). In the finite rational span of its actual line factors, this zero-degree subspace is rational: its defining functional has rational values on rational lines. Thus \(w_i\) is a real combination of rational lines of zero ray degree. Lemma~\ref{prog:integral-descent} descends those lines to the contraction base. Their combination is nef over \(S\), by lifting curves through the projective contraction. Pulling it to the positive side shows that the transformed wall stays nef. The next threshold therefore satisfies \(\lambda_{i+1}\leq\lambda_i\); equality is permitted. No interval of ampleness between successive thresholds is required. If \(E_i\geq0\) is the ordinary \(D_i\)-comparison on a common resolution, this actual wall descent gives, for every real \(t\), \[ p_i^*(D_i+tN_i) \sim_{\R}q_i^*(D_{i+1}+tN_{i+1}) +(1-t/\lambda_i)E_i. \] For \(t\leq\lambda_i\) the error is effective. This proves all the nonpositivity and fixed-pair persistence assertions used in the induction, including the lower bound on thresholds. At every working model use the \emph{wall parameter} \(t=\lambda_i\), rather than an interior ample parameter. Since \(\lambda_i\leq\lambda_j\) for every earlier step, the displayed comparisons show that \(X_i\) is a weak log canonical model of the initial fixed chart pairs representing \(D+\lambda_iN\). Its trace is nef by definition. Theorem~E counts all these marked weak log canonical models, including the ones at repeated wall parameters. The finite chart cover gives a finite global marked list. Repetition would contradict the strict ordinary \(D\)-discrepancy increase at an intervening step. Thus this program with the specified \(N\) is finite. Non-pseudoeffectivity excludes a nef endpoint for \(D\), so the last contraction is a Mori fibre contraction. The comparison at \(t=1=\lambda_i\) is crepant: such a step supplies no strict discrepancy improvement for \(D+N\), while it is still strictly negative for \(D\). \end{proof} ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03b-projective-good-model.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-base-presentations.tex ==== \subsubsection{Base presentations and multiplier ideals} The following constructions take place in the category of generalized data specified above. In particular, the whole adjoint is a Bott--Chern class, whereas the boundary and nef traces on a singular space are interpreted as currents. On a smooth carrier the structure boundary is a real divisor; its negative components may be exceptional over the space of the pair. The nef part is specified on that carrier and pulled back to higher models. \begin{lemma}[Adjoint-preserving preparation] \label{prog:base-presentation-preparation} Let $(X,B+\mathbf M)$ be a generalized pair on a normal compact K\"ahler space, with globally modified-big total boundary $B+\mathbf M_X$. Write \[ A=[K_X+B+\mathbf M_X],\qquad \mathcal J_o=\mathcal J(X,B+\mathbf M). \] There are generalized data on $X$ with the same adjoint class, a projective carrier $p:U\to X$ with $U$ smooth and compact K\"ahler, an SNC real divisor $C$ whose negative components are $p$-exceptional, and a K\"ahler class $\beta$ such that \begin{equation}\label{prog:base-presentation-prepared} p^*A=[K_U+C]+\beta,\qquad \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor) \supseteq\mathcal J_o. \end{equation} The carrier may be required to dominate any prescribed projective modification of $X$, with the structure and exceptional divisors having simultaneous SNC support. The prepared total boundary is globally modified big. In particular, the preparation preserves the gklt condition. \end{lemma} \begin{proof} Choose a common projective log resolution $p_0:U_0\to X$ of the original nef carrier and the prescribed modification, and write \[ p_0^*A=[K_{U_0}+C^0]+\mathbf M_{U_0}. \] The class $\mathbf M_{U_0}$ is nef, and the negative part $(C^0)^-$ is exceptional over $X$. The modified-big preparation \cite[Lemmas~2.10 and~2.23]{HX} can be made with the prescribed resolution. We give its divisor and ideal details. Choose an effective divisor $E_0$ with sufficiently large coefficients and full $p_0$-exceptional support so that the class \[ [(C^0)^+]+\mathbf M_{U_0}+[E_0] \] is big. Resolve a Kodaira decomposition of this class. On a further projective modification $a:U\to U_0$ with $U$ smooth one obtains \[ a^*\bigl([(C^0)^+]+\mathbf M_{U_0}+[E_0]\bigr) =\kappa+[D],\qquad D\geq0, \] where $\kappa$ is K\"ahler. The support can be made SNC together with all the other divisors. On any additional blowup, subtract a sufficiently small effective exceptional class from the pulled-back K\"ahler class and add it to the divisor; an exceptional divisor whose negative is relatively ample makes the resulting class K\"ahler again. Put $p=p_0a$ and \[ C^{\mathrm{alt}}=D-a^*(C^0)^--a^*E_0-K_{U/U_0}. \] Then \[ p^*A=[K_U+C^{\mathrm{alt}}]+\kappa, \] and every negative component of $C^{\mathrm{alt}}$ is $p$-exceptional. Let $C^{\mathrm{old}}$ be the crepant structure boundary of the original data on this same model. For $00),\qquad f_*\mathcal O_{N_1}=\mathcal O_Z/(f_*\mathcal J_1), \qquad N_1=V(\mathcal J_1). \end{equation} If $N_1$ dominates $Z$, then $f_*\mathcal J_1=0$. Otherwise there are generalized data $(Z,B_Z+\mathbf N)$, a projective carrier $\tau:T\to Z$ with $T$ smooth and compact K\"ahler, a real SNC structure divisor $C_T$, and a K\"ahler class $\beta_T$ such that \begin{equation}\label{prog:base-presentation-identities} \begin{gathered} [K_Z+B_Z+\mathbf N_Z]=\gamma,\qquad \tau^*\gamma=[K_T+C_T]+\beta_T,\qquad \mathbf N_T=\beta_T,\\ B_Z=\tau_*C_T\geq0,\qquad B_Z+\mathbf N_Z\ \text{is globally modified big}. \end{gathered} \end{equation} The negative components of $C_T$ are $\tau$-exceptional, and \begin{equation}\label{prog:base-presentation-ideal} \mathcal J_Z:=\mathcal J(Z,B_Z+\mathbf N) =\tau_*\mathcal O_T(-\lfloor C_T\rfloor) \supseteq f_*\mathcal J_1. \end{equation} In particular the base data are gklt wherever $f_*\mathcal J_1$ is the unit ideal. If the source data are gklt, the base data are gklt. All adjoint identities are preserved by pullback to higher models. \end{proposition} \begin{proof} Choose a projective resolution $\mu:W\to Z$, with $W$ smooth compact K\"ahler and reduced exceptional divisor $F$ having SNC support. Apply Lemma~\ref{prog:base-presentation-preparation} on a carrier dominating the main component of the base change. We obtain projective maps $p:U\to X$ and $q:U\to W$ with $fp=\mu q$ and \begin{equation}\label{prog:base-presentation-source} [K_U+C]+\beta=p^*A=q^*\mu^*\gamma, \qquad \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor). \end{equation} Here $U$ is smooth compact K\"ahler, $\beta$ is K\"ahler, and all negative components of $C$ are $p$-exceptional. The maps are projective by composition and factorization over the separated base $W$. The map $q$ has connected fibres: this holds over the open set where $\mu$ is an isomorphism, and its Stein factorization is finite and bimeromorphic over the normal space $W$. The integral divisor $-\lfloor C\rfloor$ has class \[ [-\lfloor C\rfloor] =[K_U+\{C\}]+\beta-(fp)^*\gamma. \] The pair $(U,\{C\})$ is klt with SNC boundary. Relative Kawamata--Viehweg vanishing, first for $p$ and then for $fp$, applies because the positive class modulo pullbacks is K\"ahler \cite[Theorem~2.2]{HX}. Equivalently, the real line bundle $-K_U-C$ is relatively ample, so the vanishing follows from \cite[Theorem~1.2]{FujinoVanishing2025} with the identity as its first morphism. Leray gives $R^if_*\mathcal J_1=0$ for $i>0$. Applying $f_*$ to the sequence defining $N_1$, and using $f_*\mathcal O_X=\mathcal O_Z$, proves \eqref{prog:base-presentation-Stein}. If $N_1$ dominates $Z$, the ideal $f_*\mathcal J_1\subseteq\mathcal O_Z$ vanishes generically and therefore vanishes. This proves that alternative. Suppose from now on that $N_1$ does not dominate $Z$. Every $q$-horizontal coefficient of $C$ is then less than one. Indeed, a component with coefficient at least one forces the multiplier ideal to be proper along its image in $X$, and its image would make $N_1$ dominate $Z$. If $Z$ is a point, the asserted base presentation is the dimension-zero presentation. We therefore assume $b=\dim Z>0$. \paragraph{The fibre metric.} Choose a smooth representative $\rho$ of $\mu^*\gamma-c_1(K_W)$, a K\"ahler form $\omega_W$, and a K\"ahler form representing $\beta$, still denoted by $\beta$. For some $a>0$, \[ \beta\geq a q^*\omega_W. \] Consider the holomorphic line bundle \[ L=\mathcal O_U(-K_{U/W}-\lfloor C\rfloor). \] Equation~\eqref{prog:base-presentation-source} gives a singular Hermitian metric $h$ on $L$ whose curvature current is \begin{equation}\label{prog:base-presentation-curvature} \Theta_h(L)=\beta+[\{C\}]-q^*\rho. \end{equation} Its weights are smooth functions plus the divisorial logarithms with coefficients in $\{C\}$. We normalize $dd^c$ so that $dd^c\log|z|^2=[z=0]$ and write squared-norm weights as $e^{-\varphi}$. Let $e$ be the canonical meromorphic section of \[ K_{U/W}\otimes L=\mathcal O_U(-\lfloor C\rfloor), \] so that $\operatorname{div}(e)=-\lfloor C\rfloor$. Choose an analytic Zariski open set $W^\circ\subseteq W$ on which $q$ is a submersion, the horizontal boundary is relatively SNC, there are no vertical boundary components, $\mu$ is an isomorphism onto an open subset of $Z$, and coherent base change holds in degree zero. On $q^{-1}(W^\circ)$, the divisor $-\lfloor C\rfloor$ is effective and exceptional over $X$. Normality and the contraction property therefore identify \[ q_*(K_{U/W}\otimes L)|_{W^\circ} =\mathcal O_{W^\circ}e. \] Here $K_{U/W}$ is viewed as the relative canonical line bundle. The fibre norm \[ I(w)=\int_{U_w}|e_w|_h^2 \] is positive and finite. The fractional SNC weights are integrable; any zeros of $e_w$ only improve integrability. Relative SNC coordinates and a partition of unity show that $I$ is smooth: parameter derivatives act on smooth factors multiplying fixed integrable monomial weights. Set $\psi=-\log I$ on $W^\circ$. Then \begin{equation}\label{prog:base-presentation-open-positivity} \rho+dd^c\psi\geq a\omega_W. \end{equation} Here are the details for the singular weights. Locally choose $\ell$ with $dd^c\ell=\rho-a\omega_W$, and put $h'=he^{-q^*\ell}$. Its weights are psh, and its fibre norm satisfies $I'=e^{-\ell}I$. Restrict the family to a small coordinate line disk $\Delta_R$, with centre $0$. The sharp $L^2$ extension theorem gives a twisted top form extending $e_0$ and satisfying \begin{equation}\label{prog:base-presentation-sharp-extension} \int_{q^{-1}(\Delta_R)}|E|_{h'}^2 \leq\pi R^2 I'(0). \end{equation} Top-form norms here are normalized to use Euclidean area in the base coordinate. We use the Stein hypersurface form of the optimal extension theorem \cite[Theorem~2.2 and \S3.5, (3.1)]{GuanZhouOptimal2015}, with defining function the disk coordinate divided by $R$. To check applicability, embed the restricted projective family into projective space over the disk and remove a relative hyperplane section that does not contain the central fibre. The complement is Stein. A holomorphic section of $L$ on this Stein space can be chosen nonzero at a point of the central fibre; deleting its zero divisor trivializes $L$ and leaves a Stein space. The central fibre is smooth and connected, hence irreducible, so both deletions leave a dense open subset of it. For a zero-dimensional fibre choose the deletions to avoid its point. In the resulting trivialization the weight is psh, and the sharp extension theorem applies to canonical forms. The extension extends across the deleted hypersurfaces in reverse order by local $L^2$ removability: in every original frame the metric is bounded below by a smooth positive metric. The extension retains its norm bound and its prescribed restriction to the central fibre. After dividing $E$ by the base differential, the rank-one direct image identifies it with $b(t)e$, where $b$ is holomorphic and $b(0)=1$. Thus the disk average of $|b(t)|^2I'(t)$ is at most $I'(0)$. Jensen's inequality and the submean inequality for $\log|b|^2$ give \[ \log I'(0) \geq\operatorname{avg}_{\Delta_R} \bigl(\log|b(t)|^2+\log I'(t)\bigr) \geq\operatorname{avg}_{\Delta_R}\log I'(t). \] The same argument applies to every sufficiently small disk at every centre and in every complex line direction. Hence $-\log I'=\psi+\ell$ is psh, which proves \eqref{prog:base-presentation-open-positivity}. \paragraph{Extension over the base.} Let $D$ be a prime divisor in $W\setminus W^\circ$. Choose a component $Q$ of $q^*D$ dominating $D$. When $D$ is not $\mu$-exceptional, choose $Q$ not $p$-exceptional: take a component over the corresponding prime on $Z$ already on $X$ and then its strict transform. Put \[ u=\operatorname{ord}_Q(q^*D),\qquad c=\operatorname{coeff}_Q(C). \] At a general point of $Q$, avoiding all other relevant divisors, there are local coordinates with \[ t_1=z_1^u,\qquad t_j=z_j\quad(2\leq j\leq b). \] Indeed the map $Q\to D$ is submersive at a general smooth point, and a local unit in the first equation is absorbed by a holomorphic $u$th root. Such points can be chosen above every point of a dense analytic open subset of $D$ by generic smoothness and properness. Integration over a fixed small polydisk in the remaining fibre coordinates gives \begin{equation}\label{prog:base-presentation-divisor-bound} I(t)\geq c_*|t_1|^{-2(c+u-1)/u} \end{equation} for some $c_*>0$ near each such base point. The section and metric contribute the exponent $c$, while division by the base differential in the relative canonical form contributes $u-1$. For a nonexceptional $D$ the chosen $Q$ has $c\geq0$, so \eqref{prog:base-presentation-divisor-bound} bounds $\psi$ from above near its general points. For the finitely many exceptional divisors, the same estimate supplies a nonnegative integer $m$ such that, if $k$ is a smooth metric on $\mathcal O_W(F)$ and $s_F$ its divisor section, \[ \varphi=\psi+m\log|s_F|_k^2 \] is locally bounded above near general points of every divisor. Adding local potentials of $\rho+m c_1(k)-a\omega_W$ to $\varphi$ gives psh weights by \eqref{prog:base-presentation-open-positivity}. These weights extend across the divisors and then across the remaining analytic set of codimension at least two, giving \begin{equation}\label{prog:base-presentation-positive-current} S=\rho+m c_1(k)+dd^c\varphi\geq a\omega_W \end{equation} on all of $W$. \paragraph{A K\"ahler carrier.} Regularize the current in \eqref{prog:base-presentation-positive-current} in its class, retaining a lower bound for the potentials. More precisely, choose an approximation with analytic singularities such that \begin{equation}\label{prog:base-presentation-regularization} S'=\rho+m c_1(k)+dd^c\varphi',\qquad S'\geq\frac a2\omega_W,\qquad \varphi'\geq\varphi-O(1). \end{equation} These simultaneous properties are the Bergman regularization properties of \cite{DemaillyRegularization1992}; see also \cite[pp.~59--60]{PopoviciRegularization2004}. Resolve the analytic singularities and the support of $F$ by a sequence of blowups with smooth centres $r:T\to W$, with simultaneous SNC support, and write \[ r^*S'=[H]+\zeta,\qquad H\geq0. \] The residual current $\zeta$ has locally bounded potentials. Removing the divisorial part from the positive current $r^*S'-(a/2)r^*\omega_W$ shows that \[ \zeta\geq\frac a2r^*\omega_W. \] The difference has vanishing Lelong numbers, so its class is nef by regularization. Choose an effective $r$-exceptional divisor $G$ with $-G$ relatively ample. For $0<\eta\ll1$, the class $(a/2)r^*\omega_W-\eta[G]$ is K\"ahler. Therefore \begin{equation}\label{prog:base-presentation-carrier-data} \begin{split} \tau&=\mu r,\qquad \beta_T=[\zeta]-\eta[G],\\ C_T&=H+\eta G-mr^*F-K_{T/W} \end{split} \end{equation} has K\"ahler $\beta_T$. If $r$ is an isomorphism, omit $G$ and its correction. The choice of $\eta$ will be decreased once more below. The exact class calculation is \[ \begin{split} [K_T+C_T]+\beta_T &=r^*c_1(K_W)+[H]+[\zeta]-m[r^*F]\\ &=r^*(c_1(K_W)+[\rho])=\tau^*\gamma. \end{split} \] All possibly negative terms in $C_T$ are $\tau$-exceptional. Consequently $B_Z=\tau_*C_T$ is effective. Specify the nef b-data by $\mathbf N_T=\beta_T$; the displayed identity defines generalized data on $Z$ with adjoint class $\gamma$. At the level of currents, pushing the identity gives the required trace identity on $Z$: exceptional divisor currents push to zero, and a $dd^c$ difference pushes to a $dd^c$ difference. Canonical traces can be computed using local meromorphic canonical frames; a change of frame changes the corresponding Chern potentials by the usual logarithmic transition term. Thus only the whole adjoint, rather than its individual traces, needs to be a Bott--Chern class. The class $\beta_T$ plus the strict transform of $B_Z$ is big on $T$, so its pushforward $B_Z+\mathbf N_Z$ is globally modified big. These are~\eqref{prog:base-presentation-identities}. \paragraph{The multiplier ideal.} Take a local holomorphic function $v\in f_*\mathcal J_1$. By the pushforward description of $\mathcal J_1$, \begin{equation}\label{prog:base-presentation-source-orders} \operatorname{div}(q^*\mu^*v)\geq\lfloor C\rfloor. \end{equation} Thus $q^*\mu^*v\,e$, tensored with a local base canonical frame, is a holomorphic $L$-valued top form on $U$. Its squared norm is locally integrable by the fractional SNC weights in \eqref{prog:base-presentation-curvature}. Over the submersion locus, fibre integration followed by base integration is precisely this top-form integral. Properness and a finite coordinate covering therefore give \begin{equation}\label{prog:base-presentation-base-integral} \int |\mu^*v|^2e^{-\psi}<\infty \end{equation} locally on $W$; the integral is taken where the original fibre metric is defined. The lower bound on $\varphi'$ in \eqref{prog:base-presentation-regularization} preserves this integrability after replacing $\psi$ by $\varphi'-m\log|s_F|_k^2$. Change variables through $r$. Along a prime divisor of $T$, let $h,j,k_D,d$ be respectively the coefficients of $H$, $\operatorname{div}(\tau^*v)$, $K_{T/W}$, and $r^*F$. The local integrand has exponent $2(j-h+md+k_D)$, up to a bounded positive factor. Its integrability implies $j-h+md+k_D>-1$, or, since $j$ is integral, \[ j\geq\lfloor h-md-k_D\rfloor. \] Therefore \[ \operatorname{div}(\tau^*v) \geq\lfloor H-mr^*F-K_{T/W}\rfloor. \] Choose $\eta>0$ sufficiently small that adding $\eta G$ does not change any of these finitely many round-downs. The resulting inequality is exactly $\operatorname{div}(\tau^*v)\geq\lfloor C_T\rfloor$, proving \eqref{prog:base-presentation-ideal} as an inclusion of ideal sheaves. In particular, if $f_*\mathcal J_1$ is the unit ideal on an open set, then so is $\mathcal J_Z$, and the base data are gklt there. Finally, the nef data descend to $T$, and the structure boundaries on higher models are defined by crepant pullback of \eqref{prog:base-presentation-identities}. On every common refinement of the source and base carriers, both adjoints are the pullback of $\gamma$. Thus the identities remain exact on higher models. \end{proof} \begin{lemma}[Contractions of closed subspaces] \label{prog:base-emc} Let $D$ be a compact complex space, possibly nonreduced, and let $\delta\in H^2(D,\R)$ have nonnegative degree on compact curves. Use the definitions of a Moishezon morphism and of a contraction defined by a class in \cite[Definitions~2.32 and~2.34]{HX}. Then the following hold. \begin{enumerate} \item If $\delta$ is endowed with a Moishezon contraction, so is its restriction to every closed analytic subspace of $D$. \item Let $g:D\to V$ be a projective contraction of compact complex spaces, so $g_*\OO_D=\OO_V$, and suppose $\delta=g^*\epsilon$ for a class $\epsilon\in H^2(V,\R)$ having nonnegative degree on compact curves. Then $\delta$ is endowed with a Moishezon contraction if and only if $\epsilon$ is endowed with one. \end{enumerate} \end{lemma} \begin{proof} Write $h:D\to S$ for a Moishezon contraction defined by $\delta$. For a closed subspace $D_0\subset D$, take the Stein factorization \[ D_0\longrightarrow S_0\longrightarrow S \] of its restriction. The first map is a contraction, and the second is finite. Thus a curve in $D_0$ is contracted by the first map exactly when its $\delta$-degree is zero. Choose a projective surjection $P\to D$ witnessing that $h$ is Moishezon, with $P\to S$ projective. The closed subspace $P_0=P\times_D D_0$ is projective over $S$ and projective and surjective over $D_0$. It is also projective over $S_0$: a relatively ample line for $P_0\to S$ remains ample on the fibres of $P_0\to S_0$. This proves the first assertion, including its scheme structure. For the descent direction of the second assertion, every curve in a fibre of $g$ has $\delta$-degree zero. A connected projective complex space is connected by chains of compact curves after passing to its reduction. Therefore $h$ is constant on the underlying set of each fibre of $g$. It factors as $h=k\circ g$ for a holomorphic map $k:V\to S$. To see that this factorization holds also for nonreduced spaces, choose a coordinate neighbourhood in $S$ of the image of a fibre. Properness permits restriction to a neighbourhood of its point in $V$ whose inverse image maps into that coordinate neighbourhood. The coordinate functions of $h$ then descend uniquely through $g_*\OO_D=\OO_V$; their analytic relations descend through the same equality. These local factorizations glue. Furthermore, \[ k_*\OO_V=h_*\OO_D=\OO_S. \] Every irreducible compact curve $C\subset V$ is dominated by a compact curve $C'\subset D$. Indeed, after base change to the normalization of $C$, take a component dominating that curve and a projective multisection. The projection formula gives \[ \delta\cdot C' =\deg(C'\to C)\,\epsilon\cdot C. \] Consequently $k$ contracts exactly the curves of $\epsilon$-degree zero. The composite $P\to D\to V$ is a projective surjection, and $P\to S$ is projective, so it witnesses that $k$ is Moishezon. Conversely, suppose $k:V\to S$ is a Moishezon contraction defined by $\epsilon$. Then $k\circ g$ is a contraction, and the projection formula verifies its curve criterion, treating separately curves contracted by $g$. If $Q\to V$ is a projective surjection witnessing that $k$ is Moishezon, then $D\times_V Q\to D$ is projective and surjective, while $D\times_V Q\to Q\to S$ is projective. This supplies the required witness. \end{proof} \begin{lemma}[Crepant transport of the multiplier closed subspace] \label{prog:multiplier-transport} Let $g:X\to Y$ be a projective bimeromorphic morphism of normal compact K\"ahler spaces. Suppose effective generalized pair data on $X$ and $Y$ have the same globally nef b-part and are crepant over $Y$. In particular their adjoint classes $A_X$ and $A_Y$ satisfy $A_X=g^*A_Y$ in Bott--Chern cohomology. Write $J_X,J_Y$ for their multiplier ideals and $N_X=V(J_X)$, $N_Y=V(J_Y)$ for their non-klt closed subspaces. Then \[ g_*J_X=J_Y,\qquad R^ig_*J_X=0\quad(i>0), \qquad g_*\OO_{N_X}=\OO_{N_Y}. \] The restriction $N_X\to N_Y$ is a projective contraction. Hence, whenever $A_Y$ is nef, the restriction of $A_X$ to $N_X$ is endowed with a Moishezon contraction if and only if the restriction of $A_Y$ to $N_Y$ is endowed with one. \end{lemma} \begin{proof} Take a projective common log resolution $r:U\to X$ carrying the nef datum, and put $s=g\circ r$. By crepancy there is one real SNC divisor $C$ such that \[ K_U+[C]+\beta=s^*A_Y=r^*A_X, \] where $\beta$ is globally nef. The multiplier ideals are \[ J_X=r_*\OO_U(-\lfloor C\rfloor),\qquad J_Y=s_*\OO_U(-\lfloor C\rfloor). \] Their direct-image equality follows at once. For vanishing, set $L=\OO_U(-\lfloor C\rfloor)$ and $\Delta=C-\lfloor C\rfloor$. The pair $(U,\Delta)$ is klt, and the actual real line \[ L-(K_U+\Delta)=-K_U-C \] has class $\beta-s^*A_Y$. It is relatively nef over $Y$ and relatively big because $s$ is bimeromorphic. The same statements hold over $X$. Relative Kawamata--Viehweg vanishing, in the form of \cite[Theorem~1.2]{FujinoVanishing2025} with the first morphism the identity and the real line $\mathcal H=-K_U-C$, therefore gives \[ R^is_*L=R^ir_*L=0\qquad(i>0). \] The real-line identity in that theorem is the displayed definition of $\mathcal H$; its relative positivity is checked by the class identity above. The klt condition means that there are no proper log canonical strata to test for relative log bigness. Leray gives $R^ig_*J_X=0$ for $i>0$. Pushing forward $0\to J_X\to\OO_X\to\OO_{N_X}\to0$ and using $g_*\OO_X=\OO_Y$ now proves the asserted structure-sheaf equality. It also identifies the scheme-theoretic image with $N_Y$, giving the projective contraction between these closed subspaces. Apply Lemma~\ref{prog:base-emc} to conclude. \end{proof} \begin{remark} The preceding lemma also transports the contraction hypothesis through a crepant projective flip: apply it to the two contractions to their common base, equipped with the descended adjoint and the common nef b-data. On that base the boundary is the effective pushforward of the source boundary, and the nef data remain on the same carrier; the descended Bott--Chern adjoint and its identity on a common resolution define the crepant base data in the whole-adjoint and current-trace category used here. All assertions concern the fixed adjoint and its multiplier ideal. No conclusion about the multiplier ideal of a different driving adjoint is used. \end{remark} \begin{corollary}[The contraction step over a lower-dimensional base] \label{prog:base-emc-induction} Use the hypotheses and notation of Proposition~\ref{prog:base-presentation}. Suppose that $A=f^*\gamma$ is nef and that its restriction to the original non-klt closed subspace $N_o=V(J_o)$ is endowed with a Moishezon contraction. Put \[ I=f_*J_1,\qquad N_*=V(I)\subset Z. \] Then $\gamma|_{N_*}$ is endowed with a Moishezon contraction. If the prepared non-klt closed subspace $N_1=V(J_1)$ dominates $Z$, then $\gamma$ and $A$ are endowed with Moishezon contractions. Otherwise the generalized pair supplied on $Z$ has a non-klt closed subspace endowed with the contraction defined by the restricted class of $\gamma$. In this latter case, if $\gamma$ is nef and NQC and the assertion $\mathsf C_{\dim Z}$ is available, then $\gamma$ and $A$ are endowed with Moishezon contractions. \end{corollary} \begin{proof} The inclusion $J_o\subset J_1$ gives $N_1\subset N_o$ as closed analytic subspaces. By Lemma~\ref{prog:base-emc}, the given contraction restricts to $N_1$. Proposition~\ref{prog:base-presentation} identifies \[ f_*\OO_{N_1}=\OO_Z/I=\OO_{N_*}. \] Thus the restriction $N_1\to N_*$ is a projective contraction. The restricted adjoint is the pullback of $\gamma|_{N_*}$, so Lemma~\ref{prog:base-emc} descends the contraction to $N_*$. Nonnegative degrees on curves of $N_*$ follow from projective multisections and nefness of $A$. If $N_1$ dominates the normal irreducible space $Z$, then $I=0$: a local function in $I$ pulls back to zero on the dominating closed subspace and hence vanishes generically on $Z$. Normality, and in particular reducedness, gives its vanishing. Therefore $N_*=Z$, which proves the assertion in this case. If $N_1$ does not dominate $Z$, the constructed base ideal satisfies $I\subset J_Z$. Hence \[ V(J_Z)\subset V(I)=N_* \] as closed analytic subspaces. Restriction by Lemma~\ref{prog:base-emc} provides the required contraction on $V(J_Z)$, with its complete scheme structure. The generalized pair on $Z$ has globally nef carrier data and globally modified-big total boundary by Proposition~\ref{prog:base-presentation}. When the stated nef and NQC conditions hold, these facts are precisely the hypotheses of $\mathsf C_{\dim Z}$, which gives the contraction on $Z$. Finally Lemma~\ref{prog:base-emc} pulls it back through the projective contraction $f$ to $X$. The case of an empty non-klt closed subspace has the same interpretation, with its contraction condition vacuous. \end{proof} ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-base-presentations.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-restricted-induction.tex ==== \subsection{A restricted dimension induction} \label{prog:dimension-induction} All generalized nef data in this subsection are globally nef on a fixed compact K\"ahler carrier. For the degree arguments we use weak NQC: a positive combination of rational degree-two classes having nonnegative degree on compact curves. Strong NQC implies this condition, and the contraction assertion admits this weak form \cite[Definition~2.43]{HX}. The nef adjoint itself remains nef in Bott--Chern cohomology throughout. A modified-big boundary means the total trace $B+\mathbf M_X$ is globally modified big. Relative assertions retain these absolute hypotheses. A marked model records its bimeromorphic map from the specified source; two markings agree only under an isomorphism commuting with those maps. Weak models below may be arbitrary normal compact K\"ahler weak log canonical models. Their generalized nef traces are understood as closed currents pushed forward from the fixed nef carrier; only the whole adjoint is required to be a Bott--Chern class. In particular no factoriality, and no separate Bott--Chern class for either $K_Y+B_Y$ or the nef trace, is required of an auxiliary weak target. This category contains both the working models and the adjunction strata used in special termination. Chosen program outputs remain globally strongly $\mathbb Q$-factorial. For dimension at most $d$, write $\mathsf B_d$ for semiampleness of a nef gklt adjoint with modified-big boundary, including a Moishezon connected contraction onto a normal compact K\"ahler target. It makes no assertion that every such contraction is projective. Write $\mathsf C_d$ for the NQC non-klt contraction assertion of \cite[Theorem~1.6]{HX}, and $\mathsf C_{d,\mathrm{big}}$ for its big-adjoint case. The relative assertions are: \begin{itemize} \item $\mathsf M_d$: a relatively pseudo-effective gklt adjoint over any proper map to a normal compact K\"ahler base $S$, with the globally nef and modified-big data just specified, has a chosen nonextracting good log terminal model over $S$. Its working space is globally strongly $\mathbb Q$-factorial and compact K\"ahler; its connected relative canonical morphism has normal compact K\"ahler target and is Moishezon. From a smooth common carrier, all Bott--Chern classes have forward traces, and the chosen model has the rational degree-two and Bott--Chern exceptional splittings proved below. \item $\mathsf F_d$: a compact polytope of these data on one fixed carrier has a polyhedral relative effective locus, finitely many relative canonical chambers with chosen good terminal models, and finitely many marked weak models, accounting for every weak model in the category specified above. \end{itemize} Passing to the proper image and then to its normal Stein factor allows all base maps to be taken surjective with connected fibres. These operations preserve the compact K\"ahler base category. The independent inputs are the analytic cone theorem, the projective results of Proposition~\ref{prog:projective-good-model}, relative vanishing, adjunction, relative-canonical positivity, and the nef-and-big K\"ahler criterion. The base presentations and multiplier-ideal descent used in the induction are proved below. The projective constructions use the actual-line lemmas of the preceding subsection. No assertion of termination of an arbitrary sequence of flips is used. \input{sections/03c-base-presentations} \subsubsection{A fixed integral grid and cohomology transport} \begin{lemma}[Integral degree-two descent for a locally log-Fano map] \label{prog:integral-h2} Let $f:X\to Z$ be a projective surjective morphism with connected fibres between normal complex analytic spaces. Assume that $f_*\mathcal O_X= \mathcal O_Z$, that $R^if_*\mathcal O_X=0$ for $i>0$, and that every point of $Z$ has a neighbourhood $U$ on which there is a klt pair $(X_U,\Delta_U)$ with $-(K_{X_U}+\Delta_U)$ ample over $U$. Then $f^*:H^2(Z,\mathbb Z)\to H^2(X,\mathbb Z)$ is injective, and its image consists exactly of the integral classes having degree zero on every compact curve contracted by $f$. \end{lemma} \begin{proof} Properness, normality and connected fibres identify both $f_*\mathcal O_X=\mathcal O_Z$ and $f_*\mathcal O_X^*=\mathcal O_Z^*$. Apply $Rf_*$ to the analytic exponential sequence. The local surjectivity of $\mathcal O_Z\to\mathcal O_Z^*$ and the stated coherent vanishing give \[ R^1f_*\mathbb Z_X=0, \qquad R^1f_*\mathcal O_X^*\simeq R^2f_*\mathbb Z_X. \] Let $\xi\in H^2(X,\mathbb Z)$ have zero degrees on contracted curves. A germ of its edge image in $R^2f_*\mathbb Z_X$ is, under this isomorphism, represented after shrinking $U$ by an actual line bundle $L$ on $X_U$. Equality of these germs means that, after further shrinking, $c_1(L)$ and $\xi|_{X_U}$ have the same class. In particular $L$ has degree zero on every curve in every fibre above this smaller neighbourhood. On a sufficiently small Stein base neighbourhood, twisting by a relatively ample line and taking a quotient of two nonzero relative sections gives a meromorphic Cartier representative for $L$. Thus the Cartier-divisor form of base point freeness applies. The line bundle $L$ is relatively nef, and $aL-(K_{X_U}+\Delta_U)$ is relatively ample for every real $a$. The projective analytic base-point-free theorem applies to the Cartier line $L$. Its conclusion is generation for \emph{every} sufficiently large integer power, not only for a divisible subsequence \cite[Theorems~6.2 and~6.5]{FujinoAnalyticBCHM}. Thus, over a further relatively compact neighbourhood, both $L^m$ and $L^{m+1}$ are relatively generated. A generated line of degree zero on every curve of a projective connected fibre defines a constant projective map on that fibre: the restriction of the generated line is the pullback of $\mathcal O(1)$, and a positive-dimensional projective image contains a curve of positive degree. The corresponding relative image is therefore finite and bimeromorphic over the normal base, and is the base itself. Consequently \[ L^m\simeq f^*M_m,\qquad L^{m+1}\simeq f^*M_{m+1} \] for line bundles on the neighbourhood in $Z$. Taking their quotient proves $L\simeq f^*(M_{m+1}\otimes M_m^{-1})$. Its germ in $R^1f_*\mathcal O_X^*$ is zero, so the edge image of $\xi$ is zero. The integral Leray spectral sequence, using $R^1f_*\mathbb Z_X=0$, gives \[ 0\longrightarrow H^2(Z,\mathbb Z) \xrightarrow{f^*}H^2(X,\mathbb Z) \longrightarrow H^0(Z,R^2f_*\mathbb Z_X). \] Exactness at the middle term proves the assertion. The argument is integral throughout and does not discard torsion. The converse follows by restriction of a pulled-back class to a fibre. \end{proof} \begin{lemma}[A fixed integral grid along existing crepant steps] \label{prog:fixed-integral-grid} Let $X_0$ be a compact K\"ahler space, and suppose that \[ \alpha_0=\sum_{j=1}^r a_j\gamma_{0j},\qquad a_j>0, \qquad \gamma_{0j}\in H^2(X_0,\mathbb Q), \] where each $\gamma_{0j}$ has nonnegative degree on compact curves. Fix integers $m_j>0$ such that $m_j\gamma_{0j}$ is the image of an integral cohomology class. Consider an existing sequence of projective birational steps \[ X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1}, \] where $f_i^+$ is the identity for a divisorial step, $f_i$ satisfies Lemma~\ref{prog:integral-h2}, and $\alpha_i$ is trivial on its contracted ray. Assume every $f_i$-contracted curve has degree proportional to that ray for the classes in the display. Then the classes $\gamma_{ij}$ descend through $f_i$ and pull back through $f_i^+$, the same integers $m_j$ remain integral multiples at every stage, and the transported classes remain nonnegative on curves. In particular, for every integral compact curve $C\subset X_i$, \[ \alpha_i\cdot C>0\quad\Longrightarrow\quad \alpha_i\cdot C\geq\delta, \qquad \delta:=\min_j\frac{a_j}{m_j}>0. \] \end{lemma} \begin{proof} Positivity of the coefficients and nonnegativity of the summands imply that every $\gamma_{ij}$ annihilates the contracted ray. Apply the integral descent lemma to the chosen integral lift of $m_j\gamma_{ij}$ and pull its descended class back to $X_{i+1}$. This keeps $m_j$ fixed. To verify nonnegativity, take a curve on $X_{i+1}$. If it is contracted by $f_i^+$, every descended summand has degree zero. Otherwise its image is a curve $\Gamma\subset Z_i$. The projective map $f_i$ has a compact curve mapping onto $\Gamma$ with positive degree: take a component of its inverse image dominating $\Gamma$ and intersect with sufficiently many relatively ample hyperplanes. Nonnegativity of the old summand on this curve implies nonnegativity on $\Gamma$, hence on the curve on $X_{i+1}$. This argument uses a positive-degree lift, not a degree-one section. Finally $m_j\gamma_{ij}\cdot C$ is a nonnegative integer, giving the stated lower bound. \end{proof} \begin{corollary}[Uniform crepancy parameter; no termination assertion] \label{prog:uniform-canonical-parameter} Suppose additionally that the working spaces are $d$-dimensional klt spaces, that $\alpha_i$ are nef, and that the ordinary canonical negative-ray length bound $0<-K_{X_i}\cdot C\leq2d$ is available. For a fixed real number $t>2d/\delta$, every negative extremal ray chosen in an existing $(K_{X_i}+t\alpha_i)$-program is $\alpha_i$-trivial. If the steps are projective ordinary canonical steps, Lemma~\ref{prog:fixed-integral-grid} therefore applies inductively with this same value of $t$. If $\alpha_0$ is not big and $\alpha_0-c_1(K_{X_0})$ is big, then $K_{X_i}+t\alpha_i$ remains non-pseudo-effective along these crepant steps. Consequently a terminating such program ends with a Mori fibre contraction. This corollary does not assert that the requisite contractions exist or that the program terminates. \end{corollary} \begin{proof} A negative ray is $K_{X_i}$-negative because $\alpha_i$ is nef. Choose its rational curve generator with the ordinary length bound. If its $\alpha_i$-degree were positive, then \[ (K_{X_i}+t\alpha_i)\cdot C\geq-2d+t\delta>0, \] a contradiction. The step is thus an ordinary canonical step with $\alpha_i$ pulled back from its base, and the preceding lemma applies. On a common resolution the pullback of $\alpha_{i+1}-c_1(K_{X_{i+1}})$ is the pullback of $\alpha_i-c_1(K_{X_i})$ plus the effective canonical comparison. Thus it stays big. Equality of the crepant pullbacks of $\alpha_i$ preserves its nonbigness. The identity \[ (t+1)\alpha_i=(c_1(K_{X_i})+t\alpha_i) +(\alpha_i-c_1(K_{X_i})) \] excludes pseudo-effectivity of its first summand. A nef endpoint is therefore impossible. \end{proof} \begin{lemma}[Cohomology splitting along detected steps] \label{prog:detected-cohomology} Let $X_0$ be a smooth compact K\"ahler manifold. Consider a finite sequence of divisorial contractions and small flips \[ X_0\dashrightarrow X_1\dashrightarrow\cdots \dashrightarrow X_r \] between normal globally strongly $\mathbb Q$-factorial compact K\"ahler spaces with rational singularities. Assume that every negative contraction $f_i:X_i\to Z_i$ is a projective contraction of one ray of the analytic cone, that $Z_i$ has rational singularities, and that each small flip is the detected flip of a global rational line $L_i$ with $L_i\cdot R_i<0$. For a divisorial contraction assume the usual single exceptional prime. Then, for every projective smooth resolution $p:U\to X_i$, \begin{align*} H^2(U,\mathbb R) &=p^*H^2(X_i,\mathbb R) \oplus\bigoplus_{E\text{ exceptional for }p}\mathbb R[E],\\ H^{1,1}(U,\mathbb R) &=p^*H^{1,1}_{\rm BC}(X_i,\mathbb R) \oplus\bigoplus_{E\text{ exceptional for }p}\mathbb R[E]. \end{align*} The first equality also holds over $\mathbb Q$. There are compatible forward linear transforms on $H^2(-,\mathbb R)$ and on Bott--Chern cohomology. They are surjective, are isomorphisms for a small flip, and preserve rational degree-two classes. On a common resolution the pullback difference is an actual exceptional divisor class; for a flip it is exceptional over both spaces. The exceptional quotient also takes the Chern class of a holomorphic line bundle on $U$ to a rational holomorphic line class on $X_i$. \end{lemma} \begin{proof} For a modification of a smooth compact K\"ahler manifold the two splittings are the degree-two modification formula. Suppose that they hold for $X=X_i$. We use only the following birational descent fact: for a proper bimeromorphic map between normal compact spaces in Fujiki's class with rational singularities, pullback is injective on $H^2(-,\mathbb R)$ and on Bott--Chern cohomology, and its image in either group consists of the classes having degree zero on every contracted curve. This is the bimeromorphic case of \cite[Lemma~2.6(1)]{DH2024}. First note that a degree-two class $\xi\in H^2(X,\mathbb R)$ has the same degrees on curves as a Bott--Chern class. Indeed, pull it to a smooth projective resolution $a:W\to X$ and take the $(1,1)$-part of its Hodge decomposition. The induction hypothesis writes that part as $a^*\beta+[F]$, with $F$ exceptional over $X$. On every $a$-contracted curve, $a^*\xi$ and its $(2,0)$ and $(0,2)$ parts have degree zero. Hence $F$ has degree zero on every such curve. Exceptional negativity applied to both signs gives $F=0$. Projective multisections lifting a curve of $X$ then show that $\xi$ and $\beta$ have the same degrees on every curve of $X$. Suppose first that $X\dashrightarrow X^+$ is a small detected flip. Write $f:X\to Z$ and $f^+:X^+\to Z$, and fix an integral curve $C$ on its negative ray. For $\xi\in H^2(X,\mathbb R)$ set \[ s=\frac{\xi\cdot C}{c_1(L)\cdot C}. \] By the preceding paragraph, $\xi-s c_1(L)$ vanishes on every $f$-contracted curve. Thus it is $f^*\eta$ for a unique $\eta\in H^2(Z,\mathbb R)$. Define \[ T\xi=(f^+)^*\eta+s c_1(L^+). \] For a Bott--Chern class, the same descent takes place in Bott--Chern cohomology, so the two transforms agree. The detected flip construction gives an effective rational divisor $E_L$ on a common projective resolution $a:W\to X$, $b:W\to X^+$ such that \[ a^*c_1(L)-b^*c_1(L^+)=[E_L]. \] Consequently \[ a^*\xi=b^*T\xi+s[E_L]. \] If $\xi$ is rational, then $s$ is rational. Since the image of $f^*$ on real degree-two cohomology is the realification of a rational subspace, its unique preimage $\eta$ is rational as well. This proves rationality of $T\xi$. The sets of prime divisors exceptional for $a$ and $b$ are identical: the two working spaces are small bimeromorphic. The old splitting and the displayed pullback identity therefore span \[ H^2(W,\mathbb R) =b^*H^2(X^+,\mathbb R) +\operatorname{span}_{\mathbb R}\{[E]:E\text{ is }b \text{-exceptional}\}, \] and similarly in bidegree $(1,1)$. The sum is direct. If a pullback from $X^+$ equals an exceptional divisor class, that divisor has degree zero on all $b$-contracted curves, so both signs of negativity make it zero; pullback injectivity then makes the original class zero. In particular the dimensions on the two sides agree, $T$ is injective by the old direct sum, and $T$ is surjective. The rational version follows either from the same argument or from realification. For a divisorial contraction $f:X\to X^+$, let $D$ be its exceptional prime. Its degree on the contracted ray is negative: otherwise $D$ would be $f$-nef, contrary to exceptional negativity. For any $\xi$, subtract \[ \frac{\xi\cdot C}{D\cdot C}[D] \] and descend the result by the same birational cohomology lemma. This defines $T\xi$. On a common resolution its pullback difference is a multiple of the total transform of $D$. The target's exceptional prime set is the old one together with the strict transform of $D$. The same spanning and directness argument gives the target splitting and surjectivity; its kernel is $\mathbb R[D]$. Rationality is again immediate from the quotient of rational intersection numbers. These arguments initially prove the splitting on a chosen common resolution. To obtain it on any projective smooth resolution $U\to X^+$, dominate that resolution and the chosen one by a smooth common projective resolution. Apply the smooth degree-two modification formula upstairs and push down to $U$. An exceptional divisor pushes either to zero or to a divisor exceptional over $X^+$. This proves spanning on $U$; negativity gives directness there. Finally let $\mathcal H$ be a holomorphic line on $U$ and put $\mathcal Q=(p_*\mathcal H)^{**}$. Global strong $\mathbb Q$-factoriality gives an $m>0$ for which $\mathcal M=\mathcal Q^{[m]}$ is invertible. The coherent evaluation comparison gives \[ \mathcal H^{\otimes m} \simeq p^*\mathcal M\otimes\mathcal O_U(F) \] for an integral $p$-exceptional divisor $F$. This comparison is obtained from the coherent direct image and evaluation map; it does not assume that $\mathcal H$ has a global meromorphic section. Dividing first Chern classes by $m$ proves the line assertion. \end{proof} \begin{corollary}[Stability under projective extraction] \label{prog:extraction-cohomology} Let $r:Y\to X$ be a projective bimeromorphic morphism of normal compact K\"ahler spaces with rational singularities. Suppose that $Y$ is globally strongly $\mathbb Q$-factorial and that $X$ has the splitting property in the preceding lemma. Then $Y$ has that property as well. More precisely, if $E_1,\ldots,E_s$ are the prime divisors exceptional for $r$, then \[ H^2(Y,\mathbb R) =r^*H^2(X,\mathbb R)\oplus \bigoplus_{j=1}^{s}\mathbb R[E_j], \] and the analogous statements hold over $\mathbb Q$ and in Bott--Chern cohomology. \end{corollary} \begin{proof} Choose any projective smooth resolution $q:W\to Y$. The composite $p=rq$ is a projective resolution of $X$. Its exceptional primes are exactly the $q$-exceptional primes together with the strict transforms $\widetilde E_j$ of the $E_j$. Since every $E_j$ is $\mathbb Q$-Cartier, \[ [\widetilde E_j]=q^*[E_j]+[F_j] \] for a rational $q$-exceptional divisor $F_j$. Substitute these identities in the splitting for $p$. The result spans $H^2(W,\mathbb R)$ by $q^*H^2(Y,\mathbb R)$ and the $q$-exceptional divisor classes. Their sum is direct: an exceptional divisor whose class is a pullback has degree zero on every $q$-contracted curve, and applying exceptional negativity to both signs makes the divisor zero. Pullback injectivity then applies. The same argument in bidegree $(1,1)$ and over $\mathbb Q$ proves the splitting property for $Y$. Now expand $q^*\xi$, for $\xi\in H^2(Y,\mathbb R)$, using the original $p$-splitting and replace the $[\widetilde E_j]$ as above. The difference between $q^*\xi$ and a class in $q^*(r^*H^2(X,\mathbb R)+\sum_j\mathbb R[E_j])$ is $q$-exceptional, and is therefore zero by the direct sum just proved. Thus the displayed formula for $H^2(Y,\mathbb R)$ spans. Its directness follows from exceptional negativity for $r$ and pullback injectivity. The rational and Bott--Chern statements follow by the identical argument. The line bundle assertion of the preceding lemma uses only global strong $\mathbb Q$-factoriality and hence applies to $Y$ as well. \end{proof} \paragraph{Support in the special-termination comparison.} \label{prog:special-support} The following support comparison applies to the detected steps in the dimension induction. Let $\psi:X\dashrightarrow X^+$ be a detected small negative step with maps $f:X\to Z$ and $f^+:X^+\to Z$, and let $L$ be its negative detector. For a sufficiently divisible $r>0$, put \[ \operatorname{im}(f^*f_*L^r\longrightarrow L^r) =\mathcal I\otimes L^r. \] The zero set of $\mathcal I$ is exactly the exceptional locus of $f$. Outside that locus $f$ is an isomorphism, so the evaluation map is surjective. On a positive-dimensional projective fibre, every section of $L^r$ vanishes: its restriction to each integral curve has negative degree, and those curves cover every positive-dimensional fibre component. Connectedness supplies the whole nontrivial fibre. On a resolution principalizing $\mathcal I$, the detector comparison is a positive multiple of the effective divisor defined by $\mathcal I\mathcal O_W$. Consequently its support is the full inverse image of the exceptional locus. The driving adjoint differs from a positive real multiple of $c_1(L)$ by a class pulled back from the contraction base. Its comparison divisor is therefore the same positive multiple of the detector comparison: their class difference is exceptional and zero, so both-sign negativity makes it zero as a divisor. The comparison for a longer negative program is at least this first-step divisor, by discrepancy monotonicity on a common resolution. Suppose the beginning and end restrictions to a surviving normalized lc stratum $S$ are isomorphic and their strict adjunction data agree. The general point of this stratum avoids every intervening surgery: a zero-discrepancy place remains such a place, whereas strict comparison would increase its discrepancy if its center were contained in the exceptional locus. Thus its strict transform $S_W$ on a common resolution is not contained in the comparison support. Iterated strict adjunction identifies the restriction of the effective comparison with the difference of the identified adjoints, so its class is zero. A nonzero effective divisor on a positive-dimensional compact K\"ahler stratum has positive K\"ahler mass. The restricted divisor is therefore zero. If the first step meeting the stratum were nontrivial there, the full inverse-image support and surjectivity $S_W\to S$ would make this restriction nonzero, a contradiction. For a zero-dimensional stratum, generic-point avoidance already excludes an intersection. The whole program is consequently an isomorphism near the stratum. This is the same strict-adjunction support argument used in the proof of Theorem~\ref{prog:special-termination}; it does not use its lower-dimensional abundance assumption. \begin{proposition}[A uniform relative NQC bound] \label{prog:relative-nqc-grid} Let $f_0:U\to Z$ be an already projective surjective morphism with connected fibres between smooth compact K\"ahler manifolds, and suppose that \[ f_0^*:H^0(Z,\Omega_Z^2)\longrightarrow H^0(U,\Omega_U^2) \] is an isomorphism. Let $\alpha_0$ be a nef class with an expression \[ \alpha_0=\sum_{j=1}^k r_j\eta_{j,0},\qquad r_j>0,\quad \eta_{j,0}\in H^2(U,\mathbb Q), \] where $\eta_{j,0}$ has nonnegative degree on every curve vertical over $Z$. Let $D_0=K_U+\Delta_U+\mathbf M_U$ be a generalized klt adjoint. Consider its projective relative program for $D_0+a\alpha_0$, under the projective local ordinary reduction described in the proof of Lemma~\ref{prog:base-point-free-induction}. If $\alpha_0=0$, triviality is immediate. Otherwise there are positive integers $m_j$, chosen once on $U$, and a number \[ \delta=\min_j\frac{r_j}{m_j}>0 \] such that, for $a\delta>2\dim U$, the entire relative program is $\alpha_0$-trivial. The same number $a$ works at every step. \end{proposition} \begin{proof} \emph{Rational classes modulo the fixed base.} Choose an $f_0$-ample line bundle with integral Chern class $H$, and put $e=\dim U-\dim Z$ and $c=f_{0*}(H^e)>0$. The rational operator \[ \Pi(\eta)=\eta- f_0^*\!\left(\frac{f_{0*}(\eta\smile H^e)}{c}\right) \] preserves type $(1,1)$ and kills types $(2,0)$ and $(0,2)$. Indeed these latter classes are pulled back from $Z$, and the projection formula applies. Thus $\Pi(H^2(U,\mathbb Q))\subset H^{1,1}(U)\cap H^2(U,\mathbb Q)$. By the Lefschetz $(1,1)$ theorem, choose an actual line bundle $L_{j,0}$ and an integer $m_j>0$ with \[ \eta_{j,0}=\frac1{m_j}c_1(L_{j,0})+f_0^*\gamma_j, \qquad \gamma_j\in H^2(Z,\mathbb Q). \] The original summands $\eta_{j,0}$ need not have type $(1,1)$. Their base components account for this. Their weighted sum gives \begin{equation}\label{prog:eq:relative-nqc} \alpha_0=\sum_j\frac{r_j}{m_j}c_1(L_{j,0})+f_0^*\Gamma, \qquad \Gamma=\sum_jr_j\gamma_j\in H^{1,1}(Z,\mathbb R). \end{equation} The final assertion follows because $f_0^*$ is an injective Hodge map and the other terms have type $(1,1)$. The identity is therefore also one of Bott--Chern classes. Each $L_{j,0}$ has nonnegative integral degree on every $Z$-vertical curve. \smallskip\noindent \emph{Induction across an actual contraction.} Suppose at a finite stage $f_i:U_i\to Z$ we have actual line bundles $L_{j,i}$, nef over $Z$, and \[ \alpha_i=\sum_j\frac{r_j}{m_j}c_1(L_{j,i})+f_i^*\Gamma. \] Assume also that $\alpha_i$ is nef and that the transformed $D_i$ is generalized klt. These assertions hold initially. Let $R$ be a $(D_i+a\alpha_i)$-negative extremal ray over $Z$. It is $D_i$-negative because $\alpha_i$ is nef. Lemma~\ref{prog:relative-hodge} and Lemma~\ref{prog:projective-cone} identify the relative curve cone with the corresponding face of the full analytic cone, at the initial model and after every step. The analytic cone theorem therefore supplies a rational generator $C$ with \[ 0<-D_i\cdot C\leq2\dim U. \] All $L_{j,i}\cdot C$ are nonnegative integers. If $\alpha_i\cdot C>0$, one of them is at least one, and hence $\alpha_i\cdot C\geq\delta$. Therefore \[ (D_i+a\alpha_i)\cdot C \geq-2\dim U+a\delta>0, \] a contradiction. We have $\alpha_i\cdot C=0$ and $L_{j,i}\cdot C=0$ for every $j$. Every curve contracted by the projective extremal contraction $h_i:U_i\to Y_i$ has class on $R$ when tested by global line bundles, so the same vanishing holds for all of them. On each fixed Stein chart of $Z$, choose the effective real ordinary representative of the initial relatively ample nef datum once, before running the program. Its actual real linear equivalence to the fixed real line-bundle representative persists under pushforward. The Bott--Chern comparison with the generalized adjoint modulo the fixed base class persists separately by exceptional negativity, as detailed below. Thus every $h_i$ is, locally on $Y_i$, a contraction with an ordinary real klt log-Fano boundary. On a relatively compact base chart, rational approximation in the finite affine system of Cartier adjoint identities gives a rational effective klt boundary with the same antiample sign, as in Proposition~\ref{prog:detected-flip}. Lemma~\ref{prog:integral-descent} therefore applies. Consequently \[ M_{j,i}:=(h_i)_*L_{j,i}\ \text{is a line bundle},\qquad h_i^*M_{j,i}=L_{j,i}. \] For a divisorial contraction set $L_{j,i+1}=M_{j,i}$. For a flip $h_i^+:U_{i+1}\to Y_i$, set $L_{j,i+1}=(h_i^+)^*M_{j,i}$. These are actual line bundles with the \emph{same} integers $m_j$. The descended Bott--Chern class \[ \alpha_{Y_i} =\sum_j\frac{r_j}{m_j}c_1(M_{j,i})+f_{Y_i}^*\Gamma \] satisfies $h_i^*\alpha_{Y_i}=\alpha_i$. It is nef by descent of nefness through the projective surjection $h_i$; its pullback to the next model is $\alpha_{i+1}$. This proves that the step is crepant for $\alpha_i$. Hence it is also a $D_i$-negative step, and the original generalized pair remains gklt. Finally each $M_{j,i}$ is nef over $Z$. For a $Z$-vertical curve $B\subset Y_i$, projectivity supplies a curve $B'\subset U_i$ mapping onto it with some positive degree $d_B$. Then \[ d_B(M_{j,i}\cdot B)=L_{j,i}\cdot B'\geq0. \] No assertion $d_B=1$ is required. Pullback preserves this relative nefness, so the induction applies on $U_{i+1}$. Its curve degrees are again integers because the transported objects are line bundles, not because any curves were lifted with degree one. This proves the fixed bound through the whole program. \end{proof} \begin{remark}[Scope of the degree grid] The transported rational classes \[ \eta_{j,i}=\frac1{m_j}c_1(L_{j,i})+f_i^*\gamma_j \] have degrees in $m_j^{-1}\mathbb Z_{\geq0}$ on curves vertical over the fixed smooth base $Z$. This relative assertion is exactly what the initial projective program requires. It does not assert nonnegativity of these individual classes on all curves of every birational model, and uses neither rational connectedness of resolution fibres nor Graber--Harris--Starr. \end{remark} \subsubsection{The negative-part comparison for a nef adjoint} We write $N_\sigma(\xi)$ for the divisorial negative part of a pseudo-effective class. We use its homogeneity, subadditivity, continuity after adding a vanishing K\"ahler perturbation, and the identities \[ N_\sigma(r^*\xi+[E])=N_\sigma(r^*\xi)+E, \qquad r_*N_\sigma(r^*\xi)=N_\sigma(\xi) \] for an effective $r$-exceptional divisor $E$. A nef class has zero negative part. These are the divisorial Zariski-decomposition properties of \cite[Section~3]{Boucksom2004} and \cite[Appendix~A]{DHY}. No identity $N_\sigma(r^*\xi)=r^*N_\sigma(\xi)$ for arbitrary $\xi$ is assumed. \begin{lemma}[The absolute negative part after a relative program] \label{prog:negative-part-program} Let $\mu:U\to X$ be a projective resolution, let $\alpha$ be nef on $X$, let $c>0$, and let $F_U\geq0$ be $\mu$-exceptional. Put \[ D_U=c\mu^*\alpha+[F_U]. \] Suppose that $\phi:U\dasharrow V$ is a finite $D_U$-negative program, possibly relative to another base, and denote its transformed class and divisor by $D_V$ and $F_V=\phi_*F_U$. Then \[ N_\sigma(D_V)=F_V. \] \end{lemma} \begin{proof} Take a common projective resolution $p:W\to U$, $q:W\to V$. The negativity comparison for the finite program is \[ p^*D_U=q^*D_V+[E], \qquad E\geq0,\quad E\text{ is }q\text{-exceptional}. \] This comparison is valid for a relative negative program: its proof uses the negativity of the contracted rays and the positive adjoint on each flipped side, not absolute nefness of the final adjoint. Since $p^*F_U$ is effective and exceptional over $X$, the exceptional identity and nefness of $(\mu p)^*\alpha$ give \[ N_\sigma(p^*D_U)=p^*F_U =N_\sigma(q^*D_V)+E. \] Pushing forward by $q$ proves the assertion. In particular, the negative part in this statement is absolute, even when the program that produced $V$ was relative. \end{proof} \subsubsection{Descent of the effective vertical divisor} \begin{lemma}[Vertical numerical triviality] \label{prog:vertical-divisor} Let $g:V\to S$ be a projective surjective morphism with connected fibres. Assume that $S$ is globally Weil $\mathbb{Q}$-factorial. Let $F\geq0$ be an effective real Cartier divisor on $V$, vertical over $S$, such that \[ F\cdot C=0 \quad\text{for every curve }C\text{ contracted by }g. \] Then there is an effective real Cartier divisor $F_S$ on $S$ such that, as actual divisors, \[ F=g^*F_S. \] \end{lemma} \begin{proof} For a prime divisor $P\subset S$, write $m_Q$ for the multiplicity of $Q$ in $g^*P$, where $Q$ ranges over the prime divisors dominating $P$. These multiplicities may be computed over the smooth generic locus of $P$, where $P$ is Cartier. Put \[ b_P=\min_{g(Q)=P}\frac{\operatorname{coeff}_Q F}{m_Q}, \qquad F_S=\sum_P b_PP. \] Only finitely many $b_P$ are nonzero, since any such $P$ is the image of a component of $F$. Thus $F_S$ is a genuine effective Weil real divisor. Global Weil $\mathbb{Q}$-factoriality makes this finite divisor real Cartier. We first check equality over codimension one in $S$. Work near a general point of $P$ and restrict to a transverse disk. Resolving the source, and cutting by general relative ample hypersurfaces if the relative dimension exceeds one, reduces to a projective surface over that disk with connected fibres. These operations can be performed over a relatively compact Stein neighbourhood; they do not require global sections on $S$. The intersection matrix of the components of the special fibre is negative semidefinite, with kernel generated by the full fibre with its multiplicities. The restriction of $F$ has zero intersection with every fibre component. Its coefficients are therefore proportional to those multiplicities. Equivalently, all the ratios in the definition of $b_P$ are equal. When $g$ is birational this assertion over the generic point of $P$ is immediate. Consequently \[ G=F-g^*F_S \] is a signed real Cartier divisor supported over a subset of codimension at least two in $S$. Moreover, $G$ is numerically trivial over $S$. We check that such a signed exceptional divisor is zero. The assertion is local on $S$. Over a relatively compact Stein neighbourhood, take $d=\dim V-\dim S$ general relative ample hypersurfaces and resolve their intersection. They give a projective generically finite morphism $H\to S$. The cuts may be chosen to meet any prescribed component of $G$ in a nonzero divisorial trace. After normalization and Stein factorization, $H\to S$ factors as a projective birational morphism $H\to S'$ followed by a finite morphism $S'\to S$. The restricted divisor $G|_H$ is exceptional for $H\to S'$ and numerically trivial over $S'$. Applying the ordinary exceptional negativity lemma to both $G|_H$ and $-G|_H$ gives $G|_H=0$. The choice of the cuts therefore excludes every nonzero component of $G$. Hence $G=0$. The surface argument above is also the usual proof of the degenerate-divisor negativity statement: after subtracting the minimum multiple of the full fibre, an effective residual fibre divisor omits a component and cannot be numerically trivial. Notice that projectivity of $g$ was a hypothesis; it was not deduced from factoriality. \end{proof} \subsubsection{The negative part on the lower-dimensional base} \begin{lemma}[Detecting the negative part by pullback currents] \label{prog:negative-part-base} Let $g:V\to S$ be a surjective morphism. Suppose that $\alpha_S$ is nef, $F_S\geq0$ is real Cartier, $c>0$, and \[ D_S=c\alpha_S+[F_S],\qquad D_V=g^*D_S,\qquad N_\sigma(D_V)=g^*F_S. \] Then $N_\sigma(D_S)=F_S$. \end{lemma} \begin{proof} Nefness gives $N_\sigma(D_S)\leq F_S$. Fix a prime divisor $P\subset S$ and a prime divisor $Q\subset V$ dominating it. Write $m=\operatorname{mult}_Q(g^*P)>0$ and $b=\operatorname{coeff}_P F_S$. At their generic smooth points, pullback of positive currents satisfies \[ \nu(g^*T,Q)=m\nu(T,P). \] Indeed, the divisorial term $\nu(T,P)[P]$ pulls back with multiplicity $m$, and the residual current has zero generic Lelong number there. Choose K\"ahler forms $\omega_S,\omega_V$ and a constant $C>0$ for which $C\omega_V-g^*\omega_S$ is K\"ahler. For $\varepsilon>0$, let $T_\varepsilon$ be any positive current in the big class $D_S+\varepsilon[\omega_S]$. Minimality of the multiplicity and monotonicity under adding a nef class give \begin{align*} m\nu(T_\varepsilon,P) &=\nu(g^*T_\varepsilon,Q)\\ &\geq\nu(D_V+\varepsilon g^*[\omega_S],Q)\\ &\geq\nu(D_V+C\varepsilon[\omega_V],Q). \end{align*} Taking the infimum over $T_\varepsilon$, and then letting $\varepsilon\downarrow0$, gives \[ m\nu(D_S,P)\geq\nu(D_V,Q) =\operatorname{coeff}_Q(g^*F_S)=mb. \] This proves the reverse inequality for every $P$. All multiplicities are unchanged by resolving away from the generic points in question, so the same argument applies to the normal spaces under consideration. The $\varepsilon$ perturbation is necessary: at a pseudoeffective boundary class, the infimum over its exact positive currents need not equal its minimal multiplicity. The argument uses that equality only for the big perturbed classes. \end{proof} \begin{lemma}[Pullback when the positive part is nef] \label{prog:negative-part-pullback} Suppose \[ \xi=P+[F],\qquad P\text{ nef},\quad F\geq0\text{ real Cartier}, \qquad N_\sigma(\xi)=F. \] For every projective resolution $p:W\to S$, \[ N_\sigma(p^*\xi)=p^*F. \] \end{lemma} \begin{proof} Write $N'=N_\sigma(p^*\xi)$. Since $p^*P$ is nef, $N'\leq p^*F$. Birational invariance gives $p_*N'=F$, so $E=p^*F-N'\geq0$ is $p$-exceptional. The positive part of $p^*\xi$ is \[ p^*P+[E], \] and is modified nef. If $E\neq0$, exceptional negativity in its covering-curve form provides a component of $E$ covered by curves $C_t$ contracted by $p$, with $E\cdot C_t<0$ \cite[Appendix~A, Lemma~A.3]{DHY}. A modified nef class has nonnegative intersection with a general member of a family of curves covering a prime divisor: use currents with arbitrarily small negative part and zero generic divisorial Lelong number, restrict to a general member, and let the negative bound tend to zero. But here \[ (p^*P+E)\cdot C_t=E\cdot C_t<0, \] a contradiction. Thus $E=0$. \end{proof} \begin{proposition}[Any lower-dimensional good model suffices] \label{prog:nef-good-comparison} Suppose the initial relative program has the data in Lemma~\ref{prog:negative-part-program}. Assume in addition that its class $\alpha_V$ is nef and agrees with $\mu^*\alpha$ on a common resolution, and that there is an already projective connected-fibre morphism $g:V\to S$ with \[ \alpha_V=g^*\alpha_S,\qquad D_V=g^*D_S. \] Suppose there is a projective small modification $s:S^{\mathrm q}\to S$ such that $S^{\mathrm q}$ is globally Weil $\mathbb Q$-factorial; the identity is allowed. Assume that the lower-dimensional adjoint $s^*D_S$ has a good model: there are a normal compact K\"ahler space $S_m$, a common projective resolution $p:W\to S^{\mathrm q}$, $q:W\to S_m$, and a nef semiample class $D_m$ such that \[ p^*s^*D_S=q^*D_m+[E], \qquad E\geq0,\quad E\text{ is }q\text{-exceptional}. \] Then $\alpha$ is semiample on $X$. If the contraction defining semiampleness of $D_m$ is Moishezon, the resulting contraction of $\alpha$ is Moishezon as well. In particular, it is unnecessary to lift a program on $S$ to $V$, or to require that a chosen program producing $S_m$ preserve $\alpha_S$ at every intermediate step. \end{proposition} \begin{proof} Nefness descends under the surjective morphism $g$, so $\alpha_S$ is nef. The class of $F_V$ is pulled back from $S$, so $F_V$ is numerically trivial over $S$. It is therefore vertical: an effective horizontal component would restrict to a nonzero effective divisor on a general projective fibre, with positive intersection against a suitable power of an ample class, contradicting numerical triviality. For a birational $g$, verticality is automatic by dimension. First take a projective resolution $\pi:\widetilde V\to V$ of the main component of $V\times_S S^{\mathrm q}$. The induced map $\widetilde g:\widetilde V\to S^{\mathrm q}$ is projective and has connected fibres. Pull back $\alpha_V,D_V,F_V$ along $\pi$, and pull back $\alpha_S,D_S$ along $s$. All displayed class identities are preserved, and $\pi^*F_V$ remains an effective vertical divisor. Lemma~\ref{prog:negative-part-program} gives $N_\sigma(D_V)=F_V$ before this modification. Because $D_V=c\alpha_V+[F_V]$ with $\alpha_V$ nef, Lemma~\ref{prog:negative-part-pullback} then gives \[ N_\sigma(\pi^*D_V)=\pi^*F_V. \] We may therefore replace $(V,S,g)$ by $(\widetilde V,S^{\mathrm q},\widetilde g)$ and suppress the new superscripts in the rest of the proof. The good-model comparison now reads $p^*D_S=q^*D_m+[E]$. No assertion that the crepant subboundary on $\widetilde V$ is effective is needed: this space is used only for classes, currents and the effective divisor $\pi^*F_V$. The lower-dimensional generalized pair is the crepant pullback to the small model $S^{\mathrm q}$. The class identity $[F_V]=g^*(D_S-c\alpha_S)$ shows that $F_V$ is numerically trivial over $S$. Lemma~\ref{prog:vertical-divisor} gives an actual effective real Cartier divisor $F_S$ with $F_V=g^*F_S$. Injectivity of pullback yields \[ D_S=c\alpha_S+[F_S]. \] The established identity $N_\sigma(D_V)=F_V$ and Lemma~\ref{prog:negative-part-base} therefore give $N_\sigma(D_S)=F_S$, and Lemma~\ref{prog:negative-part-pullback} gives \[ N_\sigma(p^*D_S)=p^*F_S. \] On the other hand $q^*D_m$ is nef, so the exceptional-divisor identity applied to the good-model comparison gives \[ N_\sigma(p^*D_S)=E. \] Thus $E=p^*F_S$ as actual divisors. Subtracting them from the comparison proves the exact class equality \[ q^*D_m=c\,p^*\alpha_S. \] This proves the needed preservation of $\alpha_S$ from the final good model, without an assumption about its intermediate models. Choose a contraction $h:S_m\to T$ to a normal compact K\"ahler space and a K\"ahler class $\kappa$ on $T$ with $D_m=h^*\kappa$. Taking a resolution of the main component of $V\times_S W$, and then a common projective resolution with $U$, produces a projective modification $r:R\to X$ and a holomorphic map $\ell:R\to T$ satisfying \[ r^*\alpha=\frac1c\ell^*\kappa. \] For every curve $C$ in a fibre of $r$ this equality implies $\ell(C)$ is a point, since a K\"ahler class has positive degree on every nonconstant image curve. The fibres of the projective modification $r$ are connected projective complex spaces, hence are connected by chains of curves. Therefore $\ell$ is constant on each fibre of $r$. The factorization theorem for a proper map onto a normal space gives a holomorphic map $f:X\to T$ with $\ell=f\circ r$. Pushing forward the class equality by $r$ gives \[ \alpha=f^*(\kappa/c). \] The maps from the main fibre-product component to $S_m$ have connected general fibres; their Stein factorizations are finite birational over the normal space $S_m$, hence have connected fibres everywhere. Together with the connected fibres of $h$, this shows that $\ell$, and therefore $f$, has connected fibres. This is the required semiampleness of $\alpha$. Finally, $R\to S_m$ is projective: it is obtained from the projective map $g$ by base change, followed by projective resolutions and the projective map $q$. If $h$ is Moishezon, choose a projective surjection $H\to S_m$ such that $H\to T$ is projective. A component of $R\times_{S_m}H$ dominating $R$ is projective and surjective over $X$, and its map to $T$ is projective. This is a Moishezon witness for $f$. \end{proof} \begin{lemma}[The contraction step in the dimension induction] \label{prog:base-point-free-induction} Assume $\mathsf C_{d-1}$, $\mathsf B_{d-1}$ and $\mathsf M_{d-1}$. Then $\mathsf C_{d,\mathrm{big}}$ and $\mathsf B_d$ hold. \end{lemma} \begin{proof} \smallskip\noindent \emph{The big non-klt contraction.} Apply the construction of \cite[Section~3]{HX}. On its smooth modification $\nu:U\to X$ it writes \[ \nu^*\alpha=[D_U]+\eta_U, \qquad D_U\geq0,\quad \eta_U\text{ K\"ahler}. \] The induction over the jumping coefficients of the multiplier ideal reduces the new reduced stratum to dimension at most $d-1$. The given contraction on the old non-klt subspace and $\mathsf C_{d-1}$ provide the Moishezon maps of those strata. Every map that is promoted to a projective map in that construction contracts exactly the curves on which $\nu^*\alpha$ vanishes. On each such curve the actual real line bundle $-D_U$ has degree $\eta_U\cdot C$. On a further projective common resolution $q:U''\to U$, choose an effective exceptional divisor $E$ with $-E$ relatively ample and choose $\varepsilon>0$ sufficiently small. Use the decomposition \[ q^*\nu^*\alpha=[D'']+\eta'',\qquad D''=q^*D_U+\varepsilon E,\qquad \eta''=q^*\eta_U-\varepsilon[E]. \] Here $\eta''$ is K\"ahler, and the actual real line bundle $-D''$ has degree $\eta''\cdot C$ on every contracted curve. Restrict this decomposition to the smooth reduced components in the gluing construction. Lemma~\ref{prog:actual-line-polarization} then supplies precisely the relative ampleness needed in place of \cite[Lemma~2.42]{HX}, including the common-resolution step in its Claim~3.5. The exceptional term $-\varepsilon[E]$ makes the positive part $\eta''$ K\"ahler on the common resolution. The exact sequences of multiplier ideals, relative vanishing, finite pushouts, and extension from sufficiently thickened $\operatorname{Supp}D_U$ in that proof then apply unchanged. They give the birational Moishezon contraction in $\mathsf C_{d,\mathrm{big}}$. In particular the gluing step uses neither generalized termination nor a projectivity criterion for an undetected ray. \smallskip\noindent \emph{Preparation for both cases of $\mathsf B_d$.} The modified-big perturbation \cite[Lemma~2.23]{HX} supplies a nef datum that dominates a K\"ahler class on its carrier. Subtracting a sufficiently small multiple of the pullback of a K\"ahler class $\omega_X$ still leaves a nef datum on that carrier. Thus $\alpha$ is a gklt adjoint plus $\varepsilon\omega_X$, and the cone argument of \cite[Lemma~2.45]{HX} makes $\alpha$ NQC. This preparation applies whether or not $\alpha$ is big. \smallskip\noindent \emph{The nef-and-big case of $\mathsf B_d$.} For a gklt pair the multiplier ideal is the unit ideal, so the preceding big non-klt assertion now applies and gives a birational contraction $h:X\to Y$. On a projective resolution $p:W\to X$ chosen projective over $Y$, relative Kawamata--Viehweg vanishing for the effective exceptional divisor $-\lfloor B_W\rfloor$ gives $R^ih_*\mathcal O_X=0$ for $i>0$. Thus $Y$ has rational singularities. Proper birational Bott--Chern descent \cite[Lemma~8.7]{DHP} gives $\alpha=h^*\gamma$. The descended adjoint is gklt, nef and big, and has no trivial curve. The criterion \cite[Theorem~4.3]{HLX} makes $\gamma$ K\"ahler. For completeness, the projectivity input in that criterion is restricted to a map $S\to Z'$ between smooth compact K\"ahler manifolds with rationally connected general fibre: $S$ is the chosen smooth divisor on a resolution and $Z'$ is the resolution of the null-locus component. The smooth-source, smooth-base argument in \cite[Theorem~3.1, Step~1]{CH} applies. Its subsequent relative program starts with this projective map. Thus this application of the criterion uses only the smooth-source, smooth-base projectivity argument and an already-projective relative program. \smallskip\noindent \emph{The non-big case of $\mathsf B_d$.} Use a projective small factorialization and the modified-big perturbation of the pair. The non-pseudo-effectivity of $K_X$ gives an MRC fibration by \cite{Ou2025}. Choose a smooth K\"ahler MRC base $Z$ and a smooth modification $\mu:U\to X$ such that $U\to Z$ is projective. Only the smooth case of \cite[Theorem~3.1, Step~1]{CH} is needed: pullback identifies the holomorphic two-forms since the general fibre is rationally connected. Lemma~\ref{prog:relative-hodge} identifies any $(1,1)$-class on $U$, modulo a class pulled back from $Z$, with an actual real line bundle. Lemma~\ref{prog:projective-cone} identifies its relative rays with rays of the analytic cone; both statements persist along the projective relative construction. Write, as in \cite[Theorem~4.1, Claim~4.1]{HX}, \[ D_U(a)=K_U+\Delta_U+\mathbf M_U+a\alpha_U =(a+1)\alpha_U+F_U, \quad \alpha_U=\mu^*\alpha,\quad \Delta_U,F_U\geq0, \] where $\Delta_U$ is klt and $F_U$ is $\mu$-exceptional. The modified-big replacement is chosen, as in the cited Claim~4.1, so that the nef datum $\mathbf M_U$ on this smooth carrier is K\"ahler. If a further projective resolution is needed, subtract a sufficiently small effective exceptional divisor from its nef part and add that divisor to the subboundary; this preserves the adjoint and the klt inequalities. Choose $a_0$ using the uniform bound of Proposition~\ref{prog:relative-nqc-grid}. It makes the relative $D_U(a_0)$-program $\alpha_U$-trivial and preserves that choice through every step. This is a \emph{projective} relative program from its first step: the nef data are represented by real line bundles modulo $Z$. Lemma~\ref{prog:relative-hodge} applies on every intermediate model over this fixed smooth base. More precisely, put $r:U\to Z$ and choose one actual global real line bundle $L$ with \[ \mathbf M_U+a_0\alpha_U=c_1(L)+r^*\gamma. \] The bundle $L$ is relatively ample, because $\mathbf M_U$ is K\"ahler and $\alpha_U$ is nef. On each of a fixed finite collection of relatively compact Stein charts of $Z$, choose an effective real divisor $\Theta$ representing $L$, with $(U,\Delta_U+\Theta)$ klt. The representatives can retain a positive relatively ample part in their boundaries. Here the equality $K_U+\Delta_U+\Theta\sim_{\R}K_U+\Delta_U+L$ is an actual real linear equivalence of line bundles; the equality with $D_U(a_0)$ modulo $r^*\gamma$ is separately an equality of Bott--Chern classes. Fix these ordinary pairs on the initial charts. Their actual linear equivalences and their Bott--Chern comparisons with $D_U(a_0)$ modulo the fixed base class persist separately throughout a global relative program. Proposition~\ref{prog:projective-good-model} applies: the source is smooth, its adjoint is represented by an actual global real line modulo $Z$, its carrier nef part is relatively ample, and $D_U(a_0)=(a_0+1)\mu^*\alpha+[F_U]$ is pseudo-effective. It gives a finite program $U\dasharrow V$ and a good relative endpoint. This is one global program controlled by fixed ordinary chart pairs. Its construction uses global line algebras, not a choice of unrelated local minimal models. Proposition~\ref{prog:relative-nqc-grid} makes every step $\alpha_U$-trivial with the same value $a_0$. Here is an explicit ample-model comparison that gives the required descent of $\alpha_V$. Both $D_V(a_0)$ and $\alpha_V$ are nef over $Z$. Put $a_1=a_0+1$ and $a_2=a_0+2$. Before the program starts, choose fixed ordinary representatives on the finite chart cover for all three relatively ample nef parts $\mathbf M_U+a_j\alpha_U$, $j=0,1,2$, on one simultaneous log resolution. The $\alpha_U$-crepancy of the constructed program makes its endpoint $V$ a weak model for each of these three fixed ordinary adjoints. Their traces $D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V$ are nef. Corollary~\ref{prog:projective-semiample} therefore makes \[ D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V \] semiample over $Z$. Their zero curves are exactly the common zero curves of $D_V(a_0)$ and $\alpha_V$. Their projective ample-model contractions therefore have the same fibres: these fibres are connected by curves, and each contraction is constant on the fibres of the other. Identify the two normal targets, writing the common contraction as $g:V\to S$. If $D_V(a_j)=g^*\lambda_j$, with $\lambda_j$ relatively K\"ahler over $Z$, subtraction gives \[ \alpha_V=g^*(\lambda_2-\lambda_1)=g^*\alpha_S. \] This is an equality of Bott--Chern classes, not merely of curve degrees. Since the initial program is $\alpha_U$-trivial, it is also $D_U(a_1)$-negative. From now on put $a=a_1$ and suppress the argument in $D_U(a)$ and $D_V(a)$. The maps $V\to Z$ and $S\to Z$ are projective. Moreover $g$ is projective: a relatively ample bundle for $V\to Z$ restricts to an ample bundle on every fibre of $g$, and hence is $g$-ample. Each $D_U(a_j)$ is not big globally: pushing a big such class to $X$ would make $(a_j+1)\alpha$ big. If it were big over the smooth MRC base $Z$, relative-canonical positivity \cite[Theorem~2.48]{HX}, together with the pseudo-effectivity of $K_Z$, would make it big globally. It is therefore not big over $Z$. This property persists to its relative semiample model, so its relative canonical morphism has $\dim S0$, with the map to $S$ retained throughout. \end{lemma} \begin{proof} After the smooth preparation in \cite[Proposition~5.12]{HX}, write the adjoint as \[ A=D+\omega,\qquad D\geq0,\quad \omega\text{ K\"ahler}, \qquad \operatorname{Supp}D=N_+(A). \] Here $N_+(A)$ denotes the reduced divisor with support $\bigcap_{0<\epsilon\ll1}\operatorname{Supp}N_\sigma(A-\epsilon\omega)$; this support is constant for sufficiently small positive $\epsilon$ and is independent of the chosen Kähler class $\omega$ \cite[Lemma~5.3]{HX}. At a nonterminal scaling step, \[ A_i\cdot R_i<0,\qquad (A_i+t_i\omega_i)\cdot R_i=0,\qquad t_i>0. \] Consequently $\omega_i\cdot R_i>0$ and $D_i\cdot R_i<0$. A component of the actual effective divisor $D_i$ supplies a global rational Cartier detector. The supporting contraction follows from $\mathsf B_d$, and its projectivity and flip follow from Proposition~\ref{prog:detected-flip}. This argument does not require $\omega_i$ to remain nef. To apply $\mathsf B_d$ to a gdlt presentation, lower its finitely many floor coefficients on the initial smooth carrier and add the same small class to the K\"ahler nef datum. The adjoint does not change, and the resulting gklt presentation persists through the same negative steps. The original gdlt presentation is retained for adjunction to the floor. Thus the special-termination proof of \cite[Lemma~5.8]{HX} applies with $\mathsf F_{d-1}$ on its smooth lower-dimensional carrier. Its other inputs are adjunction, discrepancy monotonicity, and the local Cartier-index calculation. The last neighbourhood-isomorphism step uses the evaluation-ideal support argument in the \hyperref[prog:special-support]{support comparison above}: the comparison of the first detected flip has support equal to the full inverse image of its exceptional locus, and later comparisons dominate it. One does not infer disjointness of images merely from a vanishing restriction of an arbitrary exceptional divisor. The theta induction of \cite[Proposition~5.11]{HX} now gives a weak model. Applying $\mathsf B_d$ to its nef adjoint makes it good, and Lemma~\ref{prog:weak-to-terminal} gives the log terminal model. All its steps are detected or belong to the already-projective relative extraction in that lemma. For completeness, the negative-part identity needed in this argument is only \[ N(p^*\xi+E)=N(p^*\xi)+E\quad(E\geq0\text{ $p$-exceptional}), \] not the stronger formula $N(p^*\xi)=p^*N(\xi)$ for an arbitrary pseudo-effective class. The comparison with a nef endpoint identifies its exceptional error with the negative part. The formula for $N_+$ follows by writing a K\"ahler form upstairs as $p^*\omega-F$, with $F$ positive on every exceptional prime, and applying the displayed identity to $p^*(\xi-\epsilon\omega)+E+\epsilon F$. Pushforward identifies the nonexceptional coefficients; $\epsilon F$ supplies all exceptional primes. These are exactly the uses in the theta argument. Now put $P=f^*\omega_S$ and suppose $A+cP$ is big for some $c$. Fix $\delta>0$ smaller than the $\omega_S$-degree of every compact integral curve on $S$ and enlarge $c$ so $c>4d/\delta$. Use the following direct preparation, which keeps the original unshifted globally nef datum. On a smooth projective carrier $\nu:W\to X$ write \[ \nu^*(A+cP)=[G]+\omega,\qquad G\geq0,\quad\omega\text{ K\"ahler}, \qquad [K_W+B_W]+M_W=\nu^*A, \] with the non-K\"ahler support property used in the big-class preparation. Here $\operatorname{Ex}(\nu)$ below denotes the reduced divisor formed by all $\nu$-exceptional primes. Write $B_W=B_W^+-B_W^-$. For small $\epsilon>0$ put \[ \begin{split} B'&=B_W^++\epsilon B_W^-+ \epsilon(G+\operatorname{Ex}(\nu)),\\ M'_W&=M_W+c\nu^*P+\epsilon\omega,\\ b&=B_W^++\frac{\epsilon}{1+\epsilon}\operatorname{Ex}(\nu). \end{split} \] The new boundary is klt and $M'_W$ is K\"ahler. Direct addition gives \[ A'=[K_W+B']+M'_W =(1+\epsilon)\bigl([K_W+b]+M_W+c\nu^*P\bigr). \] Moreover $B'\geq b$. In every theta branch the cumulative added boundary satisfies $0\leq C\leq1-B'$, so \[ b+\frac{C}{1+\epsilon} \leq b+\frac{1-b}{1+\epsilon}<1. \] Thus the unshifted presentation $[K_W+b+C/(1+\epsilon)]+M_W$ is genuinely gklt with the \emph{original} globally nef datum. This is the presentation used for the horizontal length bound. No base form is subtracted from an arbitrary newly prepared nef datum. A horizontal negative ray would have $P$-degree at least $\delta$; the bound $0<-A_{\rm unshifted}\cdot C\leq4d$ would contradict $c\delta>4d$. Every constructed step is therefore over $S$. Its projective connected fibres are contracted by $f$, so $f$ factors through its base and through the flip. The final finite extraction is also over $S$. The shifted nef endpoint is semiample by $\mathsf B_d$. Its zero face contains no horizontal curve: decompose such a curve by the unshifted adjoint's cone theorem; nefness of the shifted class forces all summands into the zero face, and a summand of positive $P$-degree again contradicts the length bound. The shifted class is big, so the connected semiampleness map is bimeromorphic. Its connected fibres are Moishezon and are connected by chains of compact curves. The preceding zero-face argument makes the map to $S$ constant on each such curve, hence on every fibre. The analytic factorization lemma therefore factors the map to $S$ through the semiampleness map. Subtracting $c\omega_S$ from the normalized class on its target gives the required relative K\"ahler class for the unshifted adjoint. The effective resolution error from the preparation is removed by the usual negativity comparison, giving a model of the original data. The forward traces and exceptional splittings persist by Lemma~\ref{prog:detected-cohomology}. In the last extraction, the newly extracted prime classes move from the old exceptional span into the pullback span; they are actual rational Cartier classes. Both-sign negativity proves directness as before. \end{proof} \begin{lemma}[Changing the base of a relatively trivial adjoint] \label{prog:graph-good-model} Suppose $g:X\to Y$ is projective, its general fibre is rationally connected, and a gklt adjoint satisfies $A_X=g^*A_Y$. Assume the prepared nef datum is K\"ahler on a smooth carrier. Let $Y\dashrightarrow Y_m$ be the chosen nonextracting good model supplied by $\mathsf M_{d-1}$ over $S$, with its exceptional cohomology splitting, and let $\dim Y2d/\delta$, where every integral curve on $S$ has $\omega_S$-degree greater than $\delta>0$. Set \[ A_c=A_0+cP=D+N,\qquad D=K_U+B_U,\quad N=M_U+cP. \] Since $A_c$ is not big globally, relative-canonical positivity \cite[Theorem~2.48(2)]{HX}, together with pseudo-effectivity of $K_Z$, shows that $A_c$ is not big over $Z$, whether or not it is globally pseudo-effective. The class $N$ is big over $Z$. Consequently $D$ cannot be pseudo-effective over $Z$: otherwise $D+N=A_c$ would be big over $Z$. The smooth projective RC Hodge projection represents every class modulo $Z$ by an actual real line. The ordinary global klt adjoint $D$ is therefore relatively non-pseudo-effective. The projective ordinary Mori construction of Corollary~\ref{prog:projective-mori}, with scaling of the actual relatively ample real line represented by $N$, terminates in a Mori fibre space. Lemma~\ref{prog:relative-hodge} and Lemma~\ref{prog:projective-cone} identify every relative ray with a full analytic ray. Its nef b-data retain the fixed globally nef carrier $U$. First suppose $A_c$ is not pseudo-effective globally. The retained relative-canonical positivity theorem on $h$, together with pseudo-effectivity of $K_Z$, then says that $A_c$ is not pseudo-effective over $Z$. The terminating relative program ends in a Mori contraction at a threshold $\tau>1$. At every step its scaled zero ray has $A_c$-degree negative. Inductively the map to $S$ is retained: if such a ray were horizontal, its $P$-degree would exceed $\delta$, while the gklt cone theorem for $A_0$ gives a rational generator of $-A_0$-degree at most $2d$, contradicting $A_c\cdot R\leq0$. Once $P\cdot R=0$, the step is negative or crepant for $A_0$, so its gklt presentation persists. The same argument applies to the final Mori ray. This would be an $A_c$-negative Mori contraction over $S$, contradicting the preserved pseudo-effectivity over $S$. Hence $A_c$ is pseudo-effective globally. It is not big globally. Relative-canonical positivity now implies that it is not big over $Z$. Its pseudo-effective threshold for $D+tN$ over $Z$ is exactly one: for $t>1$ the class is big, while pseudo-effectivity for $t<1$ would make $A_c$ big because $N$ is big over $Z$. The same finite relative program therefore ends in a Mori contraction $g:U'\to Y$ at threshold one. All birational steps, including any steps at threshold one, and the final zero ray are over $S$ by the preceding length argument with $A_c\cdot R\leq0$. For a zero ray with positive $P$-degree, $A_0$ would again have degree less than $-c\delta$. Thus $Y$ carries a map to $S$ and $\dim Y0$ below the $\omega_c$-degree of every compact integral curve in $X_c$. Fix $\bar\lambda<\delta/(\delta+2d)$ and for $0<\lambda\leq\bar\lambda$ put \[ A_{\lambda,i}=(1-\lambda)H_i+\lambda A_{u,i}. \] The same cone estimate shows more than nefness: for a sufficiently small $\epsilon>0$, uniform on the finite list and the chosen radial interval, $A_{\lambda,i}-\epsilon H_i$ is nef. Indeed on a horizontal $A_{u,i}$-negative rational ray its degree is greater than $(1-\lambda-\epsilon)\delta-2d\lambda>0$; on the nonnegative part of the cone and on the vertical cone it is nonnegative. It follows for every class $\xi\in\overline{\mathrm{NA}}(Y_i)$ that \[ A_{\lambda,i}\cdot\xi=0 \quad\Longleftrightarrow\quad H_i\cdot\xi=A_{u,i}\cdot\xi=0. \] In particular its null curves are exactly those contracted by $\psi_i$. The central data are crepant through these maps over $X_c$; thus the convex data defining $A_{\lambda,i}$ remain gklt with globally modified-big boundary. Apply $\mathsf B_d$ to $A_{\lambda,i}$. Its connected semiampleness map and $\psi_i$ both have Moishezon fibres, and each connected such fibre is connected by chains of compact curves. Equality of their contracted curves makes each map constant on the fibres of the other. The analytic factorization lemma identifies the two maps. Thus the selected relative canonical model is also the canonical model over $S$ on this radial interval. This proves local polyhedrality and the finite local list; compactness gives the global finite list of canonical models and chosen terminal models. \paragraph{Uniform preparation preserving every weak model.} \label{prog:uniform-preparation} Write $A_u=[K_X+B_u+\mathbf M_{u,X}]$ for the adjoints in the parameter polytope. We first record why a weak model of these gklt data extracts no divisor. On a common resolution $p:U\to X$, $q:U\to Y$ put $D=p^*A_u-q^*A_{u,Y}$, using the actual structure-boundary comparison. The weak-model discrepancy inequalities give $p_*D\geq0$, and $-D$ is $p$-nef because $A_{u,Y}$ is nef over the fixed base. Negativity therefore gives $D\geq0$. An extracted prime on $Y$ has coefficient one in the weak-model boundary, whereas its coefficient in the structure boundary of the gklt source is less than one; it would give a negative coefficient of $D$. Hence there is no such prime. Consequently $D$ is $q$-exceptional and the target is itself gklt. Fix a parameter $u_0$. The boundary plus nef part is globally modified big by hypothesis. The modified-big preparation \cite[Lemma~2.23]{HX} gives, on a fixed projective smooth carrier $a:W\to X$, an effective klt boundary $\Gamma$, a K\"ahler class $\gamma_0$, and an effective $a$-exceptional real divisor $F$ such that \begin{equation}\label{prog:eq:uniform-preparation} [K_W+\Gamma]+\gamma_0=a^*A_{u_0}+[F]. \end{equation} Here is the effectivity detail in this use of the preparation. Its new nef datum dominates a K\"ahler form on a carrier. After resolving the new structure boundary, choose an effective exceptional divisor whose negative is relatively ample and subtract a sufficiently small multiple from the pulled-back datum to make it K\"ahler. Add the same multiple to the structure boundary, then replace its negative coefficients by zero and, if needed, add small positive coefficients on the remaining exceptional primes. All coefficients stay below one. Only exceptional coefficients have been increased; their increase is the divisor $F\geq0$ in \eqref{prog:eq:uniform-preparation}. On a sufficiently small closed polyhedral neighbourhood of $u_0$ put \[ \gamma_u=\gamma_0+a^*(A_u-A_{u_0}). \] These classes remain K\"ahler by openness, and the same $\Gamma$ and the same $F$ satisfy \[ [K_W+\Gamma]+\gamma_u=a^*A_u+[F]. \] Thus this is one affine family of gklt pairs on one smooth carrier, with a fixed effective boundary and K\"ahler nef data. Let $\psi:X\dashrightarrow Y$ be \emph{any} weak model at a parameter $u$ in this neighbourhood, and resolve the induced map $W\dashrightarrow Y$ by $r:T\to W$, $q:T\to Y$. Its original comparison is \[ (ar)^*A_u=q^*A_{u,Y}+E_u, \qquad E_u\geq0\text{ is }q\text{-exceptional}. \] Since $\psi$ is nonextracting, every $a$-exceptional prime is exceptional over $Y$. Hence $r^*F$ is also $q$-exceptional, and \begin{equation}\label{prog:eq:prepared-weak} r^*\bigl([K_W+\Gamma]+\gamma_u\bigr) =q^*A_{u,Y}+E_u+r^*F. \end{equation} This proves that $W\dashrightarrow Y$ is a weak model of the prepared pair. In detail, subtract the effective exceptional error on the right from its structure boundary on $T$ and push down to $Y$. The resulting boundary is precisely the transform of $\Gamma$, it is effective, the adjoint is the already existing Bott--Chern class $A_{u,Y}$, and all discrepancies are at least those of the prepared gklt pair. This uses generalized data at the level of currents. On $T$ keep the nef datum $r^*\gamma_u$, and subtract the displayed effective $q$-exceptional error from the prepared structure boundary. Pushing forward by $q$ gives the boundary $\Gamma_Y$ and the trace of this fixed nef b-class. Their sum with the canonical current represents the already existing locally exact adjoint $A_{u,Y}$; the resolution identity defines its structure boundary. The generalized discrepancies have not decreased. This argument neither requires $K_Y+\Gamma_Y$ to be real Cartier nor asserts that the nef trace is separately a Bott--Chern class. It therefore applies to every K\"ahler weak target, including the adjunction strata used in special termination. \paragraph{Every weak model occurs in the finite canonical list.} Choose a small box in $H^{1,1}(W,\mathbb R)$ which spans that vector space and whose addition to the compact family $\{\gamma_u\}$ stays inside the K\"ahler cone. For any $Y$ above choose a K\"ahler class $\eta_Y$ and define $\eta_W=r_*q^*\eta_Y$. Smooth-source cohomology puts $\eta_W$ in $H^{1,1}(W,\mathbb R)$ and gives an actual $r$-exceptional real divisor $D_\eta$ with \[ r^*\eta_W-q^*\eta_Y=[D_\eta]. \] Its negative is $r$-nef, so $D_\eta\geq0$ by exceptional negativity. As $W\dashrightarrow Y$ extracts no divisors, $D_\eta$ is also $q$-exceptional. Scale $\eta_Y$ by a sufficiently small positive number so that $\eta_W$ lies in the fixed box. Adding this identity to \eqref{prog:eq:prepared-weak} yields an effective $q$-exceptional comparison with target class $A_{u,Y}+\eta_Y$, which is K\"ahler over the original base. Thus $W\dashrightarrow Y$ is the canonical model of a member of this one enlarged prepared polytope. Its canonical-model list is finite by the first part of the geography proof. The induced list of maps from $X$ is therefore finite as well. A finite cover of the original compact parameter polytope completes the proof for all weak models. \end{proof} \begin{theorem}[The restricted package in every dimension] \label{prog:restricted-package} The assertions $\mathsf B_d$, $\mathsf C_d$, $\mathsf M_d$ and $\mathsf F_d$ hold in every finite dimension, for the globally nef carrier data and globally modified-big boundary traces specified above. All birational steps used to prove them are detected steps or belong to an already-projective relative construction with the stated relative-line algebraicity. A general semiample morphism is asserted to be Moishezon; projectivity is asserted only when the construction provides an actual relatively ample line. \end{theorem} \begin{proof} The assertions in dimensions zero and one follow from degrees and factorization of maps of compact curves. Suppose the package holds in dimensions less than $d$. Lemma~\ref{prog:base-point-free-induction} first proves $\mathsf C_{d,\mathrm{big}}$ and then $\mathsf B_d$. The former uses $\mathsf C_{d-1}$; the nonbig part of the latter uses only a chosen $\mathsf M_{d-1}$ model and the explicit negative-part comparison. In particular, no lower program is lifted. With $\mathsf B_d$ available, Proposition~\ref{prog:ordinary-step} constructs detected negative steps in dimension $d$. Lemma~\ref{prog:big-good-model}, using the already known $\mathsf F_{d-1}$, gives the big-shift case of $\mathsf M_d$. Lemma~\ref{prog:relative-model} gives its remaining case, using $\mathsf M_{d-1}$ and the projective graph construction. This proves $\mathsf M_d$ over every proper compact K\"ahler base, with its chosen-model invariant. Lemma~\ref{prog:finite-geography}, whose hypotheses are \emph{both} $\mathsf B_d$ and $\mathsf M_d$, now proves $\mathsf F_d$. It remains to prove $\mathsf C_d$ when its nef NQC adjoint $\alpha$ is not big. First use Lemma~\ref{prog:base-presentation-preparation}, preserving the adjoint $\alpha$ and making the nef datum K\"ahler on a smooth carrier. Its multiplier-ideal inclusion allows the given contraction to restrict to the prepared non-klt closed subspace by Lemma~\ref{prog:base-emc}. Next take the projective dlt preparation in \cite[Section~6]{HX}, keeping its crepant structure data. The working space is klt and globally strongly $\mathbb Q$-factorial. The retained crepant structure boundary need not be lc; it is effective, including the extracted primes of coefficient at least one. On a common carrier the nef datum is nef and big. Consequently the total boundary trace is modified big, and $\alpha-c_1(K_X)$ is a big Bott--Chern class. The contraction on the non-klt closed subspace transports through this preparation by Lemma~\ref{prog:multiplier-transport}. Choose an NQC expression with positive coefficients and integral multiples of its rational degree-two summands. Corollary~\ref{prog:uniform-canonical-parameter} provides a single number $t$ such that each negative ray of $K_X+t\alpha$ is $\alpha$-trivial. Choose a K\"ahler scaling class $\omega$ with $K_X+t\alpha+\omega$ nef. Every selected negative ray is ordinary $K_X$-negative; the actual rational canonical line is its detector. Proposition~\ref{prog:ordinary-step} supplies its projective contraction or flip, and the fixed integral grid persists. The descended $\alpha$ remains nef by projective descent of nefness. The driving class is non-pseudo-effective throughout, by the corollary. Its initial pseudo-effective scaling threshold is positive; choose a cutoff $\varepsilon>0$ below that threshold. On the fixed initial carrier the compact segment of globally nef data \[ t\alpha+[\varepsilon,1]\omega \] has modified-big total boundary. At its scaling parameter each working model is a marked weak model of a member of this segment. Each selected step is negative for the fixed driving adjoint, so its discrepancies do not decrease and increase at a valuation affected by that step. An infinite sequence would repeat a marked model by $\mathsf F_d$; the cumulative nonzero effective comparison for the fixed adjoint would then contradict equality of that marking. Hence this program terminates. Non-pseudo-effectivity excludes a nef endpoint, leaving an $\alpha$-trivial ordinary Mori contraction $f:X'\to Z$. The structure data of the prepared contraction problem remain crepant throughout this $\alpha$-trivial program. Their effective traces and globally nef and big carrier datum therefore persist. Lemma~\ref{prog:multiplier-transport} transports the contraction on the non-klt closed subspace through each birational step. Lemma~\ref{prog:bc-descent} gives \[ \alpha_{X'}=f^*\gamma \] in Bott--Chern cohomology. The class $\gamma$ is nef by projective descent; Lemma~\ref{prog:integral-h2} descends the fixed rational NQC summands, and projective multisections preserve their nonnegative degrees. Apply Corollary~\ref{prog:base-emc-induction} to this projective Mori contraction. In its notation the prepared non-klt subspace $N_1$ has Stein space $V(f_*J_1)$ over $Z$. If $N_1$ dominates $Z$, this Stein space is $Z$ itself and its inherited contraction already supplies the contraction of $\gamma$. Otherwise the base presentation has multiplier ideal $J_Z\supset f_*J_1$, so the contraction restricts to $V(J_Z)$ as a closed subspace of $V(f_*J_1)$. Since $\dim Z0$, is finite. If the driving adjoint is not pseudo-effective, the scaling terminates with a Mori fibre space. \end{corollary} \begin{proof} Proposition~\ref{prog:ordinary-step} constructs each step and Lemma~\ref{prog:detected-cohomology} supplies the forward traces from the fixed carrier. Each reached model at a scaling parameter is a marked weak model of that parameter's adjoint. The compact parameter segment satisfies $\mathsf F_d$, so there are finitely many such markings. Strict increase for the fixed driving adjoint rules out repetition. If that adjoint is not pseudo-effective, the positive pseudo-effective scaling threshold gives a fixed positive cutoff containing every parameter of a continuing program. The finite program cannot end nef, hence ends with the claimed Mori map. \end{proof} ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-restricted-induction.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//04-boundary.tex ==== \section{Generation along the dlt boundary}\label{sec:boundary} We prove that the restrictions of a nef adjoint to the separate log canonical strata fit together to generate its restriction to the whole reduced boundary. Semiampleness on the normal components does not by itself supply compatible sections on their union. The proof has three parts. A Mori contraction reduces the obstruction to restriction from a stratum to one comparison of residues. Crepant transport respects all lower residues. Finally a finiteness argument permits us to impose every comparison simultaneously by taking products of sections. Throughout this section, \(n\geq1\) and \(\Ggood_j\) is assumed for every \(j0\), contrary to the preceding inequalities. The strict transform is finite birational over the smooth marked curve, and hence isomorphic to it. Adjunction to that smooth strict transform thus has nonnegative coefficient. This is the original coefficient by the surface-slice residue calculation. The full-weight pair is dlt and has the asserted floor. On an SNC model, at the general point of the center of a divisorial valuation \(v\), take normal parameters \(x_1,\ldots,x_a\), giving an unmarked parameter boundary weight zero. The SNC discrepancy inequality is \[ a(v;\widehat Z,\widehat B_{\widehat Z}) \geq \sum_{i=1}^a(1-b_i)v(x_i). \] Every \(v(x_i)>0\). Thus a zero-discrepancy center is an intersection of unit components. Those intersections meet the isomorphism locus, while every exceptional component has coefficient below one. To produce a dlt resolution for the restricted map, principalize its nonisomorphism locus and resolve together with the ordered SNC boundary. That locus contains no entire log canonical center, so the displayed discrepancy inequality makes every new exceptional coefficient strictly below one. This is the dlt resolution criterion. It also identifies the unit primes downstairs with the next incident strata. Finally let \(R_{\rm all}\) be the whole unit union on \(\widehat Z\). Sections on its smooth components which agree on all their SNC intersections glue by the elementary equalizer for the coordinate ideals of an SNC union. Even iterated residues make the equalities independent of the chosen chain. Apply the floor-connectedness lemma to the full-weight pair and this full unit union. It gives \((p_Z)_*\mathcal O_{R_{\rm all}}=\mathcal O_{C_Z}\). Projection formula therefore descends the glued section uniquely to the restricted invertible line on the reduced \(C_Z\). The proof with \(Z=V\) gives the assertion for \(S\). \end{proof} Each boundary stratum has dimension less than \(n\). The class of \(J_Z:=K_Z+B_Z\) is nef, since it is the restriction of the nef adjoint class. Proposition~\ref{bir:resolution-transfer}, applied using \(\Ggood_{\dim Z}\) on a log resolution, makes the actual rational line \(J_Z\) semiample. Increase \(q\) once for all so that \(L_Z:=\mathcal O_Z(qJ_Z)=\mathcal O_V(qJ)|_Z\) is generated for every boundary stratum. Its map and Stein factorization give \begin{equation}\label{floor:semiample-map} f_Z:Z\longrightarrow Y_Z,\qquad L_Z=f_Z^*N_Z, \end{equation} where \(Y_Z\) is normal projective, \(f_Z\) has connected fibers, and \(N_Z\) is ample. These are actual line identities. We call \(C_Z\) \emph{vertical} if \(f_Z(C_Z)\ne Y_Z\); the empty floor is vertical. If \(C_Z\) dominates \(Y_Z\), then for every \(k\geq1\) the restriction \[ H^0(Z,L_Z^k)\longrightarrow H^0(C_Z,L_Z^k|_{C_Z}) \] is injective. Indeed, every section comes from \(Y_Z\); if its pullback vanishes on \(C_Z\), surjectivity makes it vanish at every point of \(Y_Z\). Fix a K\"ahler class \(c\) on \(V\), and let \(c_Z\) be its restriction to each stratum. For boundary strata \(Z,Z'\) of the same dimension, an \emph{arrow} is a proper bimeromorphic comparison with a projective resolution whose source is smooth and compact K\"ahler, \[ \begin{tikzcd}[column sep=large] & T\arrow[dl,"p"']\arrow[dr,"p'"]&\\ Z&&Z' \end{tikzcd} \] such that the two pulled-back adjoint lines are equal as invertible subsheaves of meromorphic \(q\)-pluricanonical forms on \(T\), and \begin{equation}\label{floor:class-comparison} p^*c_Z-(p')^*c_{Z'}\in\NS(T)_{\R}. \end{equation} Here \(\NS(T)_{\R}\) is the real span of first Chern classes of holomorphic lines, with torsion removed. The degree-two formula for a smooth modification says that its new classes are exceptional divisor classes. It follows that Equation~\eqref{floor:class-comparison} is independent of further resolution and is preserved by composition. These arrows therefore form a groupoid on the finite set of strata of each dimension. An arrow \(\gamma:Z\dashrightarrow Z'\) transports sections by \[ (p')^*(\gamma_*s)=p^*s. \] Normality gives existence and uniqueness. Transport commutes with multiplication and identifies the complete systems and their Stein targets in Equation~\eqref{floor:semiample-map}. The equality of meromorphic lines defines this transport of adjoint sections; the class congruence will control self-arrows when the floor is vertical. \subsection{One residue comparison suffices for restriction} The only obstruction to extending a section from \(C_Z\) is its possible variation among the connected components of a general floor fiber. We show that all these components meet one general Mori fiber, whose floor is connected or consists of two points. Thus at most one residue comparison is needed. \begin{lemma}[Restriction and a Mori link]\label{floor:restriction} For each positive-dimensional boundary stratum \(Z\), there is either no comparison or one arrow between two components of \(C_Z\), allowing a component to be compared with itself, with the following property. In all sufficiently large divisible degrees \(k\), a section of \(L_Z^k|_{C_Z}\) extends to \(Z\) if its two residues agree under that arrow. With no arrow, every such section extends. The degree bound is uniform over sections. When present, the arrow compares the two coefficient-one points on general \(\PP^1\) fibers of a projective Mori contraction on a bimeromorphic compact K\"ahler model of \(Z\). \end{lemma} \begin{proof} We use the dimension-free torsion-free restriction lemma \cite[Lemma~7.1]{BL}. In its notation, for an effective compact K\"ahler lc pair \((T,G)\), klt away from its reduced floor \(C\), and a connected map \(f:T\to P\) to a normal compact base from which an actual positive adjoint multiple is pulled back, it asserts that \(R^1f_*\mathcal O_T(-C)\) is torsion-free. The ideal sequence gives \begin{equation}\label{floor:restriction-obstruction} \mathscr Q:=\coker(\mathcal O_P\longrightarrow f_*\mathcal O_C) \lhook\joinrel\longrightarrow R^1f_*\mathcal O_T(-C). \end{equation} For \(T=Z\) and \(P=Y_Z\), if \(C_Z\) is vertical, \(\mathscr Q\) has proper support and is zero. Serre vanishing for the kernel of \(\mathcal O_P\to f_*\mathcal O_C\), tensored with \(N_Z^k\), extends every restriction in a uniform large-degree tail. If \(C_Z\) dominates, the same display says that a section extends as soon as its values are constant on the reduced floor fiber over a dense open of \(Y_Z\): its class in the torsion-free \(\mathscr Q\otimes N_Z^k\) then vanishes, and the local lifts glue. They are unique because \(C_Z\) dominates \(Y_Z\), so \(\mathcal O_{Y_Z}\to(f_Z)_*\mathcal O_{C_Z}\) is injective. This uses neither reduced special scheme fibers nor base change at their points. Suppose now that \(C_Z\) dominates \(Y_Z\). There is an effective rational Cartier divisor \(D\) with \(\Supp D=\Supp C_Z\): restrict to \(Z\) the globally \(\Q\)-Cartier floor primes of \(V\) not containing \(Z\). For small rational \(\epsilon>0\), \((Z,B_Z-\epsilon D)\) is effective klt. Apply Corollary~\ref{prog:small-model} to obtain a projective small strong \(\Q\)-factorialization \(\rho:Z_0\to Z\), crepant also for the full pair. Write \(\Delta_0\) for its full boundary and \(D_0=\rho^*D\). The full adjoint \[ P=K_{Z_0}+\Delta_0 \] is the actual semiample rational line pulled back from \(Y_Z\), and the lowered klt adjoint is \(J_{\rm low}=P-\epsilon D_0\). Put \(d=\dim Z\), choose \(m>0\) with \(mP\) Cartier and generated, and take \(b>4md\). Consider \[ H=J_{\rm low}+bP=(1+b)P-\epsilon D_0. \] On a general fiber of the map to \(Y_Z\), its class is \(-\epsilon\{D_0\}\). This is the negative of a nonzero effective divisor because the floor dominates \(Y_Z\), so its pairing with a K\"ahler power is negative. If \(H\) were pseudo-effective, a positive current representing it would restrict to a positive current on almost every resolved smooth fiber, contradicting this pairing. Thus \(H\) is not pseudo-effective. Apply Lemma~\ref{prog:nef-trivial} to the lowered klt pair and the nef rational line \(P\). Its \(H\)-program with K\"ahler scaling has ordinary \(J_{\rm low}\)-negative steps, is finite, and ends in a projective Mori contraction. Every step, including the final Mori ray, is \(P\)-trivial. The same actual Cartier line \(mP\) descends at each birational step. Consequently its semiample map to the fixed base \(Y_Z\) persists on every working model. That map is constant on each connected final Mori fiber and hence factors through the Mori contraction. We obtain \[ T\xrightarrow{u}W\xrightarrow{g}Y_Z. \] Both maps have connected fibers. The full adjoint on each model is the trace of \(P\). On a common resolution of a birational step, its natural meromorphic canonical comparison is an exceptional divisor with zero class, since the two actual lines descend from the same line on the contraction base. Exceptional negativity applied to both signs makes that divisor zero. Thus the full pair remains crepant, including its meromorphic adjoint identifications. The full transformed pair \((T,G)\) is effective lc and is klt away from its full floor \(C^+\): there it agrees with the transformed lowered klt pair. The program is nonextracting, so the transformed divisor \(D^+\) has \(\Supp D^+=\Supp C^+\). Since the full adjoint is pulled back from \(Y_Z\), the lowered adjoint on a \(u\)-fiber is \(-\epsilon D^+\). Hence \(D^+\) is relatively ample over \(W\). Compare the connected components of a general floor fiber on the original small model and on \(T\). On a common projective log resolution the full crepant subboundary is the same. Its unit union maps to both floors with connected fibers by \cite[Lemma~2.1]{BL}. Proper closed surjections with connected fibers preserve connected components, also after restricting over a point of \(Y_Z\). Thus the connected components of the two underlying floor fibers correspond. Every connected component of the floor fiber on \(T\) meets one common general Mori fiber. Indeed \(C^+\to W\) is surjective because \(C^+\) is the support of the relatively ample \(D^+\). Apply Equation~\eqref{floor:restriction-obstruction} to \(u\). Surjectivity of the floor makes \(\mathcal O_W\to u_*\mathcal O_{C^+}\) injective; the displayed torsion-free cokernel then makes \(u_*\mathcal O_{C^+}\) torsion-free. Let \[ C^+\longrightarrow\overline C\longrightarrow W \] be the Stein factorization. The finite space \(\overline C\) is reduced, and every irreducible component of \(\overline C\) dominates \(W\). To see the last implication algebraically, a vertical minimal prime of a finite reduced algebra over a domain would give, by prime avoidance, a nonzero element killed by a nonzero element of the domain. After shrinking \(Y_Z\) to a dense open, the fiber-dimension theorem gives every component of \(\overline C_y\) dimension \(\dim W-\dim Y_Z\). The fiber \(W_y\) is irreducible of that dimension: a resolution of \(W\) still has connected fibers over \(Y_Z\), and a general one is smooth and connected and surjects onto \(W_y\). Finiteness now makes each component of \(\overline C_y\) dominate \(W_y\). Consequently each connected component of \(C^+_y\) meets \(u^{-1}(w)\) for the same general \(w\in W_y\). The reduced general fiber \(F\) of \(u\) is irreducible by the same resolution argument. If \(\dim F\geq2\), the support of the effective ample Cartier multiple of \(D^+|_F\), whose support is \(C^+\cap F\), is connected. One elementary proof, valid even if \(F_{\rm red}\) is not normal, cuts by general hyperplanes to an integral projective surface. If the ample divisor split into disjoint nonzero Cartier parts \(D_1,D_2\), their pullbacks to a resolution would be orthogonal, while the projection formula gives \(v^*(D_1+D_2)\cdot v^*D_i>0\) for both \(i\). Thus both \((v^*D_i)^2>0\), contradicting the surface Hodge index theorem. If \(\dim F=1\), choose the fiber also off the singular, boundary intersection, and ramification loci. It is a smooth connected curve and \[ 0=\deg(K_F+G|_F)=2g(F)-2+\deg(G|_F). \] There is a unit point, so \(F\simeq\PP^1\). Its floor has one or two points; if there are two they exhaust the horizontal boundary. In every connected case the preceding common-fiber observation makes the general floor fiber connected, so no comparison is needed. In the remaining case the two unit points represent all its connected components. Normalize the horizontal floor primes and take their Stein factorizations over \(W\). If there are two primes, each finite Stein map has degree one and is an isomorphism over normal \(W\); their main fiber product gives a proper bimeromorphic comparison. If there is one prime, its normal Stein space is a double cover of \(W\). The exchange on its general fiber extends over the branch locus: the reduced horizontal non-diagonal component of the double fiber product has finite birational projections to the normal cover and hence both projections are isomorphisms. Its graph lifts to the normalized prime. Composing with the program graphs transports the comparison to two original normal floor primes, possibly the same one. All graph projections are projective, and the primes survive birationally because the program extracts no divisors. This comparison is crepant as a meromorphic residue identity. On a smooth ruled open, order the markings after an \'etale local cover, choose a base volume \(\xi\), and put the markings at \(z=0,\infty\). A pulled-back \(q\)-pluriadjoint frame has the form \[ a(w)(dz/z)^{\otimes q}\otimes\xi^{\otimes q}. \] Its two residues are \(a(w)\xi^{\otimes q}\) and \((-1)^q a(w)\xi^{\otimes q}\), which agree. This is the two-marking Poincar\'e-residue comparison of \cite[Section~3, Definition~13 and Proposition~14]{Kollar2013Sources}. On a resolved graph both restricted actual lines are pulled back from \(Y_Z\); their meromorphic embeddings agree on this dense open and hence everywhere, including along exceptional primes. Equal residues therefore give a constant value on the entire general floor fiber. The horizontal consequence of Equation~\eqref{floor:restriction-obstruction} extends the section. Normality descends it from the small model. Its restriction is the prescribed section on the original reduced floor, since every original floor prime is covered birationally and the two restrictions agree after that pullback. It remains to check Equation~\eqref{floor:class-comparison}. Let \(\widehat T\) be a common smooth resolution of the parent and Mori models and let \(a,b:H\to\widehat T\) be the two maps from a resolved branch graph. The resolution of the Mori model is an isomorphism near a general entire \(\PP^1\) fiber: its centers have codimension at least two, hence cannot dominate the ruling base. Every holomorphic two-form on \(\widehat T\) comes from the base on that ruled open. In fact its vertical one-form terms vanish on \(\PP^1\), and its remaining coefficients are constant on that compact fiber. Thus \(a^*-b^*\) kills \(H^{2,0}\), and also \(H^{0,2}\) by conjugation. This rational Hodge map on degree two has image of type \((1,1)\). The Lefschetz \((1,1)\) theorem puts its real image in \(\NS(H)_{\R}\). Apply this to the pulled-back parent class \(c_Z\). The link is therefore an arrow as defined above. \end{proof} \subsection{Transport respects every lower residue} An arrow can contract a divisor to a higher-codimension stratum. The next local observation explains how to compare residues even in that case. All valuations in its proof are local monomial valuations in an SNC chart. \begin{lemma}[Unit strata above an SNC stratum]\label{floor:unit-strata} Let \(p:T\to Z\) be a projective log resolution of a dlt pair, with crepant SNC subboundary on the smooth \(T\). Let \(R\subseteq T\) be a unit stratum and let \(D=p(R)\) be its center. Then \(D\) is a dlt stratum. There is a unit stratum \(R'\subseteq R\) such that \(p(R')=D\) and \(R'\to D\) is birational. In the iterated residue comparison along \(R'\), the normal logarithmic Jacobian is \(\pm1\). \end{lemma} \begin{proof} The center is a dlt stratum by the SNC discrepancy calculation in the proof of Lemma~\ref{floor:adjunction}. Work over the generic SNC part of \(D\), of codimension \(a\), with normal coordinates \(x_1,\ldots,x_a\) for the unit divisors. The zero-discrepancy divisorial valuations centered there are exactly the primitive rational monomial rays in \(\R_{>0}^a\). Here is a local verification that also fixes their normalization. The discrepancy inequality in the proof of Lemma~\ref{floor:adjunction} says on every log-smooth model that the center of a zero-discrepancy valuation is exactly a unit stratum: an additional unmarked normal parameter, or a parameter with coefficient below one, would give positive discrepancy. Blow up that stratum. If the positive normal orders of the valuation are \(w_1,\ldots,w_b\), the chart indexed by a smallest order \(w_k\) replaces these by \(w_k\) and the positive members of \(w_i-w_k\). The exceptional divisor is again unit, and the center is exactly the new unit stratum. While there is more than one positive order their sum decreases. Eventually the center is the generic point of one unit divisor, where a normalized divisorial valuation has order one. Reversing these ordinary toric blowups proves monomiality and primitive normalization. Conversely this subtraction algorithm for any primitive positive integer vector terminates at order one, and reversing it extracts that monomial valuation. Comparison on a common graph gives uniqueness of the valuation with those weights and of its center on any model. For a unit component \(E\) meeting the inverse image of the generic SNC part of \(D\), put \[ v_E=(\ord_E x_1,\ldots,\ord_E x_a)\in\Z_{\geq0}^a. \] If the unit components through a unit stratum \(Q\) meeting that inverse image are \(E_1,\ldots,E_b\), define \[ \phi_Q:\R_{\geq0}^b\longrightarrow\R_{\geq0}^a,\qquad (t_1,\ldots,t_b)\longmapsto\sum_{j=1}^b t_jv_{E_j}, \qquad \sigma_Q=\phi_Q(\R_{\geq0}^b). \] The source carries its standard lattice \(\Z^b\). For the fixed resolution \(T\), these cones subdivide the orthant over its interior. On rational rays, \(\phi_Q\) sends a monomial valuation on \(T\) to the same normalized valuation on \(Z\), so it is injective on each cone. Interiors of different cones are disjoint, because the center and the weights of a monomial valuation on \(T\) are unique. Every rational ray in the orthant occurs: view its valuation on \(T\) and use the same SNC description there. There are only finitely many cones over the generic part of \(D\); density of rational rays therefore gives coverage. Taking subsets of unit components gives the faces, and uniqueness of valuations also on faces identifies intersections as common faces. Consequently \(\sigma_R\) is a face of an \(a\)-dimensional cone \(\sigma_{R'}\) with the same downstairs center, for a unit stratum \(R'\subseteq R\). Let \(E_1,\ldots,E_a\) be the unit components through \(R'\). The integer matrix \(A=(\ord_{E_j}x_i)\) of this full cone is unimodular. Indeed every positive primitive integral vector in the source cone is the normalized vector of a divisorial valuation, so its image is primitive. If a prime divided \(\det A\), a nonzero kernel vector modulo that prime, lifted to a positive primitive integral vector, would have nonprimitive image. Hence \(\det A=\pm1\). The strata \(R'\) and \(D\) have the same dimension. We next prove that their generically finite map has degree one. In local coordinates at a general point of \(R'\), \[ x_i=u_i(y,z)\prod_{j=1}^a y_j^{A_{ij}}, \] where the \(u_i\) are units, the \(y_j\) are normal coordinates, and \(z\) are coordinates along the stratum. The tangential map is generically \'etale in characteristic zero. Since \(A^{-1}\) is integral, replace \(y_j\) by \(\widetilde y_j=y_j\prod_i u_i^{(A^{-1})_{ji}}\). Then \(x_i=\prod_j\widetilde y_j^{A_{ij}}\), and the map consisting of \(\widetilde y\) and the downstairs tangential coordinates is locally biholomorphic at a general \'etale point of \(R'\to D\). Two distinct points over a general point of \(D\) would therefore give two lifts of nearby target points whose normals approach with any fixed positive rate vector in the interior of this cone. Varying the nonzero leading coefficients fills an open set of nearby torus points, so such a target can be chosen in the locus where \(p\) is an isomorphism. This contradiction proves that \(R'\to D\) is birational. Finally, after taking residue along \(R'\), the normal coefficient of \(\bigwedge_i dx_i/x_i\) is \(\det A\) times that of \(\bigwedge_jdy_j/y_j\); terms differentiating the units vanish in that residue. Since \(\det A=\pm1\), the even pluriresidue removes the sign. \end{proof} \begin{lemma}[Preservation of lower restrictions]\label{floor:transport} Fix \(0\leq d0\), choose a common log resolution \(p:T\to Z\), \(p':T\to Z'\) of the arrow. For a target floor prime \(D'\), start with its strict transform \(R\). It is a unit stratum and maps birationally to \(D'\). Its source center \(p(R)\) is a stratum by Lemma~\ref{floor:unit-strata}. If \(R\to p(R)\) is not birational, that lemma replaces \(R\) by a unit stratum inside it which maps birationally onto the same source center. The target center may then shrink. If the map to that center is not birational, apply the lemma on the target side, and continue alternately. Each necessary replacement strictly decreases dimension. The process therefore ends with one unit stratum mapping birationally to lower strata \(D_0,D_0'\) on the two sides. The image in the common system base remains unchanged throughout. Indeed the two maps from the resolved graph to the Stein targets are identified by the arrow, and a replacement keeps its entire image on the side then being treated. Initially the image was \(f_{Z'}(D')\). The terminal comparison of \(D_0\) and \(D_0'\) is an arrow: restricting the actual meromorphic identification to iterated residues gives crepancy, with the logarithmic determinant sign removed by Lemma~\ref{floor:unit-strata}; restricting the line Chern classes in Equation~\eqref{floor:class-comparison} gives the same condition modulo \(\NS_{\R}\). Lower invariance thus identifies the transported and assigned residues on \(D_0'\). This equality determines the entire section on \(D'\). The Stein map of \(L_{D'}=L_{Z'}|_{D'}\) is the Stein factor of \(f_{Z'}|_{D'}\), because the line induced on that Stein space is ample; its target is finite over \(f_{Z'}(D')\). The image of \(D_0'\) in this target is a closed irreducible subset of full dimension, since it still maps onto \(f_{Z'}(D')\), and therefore is the whole target. Both sections on \(D'\) come from it. Equality after restriction to \(D_0'\) proves equality on \(D'\). The same construction shows that the image of every target floor prime in the system base is the image of a lower source stratum, and the reverse statement follows from the inverse arrow. A floor dominates the system base on one side exactly when it does on the other. This proves the last assertion. \end{proof} \subsection{Finite actions on vertical strata} Once lower-dimensional sections are compatible and invariant, Lemma~\ref{floor:transport} preserves their restrictions under every arrow. If \(C_Z\) dominates \(Y_Z\), injectivity of restriction forces any such transported extension to equal the assigned extension. Thus only vertical strata require the following argument for finite image. \begin{proposition}[Finite image of self-arrows]\label{floor:finite-action} Let \(Z\) be a boundary stratum with \(C_Z\) vertical. For all sufficiently large divisible \(k\), the self-arrows of \(Z\) have finite image on \(H^0(Z,L_Z^k)\). \end{proposition} For \(m=qk\), a pluricanonical eigenform \(s\in H^0(Z,L_Z^k)\) with \(\int_Z|s|^{2/m}<\infty\) becomes the \(m\)-th power of a holomorphic top form on a smooth resolution of a cyclic root cover, so the auxiliary proposition below makes its eigenvalue a root of unity. A uniform bound on the middle Betti numbers of selected resolutions of these covers in the fixed degree will then bound the possible orders; boundedness of the representation will give finite image. The auxiliary proposition uses the polarization congruence in Equation~\eqref{floor:class-comparison}. \subsubsection{Top-form scalars} \begin{proposition}[A polarized top-form scalar]\label{floor:top-form-scalar} Let \(U\) be a smooth connected compact K\"ahler manifold of dimension \(d0\) sufficiently divisible that the connected semiample map and its Stein target satisfy \[ f:U'\longrightarrow Y,\qquad \mathcal O_{U'}(\ell P)=f^*N, \] where \(Y\) is normal projective and \(N\) is ample, and that Lemma~\ref{bir:sections} identifies every graded piece in \[ R_\ell(U):=\bigoplus_{j\geq0}H^0(U,j\ell K_U) \simeq \bigoplus_{j\geq0}H^0(Y,N^j). \] This is the full section ring of the ample line \(N\), so \(Y\simeq\Proj R_\ell(U)\). Passing to a further divisible Veronese gives the same Proj. We call \(Y\) the \emph{canonical Iitaka base}. A bimeromorphic self-map of \(U\) acts on this graded ring and therefore induces an automorphism of \(Y\). \begin{lemma}[Finite action on the canonical Iitaka base]\label{floor:finite-base} Let \(U\) be a smooth connected compact K\"ahler manifold of dimension \(d0\) in the interior, since \(\chi_R\) diverges at the boundary and \(\chi\) is bounded on the closed disk. At that point, \[ 0\ge\partial_z\partial_{\bar z}\log(\chi/\chi_R) \ge\frac{\kappa}{2}(\chi-\chi_R)>0, \] a contradiction. Thus the Ahlfors--Schwarz comparison gives \(\chi(0)\le4/(\kappa R^2)\) \cite{Ahlfors1938}. Letting \(R\to\infty\) forces the metric to vanish on \(\C\). The argument is local and allows zeros of the induced metric, so the local representations just constructed suffice. Consequently their period maps are locally constant along each such orbit. In any holomorphic volume frame the logarithm of the coefficient in Equation~\eqref{floor:hodge-density} is harmonic along each orbit: the top Hodge line is flat there, and \(\sigma\) is a nonzero holomorphic multiple of a flat generator. This is incompatible with finite mass. Diagonalize the one-parameter action in projective coordinates, and choose two coordinates nonzero at a general point with different weights, of difference \(a\ne0\). A small smooth transverse slice \(Q\) on which their ratio is one gives a holomorphic local biholomorphism onto its image \[ \Phi:\C^*\times Q\longrightarrow Y^* \] with fibers of size at most \(|a|\): the ratio at \(\Phi(z,t)\) is \(z^a\). The entire \(\C^*\) factor is allowed because \(Y^*\) is invariant. In the product frame \((dz/z)\wedge dt_1\wedge\cdots\wedge dt_{\dim Q}\), write \(\Phi^*\nu\) with coefficient \(\rho(z,t)>0\). For fixed \(t\), \(\log\rho(z,t)\) is harmonic on \(\C^*\). Its circular mean at \(|z|=e^u\) is \(A(t)u+B(t)\). Jensen's inequality therefore gives, with an irrelevant positive normalization, \[ \int_{\C^*}\rho(z,t)\frac{i\,dz\wedge d\overline z}{|z|^2} \ \geq\ c\int_{\R}e^{A(t)u+B(t)}\,du=\infty. \] Tonelli's theorem contradicts the finite mass of \(\Phi^*\nu\), which is at most \(|a|\nu(Y)\) by the bounded covering degree. This proves that \(h\) has finite order. \end{proof} After a power, the scalar problem can thus be restricted to a fiber with Kodaira dimension zero. The polarization congruence then passes to that fiber; the next two lattice arguments control its nonprojective factors. In the next lemma, the hypothesis on the origin of the model is what provides the degree-two exceptional splitting; rational singularities alone are not used for that splitting. \begin{lemma}[The scalar on a symplectic factor]\label{floor:symplectic-scalar} Let \(Y\) be a terminal compact K\"ahler primitive symplectic space of dimension \(2e0\). Since \(\beta\) is nef over \(Y\), exceptional negativity applied to the divisor \(E\), with \(\{E\}=p^*c-[\beta]\), gives \(E=\sum_i e_iE_i\geq0\). Pair with the positive normalization \(W_R=(p^*\eta\,p^*\overline\eta)^{e-1}\). Exceptional classes pair to zero with \(p^*c\) by projection formula, and hence for some constant \(b_e>0\) \begin{equation}\label{floor:positive-bb} q_Y(c)=b_e\left(\int_R\beta^2 W_R+\sum_i e_i\int_{E_i}\beta W_R\right)>0. \end{equation} Every term is nonnegative. The first is strictly positive on the common open where \(\beta\) is positive and the symplectic form is nondegenerate. Set \(H_\Q=H^2(Y,\Q)\) and \(D_\Q=H_\Q\cap H^{1,1}(Y,\C)\). Lefschetz \((1,1)\) on \(R\), together with the line-trace assertion of Lemma~\ref{prog:cohomology-transport}, identifies \(D_\Q\) with the rational span of line Chern classes. It is \(G\)-invariant because \(G\) is a rational Hodge automorphism. The same line-trace assertion applied to the graph congruence gives \(Gc-c\in D_\R\). The line \(\C\eta\) survives in \(H_\C/D_\C\); our immediate aim is to prove that all eigenvalues on this quotient have absolute value one. Write \(\Sigma=(H^{2,0}\oplus H^{0,2})_\R\), the positive real two-plane. There are three possibilities for the restriction of the Lorentz form to \(D_\R\). If \(D_\R\) is negative definite, its orthogonal projection \(c_0\) of \(c\) to \(D_\R^\perp\) is fixed by \(G\) and has positive square by Equation~\eqref{floor:positive-bb}. On \(D_\R^\perp\), the invariant space \(\Sigma\oplus\R c_0\) is positive definite and its complement is negative definite. All eigenvalues there have absolute value one. If \(D_\R\) contains a positive vector, it is nondegenerate with one positive direction. Its orthogonal complement is the positive plane \(\Sigma\) plus a negative-definite space, giving the same conclusion. In either case \(H_\R/D_\R\simeq D_\R^\perp\). In the remaining case \(D_\R\) is negative semidefinite with radical a rational isotropic line \(\ell\). The lattice action on \(\ell\) is \(\pm1\). On \(\ell^\perp/\ell\) the form is the positive plane \(\Sigma\) plus a negative-definite \((1,1)\) space, and the quotient \(H_\R/\ell^\perp\) is dual to \(\ell\). The invariant flag \(0\subset\ell\subset\ell^\perp\subset H_\R\) therefore shows that all eigenvalues on \(H_\R\) have absolute value one; this argument also allows unipotent parts. In all cases the eigenvalues on \(H_\C/D_\C\) have absolute value one. The rational quotient \(H_\Q/D_\Q\) carries the preserved lattice \[ \overline\Lambda_Y=\Lambda_Y/(\Lambda_Y\cap D_\Q), \] where the intersection is saturated. Thus \(\lambda\) is an algebraic integer and all its algebraic conjugates have absolute value one. Kronecker's theorem makes it a root of unity. \end{proof} \begin{lemma}[The scalar on a torus]\label{floor:torus-scalar} Let \(A\) be the integral Hodge automorphism of \(H^1(T,\Z)\) induced by an automorphism of a compact complex torus \(T\). Suppose a positive Hermitian class \(c\in H^{1,1}(T,\R)\) satisfies \((\bigwedge^2 A)c-c\in\NS(T)_\R\). Then \(\det(A|H^{1,0}(T))\) is a root of unity. \end{lemma} \begin{proof} Put \(V=H^1(T,\Q)\) and \(D_\Q=\NS(T)_\Q\). For each eigenvalue \(\theta\) of \(A_\C\), let \(V_\theta\) be its generalized eigenspace and set \(d_\theta=\dim V_\theta\), \(p_\theta=\dim(V_\theta\cap H^{1,0}(T))\). Complex conjugation gives \(p_\theta+p_{\overline\theta}=d_\theta=d_{\overline\theta}\). For \(|\theta|\ne1\), project \(c\) to the block \(V_\theta\wedge V_{\overline\theta}\), with \(\bigwedge^2V_\theta\) understood for real \(\theta\). This projection preserves \(D_{\overline\Q}\). Indeed the Chinese-remainder projectors onto the generalized spaces are polynomials in \(A\) with algebraic coefficients. Apply them to the two tensor slots separately and then alternate. The resulting maps are algebraic linear combinations of rational Hodge maps on \(\bigwedge^2V\); these preserve rational \((1,1)\) classes, hence \(D_\Q\) by Lefschetz \((1,1)\). Separate slot projections avoid any collision of products of eigenvalues in degree two. On this block \(A_2=\bigwedge^2 A\) has the single generalized eigenvalue \(\theta\overline\theta=|\theta|^2\ne1\). The inverse of \(A_2-1\) there is a finite polynomial in its nilpotent part. Project the congruence for \(c\) to the block and apply this inverse. It follows that its block \(c_{\theta,\overline\theta}\) belongs to \(D_\C\). Positivity says that this block is a perfect tensor between \(V_\theta\) and \(V_{\overline\theta}\), pairing their opposite Hodge types: in a basis of \(H^{1,0}\) the relevant matrices are positive-definite principal Hermitian blocks. For real \(\theta\) the corresponding alternating tensor is nondegenerate. The intersection of this block with \(D_\C\) is defined over \(\overline\Q\). Nondegeneracy is the nonvanishing of a determinant, so it contains a nondegenerate tensor with algebraic coefficients. For any Galois automorphism \(\gamma\), the conjugate tensor remains in \(D_{\overline\Q}\), hence is still of type \((1,1)\) for the given Hodge decomposition, and is perfect between \(V_{\gamma(\theta)}\) and \(V_{\gamma(\overline\theta)}\). Opposite-type perfection yields \begin{equation}\label{floor:torus-hodge-count} p_{\gamma(\theta)}+p_{\gamma(\overline\theta)} =d_\theta\qquad(|\theta|\ne1). \end{equation} This does not replace \(\gamma(\overline\theta)\) by \(\overline{\gamma(\theta)}\), and assumes no Galois invariance of the Hodge decomposition. The top-form scalar is the algebraic integer \(z=\prod_\theta \theta^{p_\theta}=\det(A|H^{1,0})\), also when generalized eigenspaces are nontrivial. For any Galois automorphism \(\delta\), reindex by the image roots and pair complex conjugates: \[ \begin{aligned} \log|\delta z| &=\frac12\sum_\theta \bigl(p_{\delta^{-1}(\theta)} +p_{\delta^{-1}(\overline\theta)}\bigr)\log|\theta|\\ &=\frac12\sum_\theta d_\theta\log|\theta| =\tfrac12\log|\det A|=0. \end{aligned} \] For \(|\theta|\ne1\) the second equality uses Equation~\eqref{floor:torus-hodge-count} with \(\gamma=\delta^{-1}\); the other terms vanish. The last equality is unimodularity of the integral action. Kronecker's theorem now proves the assertion. \end{proof} \begin{proof}[Proof of Proposition~\ref{floor:top-form-scalar}] Apply Lemma~\ref{floor:finite-base}. After a power, \(g\) acts trivially on the canonical Iitaka base. Use the modification \(\pi:U'\to U\) and the map \(f:U'\to Y\) in its construction, and restrict over a general point where the graph and the fibration are smooth and the fiber meets the isomorphism locus of \(\pi\). Dividing \(\alpha\) by a local base volume gives a nonzero top form on the smooth compact fiber \(F\), which has \(\kappa(F)=0\); its returning map scales the form by the corresponding power of \(\mu\). A point fiber gives scalar one. For a positive-dimensional fiber, let \(\omega_U\) be a K\"ahler form representing the class in the proposition and put \(\beta_F=(\pi|_F)^*\omega_U\). This is a smooth semipositive form, strictly positive on the dense open where \(\pi\) is an isomorphism. Pulling the original graph congruence to \(U'\) and restricting to the fiber shows that the two pullbacks of \([\beta_F]\) on a common resolved graph of the returning map of \(F\) differ by real line classes. This weaker positivity suffices below. Since \(\dim F\leq d0\) and a compatible invariant generating system of tuples through dimension \(n-1\). \end{proposition} \begin{proof} At the point strata use the same scalar in the canonical zero-form generator \(1\). The even residue convention identifies these generators along every path and under every point arrow. This gives the required system in dimension zero; if there are no points the empty system is understood. Suppose the system has been constructed through dimension \(d-1\). Replace it by powers and their span in a common degree \(k\) chosen so that Lemma~\ref{floor:restriction} applies on every \(d\)-stratum, Proposition~\ref{floor:finite-action} applies on every vertical \(d\)-stratum, and \(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) is generated for every \(d\)-stratum \(Z\). These conditions hold in a common divisible tail: there are finitely many strata, and the last condition is Serre's theorem. For a \(d\)-stratum \(Z\), Lemma~\ref{floor:adjunction} glues the lower tuple to a section on its whole \(C_Z\). Its residues satisfy the one link in Lemma~\ref{floor:restriction}, if that link is present, because the lower tuple is invariant. That lemma extends the section to \(Z\). Let \(\mathcal P_d(k)\) be the vector space of all tuples through dimension \(d\) whose lower part is in \(\mathcal V_{d-1}(k)\) and whose new components restrict to that lower part. Call its elements \emph{pre-tuples}. Its projection to \(\mathcal V_{d-1}(k)\) is surjective, since the finitely many extensions can be chosen independently. This system already generates. To check a point \(z\in Z\), put \(y=f_Z(z)\). If \(y\in f_Z(C_Z)\), choose a floor point over \(y\) and a lower tuple nonzero there. Every extension is the pullback of a section of \(N_Z^k\), so it is nonzero at every point over \(y\), including \(z\). If \(y\notin f_Z(C_Z)\), the chosen generation of \(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) supplies a base section nonzero at \(y\). It pulls back to a section vanishing on \(C_Z\); combine it with the zero lower tuple and zero components on the other new strata. This also treats an empty floor. Surjectivity to the lower system preserves generation at lower points. We impose invariance in dimension \(d\). For a dominating floor, injectivity of restriction and Lemma~\ref{floor:transport} already make every pre-tuple invariant on that stratum. Verticality is invariant under arrows, so the remaining orbits consist entirely of vertical strata. For each such orbit choose a representative \(Z\), and let \(G_Z\) be the finite isotropy image on \(H^0(Z,L_Z^k)\) given by Proposition~\ref{floor:finite-action}. For a pre-tuple with component \(s_Z\), form the norm \[ P_Z(s_Z)=\prod_{g\in G_Z}g(s_Z) \in H^0(Z,L_Z^{k|G_Z|}). \] Choose a common positive integer \(a\) divisible by all \(|G_Z|\), use \(P_Z(s_Z)^{a/|G_Z|}\), and transport it to each member of that orbit. This is independent of the chosen arrow: changing that arrow by an isotropy arrow merely permutes the factors in degree \(k\). No finiteness assertion in degree \(ak\) is needed. On strata with dominating floor use \(s_Z^a\), and on all lower strata use the componentwise \(a\)-th power of the assigned tuple. The resulting tuple is compatible. By Lemma~\ref{floor:transport}, every transported factor in a norm has exactly the prescribed lower restriction. The norm and its indicated power therefore restrict to the \(a\)-th power of that restriction, the same value used on all lower strata and in all other orbits. It is invariant in dimension \(d\) by construction, and remains invariant below it. These tuples still generate. Fix a point \(z\) in an orbit member. For each of the finitely many transported factors, choose a point over \(z\) on a resolution of its comparison graph. Equality of the pulled-back invertible adjoint lines identifies the fiber values. The factor is nonzero at \(z\) exactly when the representative component \(s_Z\) is nonzero at the corresponding point of \(Z\). Evaluation at each such point is a nonzero linear functional on the generating space \(\mathcal P_d(k)\). A finite union of their proper kernels cannot cover a complex vector space. One pre-tuple makes all factors nonzero and gives a norm nonzero at \(z\). At a lower point a nonzero assigned value stays nonzero after its \(a\)-th power. Taking the linear span of all the constructed tuples therefore gives a compatible invariant generating system in degree \(ak\). This completes the induction on \(d\). \end{proof} \begin{proof}[Proof of Theorem~\ref{floor:generation}] Take the system in Proposition~\ref{floor:tuples}. Its components on the prime components of \(S\) agree on every common lower stratum, so Lemma~\ref{floor:adjunction} descends each tuple uniquely to a section of the actual line \(\mathcal O_V(qkJ)|_S\). At any \(x\in S\), choose a prime component through \(x\) and a tuple nonzero there. It is the pullback of the descended value in the same invertible line fiber at \(x\), so that value is nonzero. The finite-dimensional span of these descended sections generates at every point of the reduced \(S\). This is the claimed global generation of an actual Cartier multiple. \end{proof} ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//04-boundary.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-projective-program-input.tex ==== \subsection{Projective programs under the assumption} \label{rank:projective-program-input} The argument for a projective base uses a terminating program to reach a nef adjoint and, after decreasing the nef data, a program ending in a Mori fiber space. The conditional input is termination in the pseudo-effective case. We first state the exact program result and then derive that case from the projective good models in Proposition~\ref{proj:good-models}. \begin{proposition}[A conditional generalized program] \label{proj:generalized-mmp} Assume Assumption~\ref{asm:log-iitaka}. Let \((V,B+\boldsymbol M)\) be a projective generalized dlt rational pair, where \(V\) is \(\Q\)-factorial, \(B\geq0\), and the nef b-divisor \(\boldsymbol M\) is determined by a nef rational Cartier divisor on a projective birational model. Put \(D=K_V+B+M_V\). There is a choice of \(D\)-MMP with ample scaling which terminates. If \(D\) is pseudo-effective, its endpoint \(V_m\) has nef adjoint \(D_m\). If \(D\) is not pseudo-effective, its endpoint has a Mori fiber contraction to a projective variety of smaller dimension. The birational steps and their endpoints stay \(\Q\)-factorial generalized dlt, and the forward birational map extracts no divisors. In the pseudo-effective case, on a common resolution one has an actual rational divisor identity \[ u^*D=v^*D_m+F,\qquad F\geq0\text{ and }v\text{-exceptional}. \] \end{proposition} For the pseudo-effective alternative, we first derive the relative weak Zariski decompositions used by the minimal-model existence theorem. For a projective morphism, first replace its image by its normal Stein factor. We use the relative convention of \cite[Section~2]{TsakanikasXie}: a divisor is pseudo-effective over the base when its restriction to a very general fiber of this surjective morphism is pseudo-effective. An NQC weak Zariski decomposition of \(D\) over the base is a birational equality \(h^*D\equiv_Z P+N\), where \(P\) is a nonnegative real combination of relatively nef rational Cartier divisors and \(N\geq0\). The decomposition below has rational Cartier parts. \begin{lemma}[Relative canonical decomposition]\label{proj:relative-wzd} Assume Assumption~\ref{asm:log-iitaka}. Let \(V\) be a smooth quasi-projective complex variety and let \(f:V\to Z\) be projective, with \(Z\) normal and quasi-projective. If \(K_V\) is pseudo-effective over \(Z\), then on a normal variety \(Y\) with a projective birational morphism \(h:Y\to V\) there are rational Cartier divisors \(P,N\) such that \[ h^*K_V\equiv_Z P+N,\qquad P\text{ nef over }Z,\qquad N\geq0. \] \end{lemma} \begin{proof} We use the construction in the proof of \cite[Proposition~3.3]{External037}. Its absolute input is a nef-plus- effective rational decomposition for the canonical divisor of a smooth projective variety with pseudo-effective canonical class. This is supplied here by Proposition~\ref{proj:good-models} and its common-resolution comparison. Replace the image of \(f\) by its normal Stein factor. This does not change relative numerical classes or contracted curves, and now \(f\) is surjective with connected fibers. Its very general fiber is smooth projective with pseudo-effective canonical divisor, hence is non-uniruled by \cite[Theorem~0.2 and Corollary~0.3]{BDPP}. Compactify the projective morphism and resolve away from \(V\). We obtain \(\bar f:\bar V\to\bar Z\), with \(\bar V\) smooth projective and \(\bar Z\) normal projective, unchanged over \(Z\). If \(\dim Z=0\), \(V\) is already projective and the absolute input proves the lemma. Suppose \(\dim Z>0\), and let \(r:\widetilde Z\to\bar Z\) be a smooth projective resolution. Choose a sufficiently positive very ample divisor \(A_Z\) on \(\bar Z\) and a general \(B\in|2A_Z|\). The pulled-back systems on both \(\widetilde Z\) and \(\bar V\) are base point free. Bertini makes \(r^*B\) and \(\bar f^*B\) smooth nonempty reduced divisors. Their double covers \[ \pi_Z:Z^\sharp\to\widetilde Z, \qquad \pi:V^\sharp\to\bar V \] are smooth integral projective varieties. The canonical formulas are \[ K_{Z^\sharp}\sim_{\Q}\pi_Z^*(K_{\widetilde Z}+r^*A_Z), \qquad K_{V^\sharp}\sim_{\Q}\pi^*(K_{\bar V}+\bar f^*A_Z). \] The first line is big when \(A_Z\) is sufficiently positive. The rational map \(V^\sharp\dashrightarrow Z^\sharp\) has the same very general fibers as \(f\). A covering family of rational curves on \(V^\sharp\) would either dominate a covering family of rational curves on \(Z^\sharp\), or cover its very general fibers by vertical rational curves. Both are impossible. Thus \(V^\sharp\) is non-uniruled and \(K_{V^\sharp}\) is pseudo-effective by BDPP. Apply the absolute decomposition to \(V^\sharp\). Moving the rational principal difference in the canonical formula into the nef part gives an actual rational divisor decomposition for a pullback of \(\pi^*(K_{\bar V}+\bar f^*A_Z)\). Take a common equivariant resolution of that model and its conjugate under the covering involution. On this smooth projective resolution \(g:T\to V^\sharp\), average the two decompositions. We obtain \[ g^*\pi^*(K_{\bar V}+\bar f^*A_Z)=\bar P+\bar N, \qquad \bar P\text{ nef},\quad \bar N\geq0, \] where \(\bar P\) and \(\bar N\) are invariant rational Cartier divisors. Let \(q:T\to\bar Y\) be the finite quotient by the involution. The invariant composite \(\pi g\) induces a projective birational morphism \(\bar h:\bar Y\to\bar V\), with \(\bar Y\) normal and projective. An invariant rational divisor descends by dividing its coefficient at an upstairs prime by the ramification index. The descended divisor is rational Cartier when the original divisor is: after clearing denominators, choose one Cartier equation on a semilocal neighborhood of the finite orbit above a point (a Cartier divisor is principal on a semilocal ring), and multiply all its translates. The product is invariant and has divisor equal to the group order times the original divisor. It descends to a local equation for a multiple of the downstairs divisor. This argument includes stabilizers and ramification primes. Accordingly \(\bar P=q^*P\) and \(\bar N=q^*N\) for rational Cartier divisors on \(\bar Y\). Lifting curves through the finite map shows that \(P\) is nef, and coefficientwise descent shows that \(N\) is effective. Finite pullback is injective on rational divisors, so the displayed identity descends. Restrict it to \(Y=\bar h^{-1}(V)\). The term pulled back from \(A_Z\) is numerically trivial over \(Z\), giving \(h^*K_V\equiv_ZP+N\), as required. \end{proof} \begin{proof}[Proof of Proposition~\ref{proj:generalized-mmp}] Lemma~\ref{proj:relative-wzd} and \cite[Theorem~5.2]{TsakanikasXie}, applied to the generalized pair \((U/Z,0+0)\) with \(U\) smooth, prove relative minimal-model existence for smooth varieties by dimension induction. The hypothesis of that theorem is relative smooth minimal-model existence one dimension lower; the lemma supplies its other hypothesis, an NQC weak Zariski decomposition. The induction begins in dimension zero. Then \cite[Theorem~5.4]{TsakanikasXie} supplies minimal models for every pseudo-effective NQC generalized lc pair in the relevant dimension. Their Theorem~2.7 gives existence in the Birkar--Shokurov sense. The nef data in the statement are NQC, because a nef rational Cartier divisor is a positive multiple of a nef Cartier divisor. In the pseudo-effective case the preceding paragraph therefore gives the Birkar--Shokurov model required by \cite[Theorem~4.2]{TsakanikasXie}. In the other case, the non-pseudo-effective alternative of that theorem applies directly. To meet its scaling hypothesis, take an effective sufficiently positive ample rational divisor \(A\) whose general components have sufficiently small coefficients. On a fixed log resolution carrying \(\boldsymbol M\), these components are transverse to the boundary and preserve generalized log canonicity, while their ample class can be chosen large enough that \(D+A\) is nef. The theorem supplies a terminating MMP starting on \(V\). We verify the step types inductively. Suppose \(V_i\) is \(\Q\)-factorial generalized dlt. By \cite[Remark~2.15]{TsakanikasXie}, each birational step has the form \[ V_i\xrightarrow{g_i}Z_i\xleftarrow{h_i}V_{i+1}, \] where \(g_i\) contracts a negative extremal ray and \(h_i\) is small and projective. Suppose \(g_i\) is divisorial, and write \(R\) for its ray. An exceptional prime \(E\) has \(E\cdot R<0\): otherwise it would be effective, exceptional, and \(g_i\)-nef, contrary to negativity. There is only one exceptional prime. Indeed, a combination of two distinct exceptional primes can be chosen to have degree zero on \(R\); negativity applied to both signs of that combination would make it zero. We show directly that \(Z_i\) is \(\Q\)-factorial. For a prime divisor \(T\) on \(Z_i\), let \(\widetilde T\) be its strict transform, choose a curve \(C\) spanning \(R\), and put \[ a=-\frac{\widetilde T\cdot C}{E\cdot C}\in\Q. \] Choose a positive integer \(m\) for which \(m(\widetilde T+aE)\) is Cartier. Its line has degree zero on \(R\), so \cite[Theorem~1.5]{Xie2022} gives an actual Cartier line \(\mathcal L_T\) on \(Z_i\) and an isomorphism \[ \OO_{V_i}\bigl(m(\widetilde T+aE)\bigr)\simeq g_i^*\mathcal L_T. \] On an open set trivializing \(\mathcal L_T\), the Cartier divisor \(m(\widetilde T+aE)\) is principal upstairs. Pushing this principal divisor identity down through the birational \(g_i\) gives a principal divisor \(mT\) on that open set. Thus \(T\) is rational Cartier. It follows that \(h_i\) is an isomorphism: the pushforward of an \(h_i\)-ample Cartier divisor is rational Cartier on \(Z_i\), and its pullback is the original divisor because \(h_i\) is small. It has degree zero on every contracted curve, so relative ampleness rules out positive-dimensional fibers; a finite birational map to the normal \(Z_i\) is an isomorphism. If \(g_i\) is small, the relatively ample canonical model \(h_i\) is the flip of \cite[Definition~3.2]{HanLi}. Thus the chosen birational steps are divisorial contractions and flips. The underlying variety of a generalized dlt pair is klt, and these steps preserve the \(\Q\)-factorial generalized dlt category \cite[Remark~2.3 and Lemma~3.7]{HanLi}. They introduce no prime divisors on the new models. For completeness, on a common resolution of one step \(V_i\dashrightarrow V_{i+1}\) carrying \(\boldsymbol M\), taking the difference of the two generalized discrepancy formulas cancels their common nef divisor. The resulting difference of the adjoint pullbacks is exceptional over the new model and anti-nef over it. The negativity lemma makes this difference effective. Composing these actual rational divisor comparisons gives \(u^*D=v^*D_m+F\) with \(F\) effective and exceptional over the endpoint. The endpoints in the two cases are the nef and Mori fiber endpoints supplied by the terminating program theorem. \end{proof} ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-projective-program-input.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-rank-one.tex ==== \section{Descent along a fibration} \label{sec:rank-one} Throughout this section we assume Assumption~\ref{asm:log-iitaka}. We prove Proposition~\ref{rank:fibration-case}. Its geometric reduction leads to a fibration whose very general fiber has logarithmic Kodaira dimension zero. In sufficiently divisible degrees the log pluricanonical systems on that fiber are one-dimensional, and a relative generating form determines an adjoint line on the base. We prove the resulting rank-one proposition by a secondary induction on the positive base dimension: after proving pseudo-effectivity on the base, we either lower its dimension or lift a nef line and its entire fixed divisor. The last step proves generation of the remaining nef line. \subsection{Reduction to the rank-one case} \begin{proposition}[The rank-one case] \label{rank:rank-one} Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume \(\Ggood_d\) for every \(d0\). If \(a(X)=n\), then \(X\) is Moishezon and Kähler, hence projective, and Proposition~\ref{proj:good-models} applies. Otherwise resolve the algebraic reduction as a fibration \(g:X'\to W\) to a smooth projective variety of dimension \(a(X)\). Put \(J'=K_{X'}+B'\) for the modified log adjoint. Its restriction to a very general smooth fiber is pseudo-effective. Indeed, one may choose a sequence of positive currents in \(c_1(J')+\varepsilon_j[\omega]\), with analytic singularities and \(\varepsilon_j\downarrow0\), and restrict all of them outside a countable union of proper analytic subsets of the base. By \(\Ggood_{\dim F}\), the restricted adjoint has nonnegative Kodaira dimension. That dimension must be zero. If it were positive, generic coherent base change would give a divisible \(q\) for which \(\mathcal E=g_*\OO_{X'}(qJ')\) has a fiber system with a positive-dimensional image on very general fibers. A sufficiently large ample twist of the coherent sheaf \(\mathcal E\) is generated at the generic point of the projective \(W\). Evaluation would therefore give two sections of \(\OO_{X'}(qJ')\otimes g^*\OO_W(A)\), for one ample \(A\), whose ratio is nonconstant on a general \(g\)-fiber. This ratio is a meromorphic function on \(X'\). Every meromorphic function on \(X'\) factors through its algebraic reduction, a contradiction. Proposition~\ref{rank:rank-one} now applies. Suppose next that \(a(X)=0\), and resolve the fibration in the hypothesis as \(f:(X',B')\to Y\). The base may be replaced by a smooth compact Kähler model, and \(a(Y)=0\), since its meromorphic functions pull back to \(X'\). Write \(d=\dim X'-\dim Y>0\). The same restriction argument and \(\Ggood_d\) show that \[ \kappa(F,K_F+B'_F)\geq0 \] on very general smooth fibers. If this integer is \(j0\), then \(K_S+a_0P_S\) is big for some rational \(a_0>0\); if \(S\) is a point, \(M\sim_{\Q}0\). There is a rational divisor \(A_*\) on \(X\) giving an actual rational line identity \begin{equation} J\sim_{\Q}g^*H+A_*. \label{rank:line-comparison} \end{equation} These data satisfy the following properties. \begin{enumerate} \item The horizontal part \(A_{*,\mathrm{hor}}\) is effective. On a very general smooth fiber it is the zero divisor of a relative generating section of a divisible multiple of \(K_F+B_F\), divided by that multiple. \item For each prime \(D\subset W\), let \(E\) run over the primes of \(X\) dominating \(D\), and put \(a_E=\ord_E(g^*D)\). Then \begin{equation} (A_*)_E\geq0,\qquad \min_{E\to D}\frac{(A_*)_E}{a_E}=0. \label{rank:zero-minimum} \end{equation} There is no assertion here about the sign at a prime whose image has codimension at least two. \item For all degrees \(q\) divisible by one positive integer, multiplication by the relative generating form identifies \[ H^0(W,qH)\simeq H^0(X,qJ) \] and preserves ratios of sections. In the same degrees, for every proper holomorphic \(h:W\to V\), there is a natural injection \begin{equation} (hg)_*\OO_X(qJ)\ \lhook\joinrel\longrightarrow\ h_*\OO_W(qH). \label{rank:relative-injection} \end{equation} \item Let \(\tau:W'\to W\) be a further smooth modification included in another diagram with the preceding properties over the same reference \(X_0\), using the same source-boundary convention. Let \(H'\) be the base line produced by that diagram. Then, as actual rational lines, \begin{equation} H'\sim_{\Q}\tau^*H+E_H, \qquad E_H\geq0\ \text{\(\tau\)-exceptional}. \label{rank:higher-base} \end{equation} \end{enumerate} \end{lemma} \begin{proof} The flattening construction of \cite[Lemma~2.4]{CI} gives the stated reference property, and preserves it after any finite sequence of further base modifications. Its source boundary is the one in \cite[Lemma~2.1]{CI}. We may resolve the discriminant and the horizontal divisor of the relative generator so that all relevant strata are smooth over the dense smooth log locus. Proposition~2.7 of \cite{CI} then gives \(T\) and a rational parabolic Hodge line \(M\). A positive multiple of that line is the highest rank-one Hodge step of a complex summand of a pure real-polarizable variation with an integral lattice. Theorem~3.1 of \cite{CI} gives the factorization \(M=p^*P_S\) and the stated positivity, after a further modification and a positive rational rescaling. These are identities in \(\Pic\otimes\Q\), not merely identities of classes. Here is the order computation that gives Equations~\eqref{rank:line-comparison}--\eqref{rank:relative-injection}. Choose a degree \(m\) for which \(g_*\OO_X(m(K_{X/W}+B))\) has generic rank one and all boundary denominators are cleared. Its reflexive hull is a line on the smooth \(W\). Let \(s\) be a meromorphic relative generating form in a local frame of this line. At the generic point of a prime \(D\), choose a nonvanishing ordinary base volume \(\omega_W\), and set \[ l_E=\frac1m\ord_E(s\wedge g^*\omega_W^m),\qquad \delta_E=B_E,\qquad \alpha_D=\min_{E\to D}\frac{l_E+\delta_E}{a_E}. \] The relative-times-base identification in this formula is taken in the \(m\)-th canonical tensor power. The source is resolved over the generic point of \(D\), so all the primes needed for the divisorial test occur in this minimum. In the notation of the proof of \cite[Proposition~2.7]{CI}, the parabolic order is \[ \beta_D=\min_{E\to D}\frac{l_E+1-a_E}{a_E}, \qquad T_D=\alpha_D-\beta_D. \] Thus the corresponding local frame of \(T+M\) has order \(\alpha_D\). Comparing its pullback with the log adjoint form defines \(A_*\), with \begin{equation} (A_*)_E=l_E+\delta_E-a_E\alpha_D . \label{rank:vertical-order} \end{equation} Changing the generating form multiplies it by a meromorphic base function. This changes both \(\alpha_D\) and \(\beta_D\) by its normalized base order and leaves the comparison invariant. The local comparisons therefore give the rational divisor and the line identity globally. Equation~\eqref{rank:vertical-order} proves Equation~\eqref{rank:zero-minimum}. Horizontally the comparison is precisely the zero divisor of the fiberwise log section, so it is effective and has the asserted restriction. The same order computation proves the relative injection. In a divisible degree \(q\), a meromorphic base coefficient \(\varphi\) gives a regular base section at \(D\) exactly when \[ \ord_D(\varphi)+q\alpha_D\geq0. \] Its corresponding upstairs orders at the primes \(E\to D\) are \[ a_E\ord_D(\varphi)+q(l_E+\delta_E). \] Their simultaneous nonnegativity is equivalent to the preceding inequality. Conversely, an upstairs section restricts to a multiple of the generator on general connected fibers, so its ratio with the generator descends meromorphically to \(W\). The order inequalities make it regular outside codimension two on \(W\), and normality extends it. Consequently \[ g_*\OO_X(qJ)\ \lhook\joinrel\longrightarrow\ \OO_W(qH). \] Applying the left exact functor \(h_*\) proves Equation~\eqref{rank:relative-injection}. The reverse global comparison follows from the reference property: its only initially untested poles are on \(g\)-contracted primes, all exceptional over \(X_0\); pushing to \(X_0\), extending in codimension two, and pulling back with the reduced-exceptional boundary removes them. This is the global comparison in \cite[Proposition~2.7]{CI}. Finally consider \(\tau:W'\to W\). The parabolic rational line pulls back under SNC modifications, as in the construction preceding \cite[Theorem~3.1]{CI}. The coefficient \(T_D\) at an old strict transform is unchanged. Indeed the sheaf of regular relative adjoint forms inside the common meromorphic generator line is unchanged over the generic point of \(D\) by the log modification formula. Equivalently, the old inequalities above already test all vertical primes there; in SNC coordinates logarithmic pullback preserves those inequalities, so new source exceptional primes impose no stronger one. This keeps \(\alpha_D\) unchanged in compatible frames, while parabolic pullback keeps \(\beta_D\) unchanged. Over a new \(\tau\)-exceptional prime every source prime is exceptional over \(X_0\): a pre-existing one had image of codimension at least two on \(W\), and a new one is exceptional by construction. All therefore have boundary coefficient one. The formula for \(T_D=\alpha_D-\beta_D\) then gives coefficient one on that new base prime. Thus \[ T'=\tau_*^{-1}T+\operatorname{Exc}(\tau)_{\mathrm{red}}. \] The usual log discrepancy formula for the SNC pair \((W,T)\) now gives \(K_{W'}+T'=\tau^*(K_W+T)+E_H\), with \(E_H\) effective and exceptional. Adding the pulled-back parabolic line proves Equation~\eqref{rank:higher-base}. \end{proof} \begin{lemma}[Pseudo-effectivity of the base line] \label{rank:base-psef} In the setting of Lemma~\ref{rank:comparison}, suppose \(\dim X=n\), \(0<\dim W0\), Equation~\eqref{rank:first-nef-endpoint} preserves the divisible section spaces. For \(\kappa(H)=k\) the nef \(H_{\mathrm{nef}}\) is big, and the second alternative holds with \(W_m=W_{\mathrm{nef}}\) and \(H_m=H_{\mathrm{nef}}\). If \(0<\kappa(H)0\). For a rational \(c\in(0,1)\), put \(H_c=K_W+T+cM\). We claim \begin{equation} H_c\ \text{is not pseudo-effective} \quad(00\) for which \(H-\delta p^*U\) has an effective representative, forcing \(\kappa(H)\geq\dim S>0\). Choose a rational \(a>\max\{1,a_0\}\) and a small rational \(\lambda>0\) such that \(K_S+aP_S-\lambda U\) is big. This is possible because \(P_S\) is nef and \(K_S+a_0P_S\) is big. Choose \[ 0<\eta<\lambda\frac{1-c}{a-1} \quad\text{rational}. \] The rational line \(cP_S+\eta U\) is ample. A sufficiently high general member divided by its degree pulls back to a rational boundary transverse to \(T\) with subunit coefficient. Applying the nonvanishing part of Proposition~\ref{proj:good-models} to this ordinary projective log pair gives \[ E_c\sim_{\Q}H_c+\eta p^*U,\qquad E_c\geq0. \] A nonzero section defining \(E_c\) restricts nontrivially to very general \(p\)-fibers in a fixed divisible degree. The hypotheses of the weak-effectivity statement \cite[Lemma~3.3]{CI} are now satisfied: the source is smooth in class \(\mathcal C\), \(T\) is rational SNC, the base \(S\) is smooth projective, and a relative log system is nonzero. Applied to the big line \(K_S+aP_S-\lambda U\), it gives \[ E_a\sim_{\Q}K_W+T+aM-\lambda p^*U,\qquad E_a\geq0. \] For \[ \theta=\frac{1-c}{a-c},\qquad \delta=\theta\lambda-(1-\theta)\eta>0, \] the rational interpolation is \[ (1-\theta)E_c+\theta E_a \sim_{\Q}H-\delta p^*U. \] Multiplying pullbacks of sections of \(U\) by a section of the effective left-hand side yields \(\kappa(H)\geq\kappa(p^*U)=\dim S>0\). This contradicts Equation~\eqref{rank:nontrivial-nef-data} and proves Equation~\eqref{rank:lowered-nonpsef}. We next run a program to fiber type while descending the actual nef line \(H_{\mathrm{nef}}\) through every contraction. Choose the common resolution in Equation~\eqref{rank:first-nef-endpoint} to determine the nef data. The negativity lemma gives \[ F_M:=\sigma^*M_{\mathrm{nef}}-\mu^*M\geq0 \quad\text{\(\sigma\)-exceptional}. \] Indeed its negative is \(\sigma\)-nef and exceptional. For every rational \(00\) with \(rH_{\mathrm{nef}}\) Cartier, and choose an integer \(d_0>2kr\). This coefficient will force every ray of the second program to have zero degree for the descended nef line. Decreasing the nef data to \((1-\varepsilon)\boldsymbol M\) preserves generalized dlt singularities: on the displayed resolution the crepant boundary changes by \(-\varepsilon F_M\), so discrepancies do not decrease. Add \(d_0H_{\mathrm{nef}}\), determined as nef data on \(W_{\mathrm{nef}}\). It changes no discrepancies, and the resulting adjoint is \[ (1+d_0)H_{\mathrm{nef}}-\varepsilon M_{\mathrm{nef}} =(1+d_0)\left( H_{\mathrm{nef}}-\frac{\varepsilon}{1+d_0}M_{\mathrm{nef}}\right). \] It is not pseudo-effective by the preceding paragraph. The non-pseudo-effective case of Proposition~\ref{proj:generalized-mmp} therefore gives a chosen terminating program to a Mori fiber contraction in the \(\Q\)-factorial generalized dlt category. At any stage of this second program, let \(H_i\) be the descended rational line, assuming inductively that it is nef and \(rH_i\) is Cartier, and write \[ G_i=K_{W_i}+T_i+(1-\varepsilon)M_i. \] An extremal ray negative for \(G_i+d_0H_i\) is negative for \(G_i\). The pair with adjoint \(G_i\) is still generalized dlt: the added nef b-divisor \(d_0H_i\) descends to the current model, so removing it changes no discrepancy. The generalized length bound \cite[Proposition~3.13]{HanLi} gives a curve \(C\) on that ray with \(0<-G_i\cdot C\leq2k\). If \(H_i\cdot C>0\), Cartier integrality would give \[ \frac{d_0}{r}\leq d_0H_i\cdot C <-G_i\cdot C\leq2k, \] a contradiction. Thus \(H_i\cdot C=0\). The exact contraction statement \cite[Theorem~1.5]{Xie2022} descends the Cartier line \(rH_i\) along this contraction as an actual Cartier line. In a flip we pull that same line to the flipped side. The descended line is nef: every curve on the contraction base has a curve above it mapping with positive degree, so its degree is nonnegative; pullback preserves nefness. Consequently \(r\) and nefness persist, and the argument applies inductively to all steps and to the final contraction. Each step is crepant for \(H_i\). Let \(W_m\) be the last birational model of this second program and let \(H_m\) be the descended line on it. Crepancy for the \(H_i\), together with the absence of extraction, turns Equation~\eqref{rank:first-nef-endpoint} into Equation~\eqref{rank:base-endpoint} on a common resolution, with an effective divisor exceptional over \(W_m\). The final contraction \(h:W_m\to V\) has connected fibers and, by the same exact descent, \[ H_m\sim_{\Q}h^*H_V \] for a rational line \(H_V\) on the normal projective \(V\). If \(V\) is a point, the second alternative holds with \(H_m\sim_{\Q}0\). Assume \(\dim V>0\). To obtain the first alternative, it remains to show that the log adjoint on the new very general fibers has Kodaira dimension zero. Resolve the base program by \(\mu:W''\to W\) and \(\nu:W''\to W_m\), and prepare the source over \(W''\), calling it \(g'':(X'',B'')\to W''\). The diagram we use is \[ \begin{tikzcd} X'' \arrow[r,"g''"] & W'' \arrow[r,"\nu"] \arrow[d,"\mu"'] & W_m \arrow[r,"h"] & V\\ & W \end{tikzcd} \] with composite \(f= h\nu g''\). Equation~\eqref{rank:higher-base} and Equation~\eqref{rank:base-endpoint} give \[ H''\sim_{\Q}\nu^*H_m+E'', \qquad E''\geq0\ \text{\(\nu\)-exceptional}. \] To see the exceptionality, the error in Equation~\eqref{rank:base-endpoint} is already exceptional over \(W_m\), and every \(\mu\)-exceptional prime is exceptional over \(W_m\) as well: otherwise \(W_m\) would extract a divisor over \(W\). For every degree divisible by one fixed integer, Equation~\eqref{rank:relative-injection}, exceptional descent, and projection formula now yield \begin{align*} f_*\OO_{X''}(q(K_{X''}+B'')) &\lhook\joinrel\longrightarrow (h\nu)_*\OO_{W''}(qH'')\\ &=h_*\OO_{W_m}(qH_m)=\OO_V(qH_V). \end{align*} Here \(\nu_*\OO_{W''}(qE'')=\OO_{W_m}\) by normality and \(h_*\OO_{W_m}=\OO_V\). Generic coherent base change, simultaneously for the countably many divisible degrees, bounds the dimension of every such system on a very general \(f\)-fiber by one. The log adjoint on that fiber is pseudo-effective by restriction of a countable sequence of perturbed currents with analytic singularities. Its dimension is less than \(n\), so the lower-dimensional \(\Ggood\) supplies a nonzero section in some divisible degree. Taking a common multiple shows that the fiber log Kodaira dimension is zero. All maps in the composite have connected fibers; resolving \(V\) and the source keeps this property and gives a smooth projective base of dimension \(0<\dim V0\), since at least one component of \(Q\) has image dimension \(l\). Thus \(Q\) is not in the radical, and its square is strictly negative, contrary to Equation~\eqref{rank:nonnegative-square}. It remains to treat \(l=k-1\). Terms from primes whose image has smaller dimension vanish in this form, as do cross terms from different divisorial images. Fix a prime divisor \(D\subset Y\) and take all the primes \(E_1,\ldots,E_s\subset X\) dominating it. Write \(a_i=\ord_{E_i}(b^*D)>0\) and \(C_{ij}=\langle E_i,E_j\rangle_{k-1}\). The divisor \(D\) is rational Cartier, also when \(Y=W_m\). Consequently \[ C_{ij}\geq0\ (i\ne j),\qquad \sum_j C_{ij}a_j=0. \] The first assertion is positivity of the proper intersection of distinct effective primes. The second pairs \(E_i\) with the pulled-back divisor \(D\); the additional base factor makes the intersection vanish by dimension. Components of \(b^*D\) mapping into a proper subset of \(D\) make no contribution. The graph with edges \(C_{ij}>0\) is connected. Indeed over a very general point of \(D\) the fiber is connected, all its points lie in the components \(E_i\), and an intersection contributes a positive entry precisely when it dominates \(D\). For any real vector \(x=(x_i)\), the two displayed properties give the exact identity \[ x^{\mathsf T}Cx =-\sum_{i1\) for which \(uL|_{D_Z}\) is generated, and consider the integral line \begin{equation} \mathscr N :=\OO_U(u\ell+E-C) =\OO_U\bigl(K_U+\Delta_F+(u-1)\ell\bigr). \label{rank:vanishing-line} \end{equation} Although the expression on the right uses rational divisors, its sum is the integral line on the left. The restriction sequence is \[ 0\longrightarrow\mathscr N \longrightarrow\OO_U(u\ell+E) \longrightarrow\OO_C(u\ell+E)\longrightarrow0. \] The direct image of the last sheaf is supported on the proper analytic subset \(f(C)\), so it is torsion. Consequently, the two assertions \begin{equation} R^1f_*\mathscr N\ \text{torsion-free}, \qquad H^1(Y,f_*\mathscr N)=0 \label{rank:vanishing} \end{equation} will give surjectivity on global sections: torsion-freeness makes the connecting map to \(R^1f_*\mathscr N\) zero, and the \(H^1\)-vanishing then removes the obstruction to lifting a global section. We prove Equation~\eqref{rank:vanishing} using analytic injectivity. Equation~\eqref{rank:big-pullback} identifies \(\mathscr N\otimes K_U^{-1}\) with \(\Delta_F+(u-1)f^*H_Y\) in \(\Pic(U)\otimes\Q\); this comparison need not be an isomorphism of the corresponding integral lines. Fix a very ample line \(A_Y\) on \(Y\) and its Fubini--Study metric. A Kodaira decomposition of the big nef \(H_Y\), given arbitrarily small weight as in the proof of Lemma~\ref{rank:lift}, writes \((u-1)H_Y\) as a positive rational multiple of \(A_Y\) plus an effective rational divisor whose fixed singular contribution is arbitrarily small and whose remaining contribution is a general divided ample member. Choose the small weight below the log canonical threshold of its pullback relative to the klt SNC boundary \(\Delta_F\), and choose the ample member generally. Then the resulting boundary on \(U\) is klt. Choose a positive integer \(q_0\) clearing all its denominators and realizing both the chosen Kodaira decomposition and the preceding rational-line comparison as line isomorphisms, in particular \[ (\mathscr N\otimes K_U^{-1})^{\otimes q_0} \simeq \OO_U(q_0\Delta_F)\otimes f^*\OO_Y\bigl(q_0(u-1)H_Y\bigr). \] The divisor metrics and the Fubini--Study metric define a singular Hermitian metric on \((\mathscr N\otimes K_U^{-1})^{\otimes q_0}\). Its \(q_0\)-th root is a metric on the actual line \(\mathscr N\otimes K_U^{-1}\), with multiplier ideal \(\OO_U\) and curvature dominating a positive multiple of the pullback Fubini--Study form. The same statements hold after every nonnegative twist by \(f^*A_Y\). The injectivity theorem \cite[Theorem~A]{FujinoMatsumura} applies on the compact Kähler \(U\): multiplication by the pullback of any nonzero section of a positive power of \(A_Y\) is injective on \[ H^1(U,\mathscr N\otimes f^*A_Y^j) \quad(j\geq0) \] into the correspondingly higher twist. Its assumptions are exactly the semipositive smooth metric on the multiplying line, the preceding positive lower curvature bound, and the trivial multiplier ideal. This injectivity implies Equation~\eqref{rank:vanishing}. For the second assertion, the Leray edge map injects \(H^1(Y,f_*\mathscr N)\) into \(H^1(U,\mathscr N)\). Multiplication by a section of a sufficiently high \(A_Y\)-power sends this subspace to zero, since Serre vanishing kills \(H^1(Y,f_*\mathscr N\otimes A_Y^j)\) for large \(j\). Injectivity upstairs forces the original subspace to be zero. For the first assertion, suppose \(R^1f_*\mathscr N\) has a nonzero torsion subsheaf. A section of a sufficiently high power of \(A_Y\) annihilates a nonzero such subsheaf. After a further sufficiently large twist it has a nonzero global section, and Serre vanishing and Leray identify that section with a nonzero class in \(H^1(U,\mathscr N\otimes f^*A_Y^j)\). Multiplication annihilates this class, again contradicting injectivity. This proves both assertions. The resulting surjectivity extends the floor sections as follows. Pull any section of \(uL|_{D_Z}\) to \(C\), multiply by the canonical section of \(E|_C\), and extend by this surjectivity. The extended section descends to \(Z\), since \(r_*\OO_U(E)=\OO_Z\) by exceptionality and normality. On the strict transforms of the components of \(D_Z\), division by the canonical section of \(E\) recovers the prescribed section. The descended restriction therefore equals it on the whole reduced \(D_Z\), since equality holds generically on every component. Since \(uL|_{D_Z}\) is generated, this proves Equation~\eqref{rank:floor-avoidance}. We finish by showing that this stable base locus is empty. The base loci of factorial Cartier multiples form a descending sequence of compact analytic subsets and hence stabilize. Sections exist because \(H_Y\) is big and Equation~\eqref{rank:big-pullback} descends their pullbacks. Choose a divisible degree \(v\) whose base locus is \(\mathbf B(L)\), and let \(\mathfrak b\) be its base ideal. If \(\mathbf B(L)\ne\varnothing\), the ideal is a unit near \(D_Z\), while \((Z,\Delta_Z)\) is klt outside \(D_Z\). Its log canonical threshold \[ t=\operatorname{lct}_{(Z,\Delta_Z)}(\mathfrak b) \] is therefore a positive rational number, attained by a divisor whose center is contained in \(\mathbf B(L)\). This follows directly on a simultaneous log resolution of the pair and the ideal: the threshold is the minimum of the positive rational discrepancies divided by the positive integral ideal orders. Choose an integer \(s>t\) and general divisors \(D_1,\ldots,D_s\in|vL|\). On the same resolution their fixed part is the divisor of \(\mathfrak b\); their free transforms are jointly transverse to the SNC data. Hence \[ \left(Z,\ \Delta_Z+\frac{t}{s}\sum_{i=1}^s D_i\right) \] is log canonical: the fixed part is at its log canonical threshold and the free components have coefficients \(t/s<1\). There is a new lc place centered in \(\mathbf B(L)\), and the new adjoint satisfies the actual identity \[ L_{\mathrm{new}}\sim_{\Q}(1+tv)L. \] Apply \cite[Theorem~3.3]{HLX} to this compact Kähler lc rational pair. It gives a projective strongly \(\Q\)-factorial dlt modification \(d:Z'\to Z\). Since the pair is lc, all extracted divisors have log discrepancy zero in our convention, so the modification is crepant. The source is again compact Kähler. Choose its defining dlt log resolution, with exceptional crepant coefficients strictly below one and an isomorphism at the general point of every lc stratum; resolving the remaining data away from those points gives the resolution required by Definition~\ref{def:lc-strata-resolution}. The floor on \(Z'\) has a point over the center of this lc place, hence over \(\mathbf B(L)\). The new nef adjoint is \(d^*L_{\mathrm{new}}\). On a common resolution with \(U\), it satisfies Equation~\eqref{rank:big-pullback} with the big nef line \((1+tv)H_Y\). The previously proved assertion \eqref{rank:floor-avoidance} therefore makes its stable base locus disjoint from its floor. On the other hand, normality and projection formula identify all divisible sections under \(d\), and give \[ \mathbf B(d^*L_{\mathrm{new}}) =d^{-1}\mathbf B(L). \] This locus contains the stated floor point, a contradiction. Thus \(\mathbf B(L)=\varnothing\); factorial stabilization provides an actual globally generated multiple of \(L\). \end{proof} \subsection{Completion of the rank-one case} \begin{proof}[Proof of Proposition~\ref{rank:rank-one}] By Proposition~\ref{bir:resolution-transfer} we can begin on the resolved source in the statement. We use a secondary induction on the positive base dimension \(k\). Apply Lemma~\ref{rank:comparison} and then Lemma~\ref{rank:base-psef} to obtain the prepared data and the pseudo-effective line \(H\). If \(a(W)=0\), Lemma~\ref{rank:zero-base} and the first case of Lemma~\ref{rank:lift} give \(J\sim_{\Q}N(J)\), which is already the required decomposition with trivial positive part. If \(W\) is projective, apply Lemma~\ref{rank:base-reduction}. Its first alternative has a strictly smaller positive base dimension, so the secondary induction and birational transfer finish the proof. In its second alternative, take a common smooth base resolution in Equation~\eqref{rank:base-endpoint} and prepare the source over it. Equation~\eqref{rank:higher-base} adds an effective divisor exceptional over the old base. It is also exceptional over \(W_m\), because the base program extracted no divisors. Thus the second case of Lemma~\ref{rank:lift} applies. We have obtained \[ J\sim_{\Q}P_0+A,\qquad A=N(J)\geq0, \] with \(P_0\) nef. A rationally trivial \(P_0\) finishes the proof. In the remaining case \(P_0=b^*H_m\), where \(b:X\to W_m\) is proper and \(H_m\) is big and nef on the projective \(W_m\). Apply Proposition~\ref{prog:contract-negative} to this nef-plus-negative presentation. It gives a globally strongly \(\Q\)-factorial dlt compact Kähler model \((Z,\Delta_Z)\) with nef adjoint \(L_Z\). The ordinary dlt endpoint has the resolution property of Definition~\ref{def:lc-strata-resolution}. On a common smooth resolution \(\mu:U\to X\), \(r:U\to Z\) its exact positive-line comparison is \[ r^*L_Z\sim_{\Q}\mu^*P_0 =(b\mu)^*H_m . \] Proposition~\ref{rank:big-nef-generation} makes \(L_Z\) semiample. Hence \(\mu^*P_0\) is semiample. Lemma~\ref{bir:pullup} identifies \(\mu^*A=N(\mu^*J)\), so this is precisely \(\Ggood_n\) on \(U\). Finally Proposition~\ref{bir:resolution-transfer} transfers the decomposition back through all the prepared source modifications. \end{proof} ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-rank-one.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//06-simple-nonvanishing.tex ==== \section{Meromorphic nonvanishing on simple spaces}\label{sec:nonvanishing} The simple case of the induction needs a divisor representing a multiple of the canonical bundle; its coefficients need not be nonnegative. The following result supplies precisely this starting point. \begin{theorem}\label{nv:meromorphic} Let \(X\) be a smooth connected compact Kähler manifold. Suppose that \(a(X)=0\) and that no positive-dimensional proper compact analytic subvariety passes through a very general point of \(X\). Then \(K_X^{\otimes m}\) has a nonzero meromorphic section for some integer \(m>0\). \end{theorem} The proof compares two ways of measuring poles. After normalizing a big class on \(X\) to have volume one, its pole order at a very general point is bounded. We construct a projective bundle over \(X^2\) whose volume grows linearly with a parameter \(q\). Restriction to the exceptional divisor of the diagonal in the square of this bundle then produces a high-rank subsheaf of a symmetric cotangent power. A second diagonal forces substantial vanishing of its determinant; descending that determinant produces a point pole of order comparable to \(q^{1/(2\dim X+1)}\), contradicting the point bound. The argument uses only meromorphic sections of line bundles; it does not require nonconstant meromorphic functions on \(X\). For the proof, suppose that the conclusion fails. Write \[ n=\dim X,\qquad \mathscr L=K_X,\qquad L=c_1(\mathscr L) \in H^{1,1}_{\BC}(X,\R). \] We may assume \(n\ge2\): a compact complex curve is projective, and the assertion for a point is immediate. Throughout this Section, an inequality between real \((1,1)\)-classes is in pseudo-effective order: \(\alpha\le\beta\) means that \(\beta-\alpha\) is pseudo-effective. For a divisor \(D\), \(\{D\}\) denotes \(c_1(\OO(D))\). Pullbacks of classes will occasionally be suppressed when the map is evident. \subsection{Line subsheaves and point poles} We first record the consequence of simplicity that controls all the determinant lines used below. \begin{lemma}\label{nv:slope} Under the contrary hypothesis above, \(H^1(X,\OO_X)=0\), the group \(\Pic(X)\) is countable, and \(L\ge0\). If \(\mathscr H\) is a holomorphic line bundle and \(\mathscr H\longrightarrow\Omega_X^{\otimes k}\) is nonzero, where \(k\ge0\), then \begin{equation}\label{nv:slope-X} c_1(\mathscr H)\le kL . \end{equation} There is a complement of a countable union of proper analytic subsets of \(X\) with the following further property. For every point \(x\) in this complement, let \[ a:Y=\Bl_xX\longrightarrow X,\qquad F=a^{-1}(x). \] Every nonzero line map \(\mathscr H_Y\longrightarrow (a^*\Omega_X)^{\otimes k}\) satisfies \begin{equation}\label{nv:slope-blowup} c_1(\mathscr H_Y)\le k a^*L . \end{equation} \end{lemma} \begin{proof} If the Albanese map of \(X\) is nonconstant, simplicity makes its image \(n\)-dimensional: a positive-dimensional general fiber of smaller dimension would be a forbidden subvariety. At a point where the Albanese map has rank \(n\), some \(n\) invariant one-forms have nonzero wedge after pullback. This gives a nonzero holomorphic section of \(K_X\), contrary to our assumption. Thus \(H^1(X,\OO_X)=0\). The exponential sequence embeds \(\Pic(X)\) into the countable group \(H^2(X,\Z)\). The manifold \(X\) is not uniruled, by simplicity. Ou's non-pseudo-effectivity criterion for the canonical class therefore gives \(L\ge0\) \cite[Theorem~1.1]{Ou2025}. We recall how the slope and foliation results in the same paper give the more precise inequality in Equation~\eqref{nv:slope-X}. Let \(\gamma\) be any class in the full dual of the pseudo-effective cone, and use the slope \[ \mu_\gamma(\mathscr E) =\frac{c_1(\mathscr E)\cdot\gamma}{\rk\mathscr E} \] for torsion-free sheaves. If \(\mu_{\gamma,\min}(\Omega_X)<0\), the first Harder--Narasimhan piece \(\mathscr T\subset T_X\) has strictly positive slope. It is saturated and semistable. The tensor slope inequality shows that the bracket \(\bigwedge^2\mathscr T\to T_X/\mathscr T\) is zero: the minimum slope of its source is at least \(2\mu_\gamma(\mathscr T)\), whereas the maximum slope of its target is smaller than \(\mu_\gamma(\mathscr T)\). Consequently \(\mathscr T\) is a foliation. Its dual has strictly negative maximum slope and is not pseudo-effective by \cite[Proposition~4.5]{Ou2025}. Ou's foliation theorem \cite[Theorem~1.4]{Ou2025} makes this foliation the tangent foliation of a meromorphic fibration with compact general leaf closures. Its rank is strictly between zero and \(n\): rank \(n\) would give \(\mu_\gamma(T_X)=-L\cdot\gamma/n\le0\). The general leaf closure is therefore a positive-dimensional proper subvariety through a general point, again contradicting simplicity. We have proved \(\mu_{\gamma,\min}(\Omega_X)\ge0\). In its Harder--Narasimhan filtration all quotient slopes are now nonnegative, so \(\mu_{\gamma,\max}(\Omega_X)\le L\cdot\gamma\). The tensor slope inequality \cite[Lemma~4.4]{Ou2025} gives \[ c_1(\mathscr H)\cdot\gamma \le \mu_{\gamma,\max}(\Omega_X^{\otimes k}) \le kL\cdot\gamma . \] Separation by the dual cone proves Equation~\eqref{nv:slope-X}. For \(k=0\) this is also immediate from the effective zero divisor of a nonzero map \(\mathscr H\to\OO_X\). It remains to choose \(x\) uniformly. For any vector bundle \(\mathscr V\), independent global sections are generically pointwise independent. Indeed, choose a maximal pointwise independent subfamily on a dense open set. Expressing any other section in that family by minors gives meromorphic function coefficients. They are constant because \(a(X)=0\), so maximality among a linearly independent family forces all its members to be pointwise independent. Thus evaluation on \(H^0(X,\mathscr V)\) is injective away from a proper analytic subset (unless that vector space is zero, in which case there is no restriction). Apply this to \(\mathscr V=\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) for every \(\mathscr H\in\Pic(X)\) and \(k\ge0\). There are countably many such bundles. For \(x\) outside the resulting exceptional set, write any line bundle on \(Y=\Bl_xX\) uniquely as \(a^*\mathscr H\otimes\OO_Y(mF)\), \(m\in\Z\). A line map from this bundle to \((a^*\Omega_X)^{\otimes k}\), restricted away from \(F\), extends over \(x\) to a section of \(\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) by Hartogs' theorem. If \(m>0\), this section vanishes at \(x\), since \(a_*\OO_Y(-mF)=\mathcal I_x^m\). The choice of \(x\) rules this out. Thus \(m\le0\), and Equation~\eqref{nv:slope-X} yields \[ c_1(a^*\mathscr H\otimes\OO_Y(mF)) \le k a^*L+m\{F\}\le k a^*L. \] If a target has a finite filtration whose graded pieces are direct sums of the indicated cotangent tensors, take the first nonzero associated graded component of the line map and then a nonzero direct-sum component. The preceding argument applies with the tensor order of that component. \end{proof} We will use several standard facts about volumes of real \((1,1)\)-classes. Our normalization is \(\vol(\alpha)=\int\alpha^e\) for a nef class on an \(e\)-fold. Volume is continuous, homogeneous, monotone in pseudo-effective order, invariant under modification, and its \(e\)-th root is concave on the big cone. Analytic Fujita approximation computes it by Kähler parts on smooth projective modifications. Here a \emph{smooth projective modification} means a projective modification whose source is smooth; the sources used here are compact Kähler manifolds obtained by resolving coherent analytic ideals. For a smooth irreducible divisor \(D\) and a big class \(\alpha\), write \(\operatorname{vol}_{\,|D}(\alpha)\) for the numerical restricted volume. It is zero when \(D\) is contained in the non-Kähler locus \cite[Theorem~1.1]{Vu2023}; otherwise it is the supremum of the masses on \(D\) of restrictions of Kähler currents with analytic singularities that are not generically singular on \(D\) \cite[Lemma~2.7]{CollinsTosatti2018}. The divisorial derivative and its continuity on the big cone are \begin{equation}\label{nv:volume-derivative} \frac{\dd}{\dd u}\vol(\alpha-u\{D\}) =-e\,\operatorname{vol}_{\,|D}(\alpha-u\{D\}); \end{equation} see \cite[Theorem~1.1]{Vu2023}. We apply this formula only on smooth manifolds while the varying class is big. Here is a useful precise form of the approximation in the restricted volume formula. Resolve the log-ideal singularities of a current restricted to \(D\). Its pullback is an effective real divisor plus a positive residual current with locally bounded potentials. The latter dominates a positive multiple of the pulled-back Kähler form. A current with locally bounded potentials has zero Lelong numbers, so Demailly regularization makes its class, after subtracting that multiple, nef \cite[Theorem~1.1]{DemaillyRegularization1992}. On a projective modification there is an effective exceptional divisor whose negative is relatively ample. Subtracting a sufficiently small multiple of this divisor from the residual class therefore makes that class Kähler; add the same multiple to the divisor part. Round all divisor coefficients slightly upwards to rational numbers. Openness of the Kähler cone preserves the Kähler property. These changes can be arbitrarily small in top intersections, which compute the original mass by the bounded-potential product formula. Thus we may approximate a restricted mass by \begin{equation}\label{nv:rational-fujita} h^*(\alpha|_D)=\beta+\{D'\},\qquad \beta\ \text{Kähler},\quad D'\ge0\ \text{a rational divisor}. \end{equation} Moreover, \(D'\) has at least the log-ideal orders of the original restriction on every further resolution. We will use this last property to turn poles into vanishing conditions. Only \(D'\) is made rational; the class \(\beta\) and the horizontal part of \(\alpha\) remain real. \begin{lemma}[A point pole bound]\label{nv:pole-threshold} Let \(W\) be a smooth compact Kähler manifold of dimension \(e\ge2\), \(\alpha\) a big real \((1,1)\)-class, and \(z\in W\). Suppose a smooth projective modification \(\mu:W'\to W\), which is an isomorphism near \(z\), admits a decomposition \[ \mu^*\alpha=K+\{D\},\qquad K\ \text{Kähler},\quad D\ge0, \] where \(D\) misses the point \(z'\) over \(z\). If the ordinary analytic Seshadri constant \(\epsilon(K,z')\) is at least \(\eta>0\), then, on the blowup \(b:\widehat W=\Bl_zW\to W\) with exceptional divisor \(G\), \begin{equation}\label{nv:pole-formula} \sup\{u\ge0:b^*\alpha-u\{G\}\ge0\} \le \frac{\eta}{2} +2^{e-1}\vol(\alpha)\eta^{-(e-1)} . \end{equation} \end{lemma} \begin{proof} Set \(u_0=\eta/2\). Blowing up \(z'\), the class \(K-u_0\{G'\}\) is Kähler. The Fujita decomposition is unchanged near \(G'\), so it supplies a Kähler current for \(\alpha_{u_0}=b^*\alpha-u_0\{G\}\) that is smooth near \(G\). Its restriction there has class \(u_0c_1(\OO_{\PP^{e-1}}(1))\). The restricted volume is consequently \(u_0^{e-1}\): the current gives this lower bound, and the volume of the restricted class gives the opposite bound. Let \(\tau\) denote the left side of Equation~\eqref{nv:pole-formula}. The classes \(\alpha_u=b^*\alpha-u\{G\}\) are big for \(0\le u<\tau\), since \(\alpha_0\) is big and the pseudo-effective cone is convex. The concave function \(f(u)=\vol(\alpha_u)^{1/e}\) satisfies, by Equation~\eqref{nv:volume-derivative}, \[ f'(u_0)=-\frac{u_0^{e-1}}{f(u_0)^{e-1}}. \] Its tangent line at \(u_0\) must remain positive up to \(\tau\). Therefore \[ \tau\le u_0+\frac{f(u_0)^e}{u_0^{e-1}} \le u_0+\frac{\vol(\alpha)}{u_0^{e-1}}, \] which is Equation~\eqref{nv:pole-formula}. \end{proof} Fix a Kähler class \(\omega\) on \(X\). The class \(L\) is not big: a big holomorphic line bundle would make \(X\) Moishezon, contrary to \(a(X)=0\). For \(t>0\) define \begin{equation}\label{nv:normalization} P=r(L+t\omega),\qquad r=\vol(L+t\omega)^{-1/n}. \end{equation} Then \(P\) is a big real class, \(\vol(P)=1\), \(L\le P/r\), and \(r\to\infty\) as \(t\downarrow0\). Choose a smooth Fujita model \(\mu:Y_P\to X\) with Kähler part \(P'\) satisfying \begin{equation}\label{nv:P-prime} \mu^*P=P'+\{D_P\},\qquad v:=\int_{Y_P}(P')^n>\frac12. \end{equation} At a very general point \(y\) of this model there is no positive-dimensional proper subvariety: its image would be one through a very general point of \(X\), and \(y\) avoids the exceptional locus. The ordinary Seshadri formula for a Kähler class, \[ \epsilon(K,y) =\inf_{\substack{W\ni y\\ \dim W>0}} \left(\frac{\int_W K^{\dim W}} {\operatorname{mult}_y W}\right)^{1/\dim W}, \] therefore gives \(\epsilon(P',y)=v^{1/n}\ge2^{-1/n}\). This is the ordinary nef/Kähler formula \cite[Theorem~2.8]{Tosatti2016}; see also \cite[Equation~(1.5)]{CollinsTosatti2018} and the Kähler cone criterion of \cite{DemaillyPaun2004}. We use this ordinary formula in both applications below. Lemma~\ref{nv:pole-threshold}, with \(\vol(P)=1\), now yields a constant \(C_n\) depending only on \(n\) such that, for a very general \(x\), \begin{equation}\label{nv:point-bound} \tau(P,x):=\sup\{u\ge0:a^*P-u\{F\}\ge0\}\le C_n, \qquad a:\Bl_xX\to X. \end{equation} Here and below positive constants denoted \(c_n,C_n\) may be decreased or increased from one occurrence to the next. They are independent of all parameters and choices of currents and modifications. \subsection{A projective bundle with controlled volume} We now produce a class whose volume grows, while its point pole threshold grows much more slowly. On \(X^2\), let \(\mathscr L_i\) and \(L_i\) denote the pullbacks of \(\mathscr L\) and \(L\) from the \(i\)-th factor. Use the quotient convention and put \[ \pi:Z=\PP_{X^2}(\mathscr L_1\oplus\mathscr L_2)\longrightarrow X^2, \qquad \xi=c_1(\OO_Z(1)),\qquad d=\dim Z=2n+1 . \] Thus \(\pi_*\OO_Z(m)=\Sym^m(\mathscr L_1\oplus\mathscr L_2)\) for \(m\ge0\). The zero divisor \(A_i\) of \(\pi^*\mathscr L_i\to\OO_Z(1)\) has class \(\xi-L_i\); it is the section on which \(\xi\) restricts to \(L_{3-i}\). In particular \(\xi=\{A_i\}+L_i\ge0\). \begin{lemma}\label{nv:subvarieties} For \(X\) and \(Z\) above, assume that \(K_X^{\otimes m}\) has no nonzero meromorphic section for every integer \(m>0\). Through a very general point of \(X^2\), the only positive-dimensional proper compact irreducible analytic subvarieties are the two factor slices. Through a very general point of \(Z\), the only positive-dimensional compact irreducible analytic subvarieties are a fiber of \(\pi\), the full inverse images of the two factor slices, and \(Z\) itself. \end{lemma} \begin{proof} Let \(C\subset X^2\) be irreducible through a pair whose coordinates are very general in \(X\). Each projection image is a point or all of \(X\), by properness and simplicity. If exactly one is a point, \(C\) is the corresponding full slice. If both projections are surjective, test their fibers at a general point of \(C\) whose coordinates are still very general. A positive-dimensional fiber must be the full other factor. Thus either \(C=X^2\), or \(\dim C=n\) and both projections are generically finite. We show that subvarieties of the latter kind cannot sweep \(X^2\). The space of compact \(n\)-cycles on a compact Kähler manifold has countably many irreducible components, each compact \cite[Theorem~1.1]{LiebermanCycles}. On a component containing integral cycles generically finite over both factors, integrality holds on a dense open set and the two projection degrees remain positive, as can also be tested by intersection with pulled-back Kähler forms. Suppose the incidence over one such component dominates \(X^2\). Resolve the parameter space and the dominating incidence component, obtaining maps \[ h:\mathcal Y\longrightarrow\mathcal S,\qquad F_i:\mathcal Y\longrightarrow X \] between smooth spaces. For general \(t\in\mathcal S\), the fiber \(Y_t=h^{-1}(t)\) is smooth, compact, and bimeromorphic to the integral cycle, hence irreducible. Its maps \(f_i=F_i|_{Y_t}:Y_t\to X\) are generically finite. Choose \(t\) also so that \(Y_t\) meets the dense open set where \(d(F_1,h)\) is invertible and \(d(F_1,F_2)\) has rank \(2n\). The first assertion follows from generic finiteness of \(f_1\), and the second from the assumed dominance in \(X^2\). Our immediate aim is to turn this dominance into a meromorphic frame of \(f_2^*T_X\). Its determinant would meromorphically trivialize \(f_2^*K_X^{-1}\), and hence also \(f_2^*K_X\). On \(Y_t\), the equal-rank bundle map \[ d(F_1,h)|_{Y_t}:T_{\mathcal Y}|_{Y_t} \longrightarrow f_1^*T_X\oplus (\OO_{Y_t}\otimes T_t\mathcal S) \] has a meromorphic inverse, given by its adjugate and determinant. For fixed \(v\in T_t\mathcal S\), apply this inverse to \((0,v)\) and then apply \(dF_2\). The result is a meromorphic section of \(f_2^*T_X\). At a general point where \(d(F_1,F_2)\) has rank \(2n\), the resulting map \(T_t\mathcal S\to f_2^*T_X\) is surjective: in local coordinates \((F_1,h)\), it is the derivative of \(F_2\) in the parameter directions with \(F_1\) fixed. Choose \(n\) fixed vectors \(v_i\) whose values there are independent. Their wedge is a nonzero meromorphic section of \(f_2^*K_X^{-1}\) on the irreducible \(Y_t\); its inverse trivializes \(f_2^*K_X\) meromorphically. Factor the proper generically finite map \(f_2\) as \[ Y_t\longrightarrow \overline Y_t \xrightarrow{\nu}X \] by Stein factorization. Here \(\overline Y_t\) is normal and irreducible, the first map is a modification, and \(\nu\) is finite of degree \(N>0\). Meromorphic sections descend across a modification of a normal space, so the section descends to a nonzero meromorphic section of \(\nu^*K_X\). Lemma~\ref{bir:finite-norm} gives a nonzero meromorphic section of \(K_X^{\otimes N}\). This norm is defined in local finite analytic fraction algebras, so it remains available even though the global meromorphic function field of \(X\) is just \(\C\). It contradicts our assumption. Consequently the incidence image of each such cycle component is a proper compact analytic subset of \(X^2\). There are only countably many components, so their union misses a very general pair. This proves the assertion for \(X^2\). Now let \(W\subset Z\) be irreducible through a very general point. Its image is a point, a full factor slice, or \(X^2\). If its generic relative dimension is one, it is the full inverse image of that image. Otherwise it is a multisection of the restricted \(\PP^1\)-bundle. Over a slice or over \(X^2\), a multisection is a divisor with line bundle \(\OO(k)\otimes\pi^*\mathscr H\), \(k>0\). Its homogeneous equation has coefficients in \[ H^0\bigl(S,\mathscr H\otimes \mathscr L_1^{\otimes i} \otimes\mathscr L_2^{\otimes(k-i)}\bigr), \qquad 0\le i\le k, \] where \(S\) is the relevant slice or \(X^2\), and a fixed-factor line is understood as constant. A single nonzero monomial cuts out only the axis sections and vertical divisors. A multisection not on an axis therefore has two nonzero coefficients. Their quotient is a meromorphic section of a nonzero power of \(\mathscr L_1\otimes\mathscr L_2^{-1}\). Over a slice this is, up to inversion and a constant line, a positive power of \(K_X\). Over \(X^2\), restriction to a general factor slice gives the same conclusion. This is impossible. The axes themselves miss a very general point of \(Z\), proving the claim. \end{proof} Let \(q\) tend to infinity through positive integers. For each sufficiently large \(q\), choose \(t=t(q)\) in Equation~\eqref{nv:normalization} so that \(r=r(q)\ge q^2\), and use the resulting normalized class \(P=P(q)\). Define the real class and the scale \begin{equation}\label{nv:M-definition} M=P_1+P_2+q\xi,\qquad A_q=q^{1/d}. \end{equation} \begin{lemma}\label{nv:two-slot-volume} For these choices, \(M\) is big and \begin{equation}\label{nv:M-bounds} c_nq\le\vol(M)\le C_nq,\qquad \tau(M,z):=\sup\{u\ge0:b_z^*M-u\{F_z\}\ge0\}\le C_nA_q \end{equation} at a very general point \(z\in Z\), where \(b_z:\Bl_zZ\to Z\) has exceptional divisor \(F_z\). \end{lemma} \begin{proof} Pull \(Z\) to the Fujita model \(Y_P^2\) from Equation~\eqref{nv:P-prime}, and put \(H=P'_1+P'_2\) on this pulled-back bundle. There is a further smooth projective modification \(\nu:\widehat Z\to\PP_{Y_P^2}(\mu^*\mathscr L_1\oplus \mu^*\mathscr L_2)\) and a Fujita decomposition of the pullback of \(M\) whose Kähler part is exactly \begin{equation}\label{nv:exact-K} K=\frac12\nu^*H+\Theta , \end{equation} where \(\Theta\) is Kähler and has degree \(q\) on a general vertical line. The modification and the divisor part miss that line. Here is the construction. It suffices to decompose \(H/2+q\xi\) as \(\Theta\) plus an effective divisor, clean on a whole general vertical line. Since \(\OO(1)\) is relatively ample, a small positive class of the form \(\delta(\xi+cH)\) is Kähler for some \(c>0\). Choose \(\delta>0\) so small that \(\delta0 \] on a general fiber. For sufficiently large divisible integers \(j\), let \(\mathcal F_j=f_*\OO_U(j\Lambda)\) and \(R_j=\rk\mathcal F_j\). Here \(j\Lambda\) is an actual line bundle and \(c_1(\mathcal F_j)\) means \(c_1(\det\mathcal F_j)\). Then \begin{align} R_j&=\frac{w}{e!}j^e+O(j^{e-1}),\label{nv:GRR-rank}\\ B_0:=G+\lim_j\frac{c_1(\mathcal F_j)}{jR_j} &=\frac{f_*\beta^{e+1}}{(e+1)w}\ge0,\label{nv:GRR-base}\\ \int_U\beta^{b_0+e} &\le (e+1)^{b_0}w\,\vol(B_0).\label{nv:mixed-bound} \end{align} The limit in Equation~\eqref{nv:GRR-base} is a limit of real Bott--Chern classes. \end{lemma} \begin{proof} The restriction of \(c_1(\Lambda)\) to each fiber is represented by the restriction of the Kähler form \(\beta\). The fiberwise criterion for relative ampleness makes \(\Lambda\) relatively ample after clearing denominators. Relative Serre vanishing then kills the higher direct images for all sufficiently large divisible \(j\). Analytic Grothendieck--Riemann--Roch \cite{LevyGRR}, in degrees zero and two, gives \[ R_j=\frac{f_*c_1(\Lambda)^e}{e!}j^e+O(j^{e-1}),\qquad c_1(\mathcal F_j) =\frac{f_*c_1(\Lambda)^{e+1}}{(e+1)!}j^{e+1}+O(j^e). \] The degree-zero pushforward is \(w\). Expanding \(\beta=f^*G+c_1(\Lambda)\) shows that \[ f_*\beta^{e+1} =f_*c_1(\Lambda)^{e+1}+(e+1)wG, \] because terms with at least two horizontal factors have negative fiber degree after pushforward. This proves the equality in Equation~\eqref{nv:GRR-base}; the cohomological GRR equality is an equality in Bott--Chern cohomology by the \(\partial\bar\partial\)-lemma on compact Kähler manifolds. Pushforward of the positive form \(\beta^{e+1}\) is a positive closed \((1,1)\)-current, so \(B_0\) is pseudo-effective. To prove Equation~\eqref{nv:mixed-bound}, we need a nef class on the base. The class \(B_0\) is presently only known to be pseudo-effective. We flatten \(f\) to obtain a nef class above the base and compare it with the pullback of \(B_0\). Take a smooth projective flattening modification \(p:Z'_0\to Z_0\), the equidimensional main transform \(\overline U\) of \(U\times_{Z_0}Z'_0\), and a resolution \(U'\to\overline U\). Write \(f':U'\to Z'_0\) and \(q:U'\to U\) for the maps and \(\beta'=q^*\beta\). The class \[ B'_0=\frac{f'_*(\beta')^{e+1}}{(e+1)w} \] is nef. It is enough to show that the positive pushforward current representing this class has zero Lelong numbers. This current can be computed by integration on the cycle \(\overline U\) of the smooth form pulled back from \(U\). At a base point, cover the compact fiber by finitely many coordinate neighborhoods, each embedded in a product of base coordinates and ambient coordinates. In the mass over a base ball of radius \(\delta\), after wedging with a base Euclidean form to power \(b_0-1\), the integrand is bounded by a finite sum of projection volume forms using \(b_0-1\) base coordinates and \(e+1\) generic linear combinations of all coordinates of the ambient product, including the remaining base direction. These projections can be chosen finite on the neighborhoods: fixing the \(b_0-1\) base coordinates leaves local dimension at most \(e+1\), by equidimensionality, and generic ambient coordinates finish a finite projection. After shrinking to compact subneighborhoods their degrees are bounded. Change of variables therefore bounds each integral by \(O(\delta^{2(b_0-1)})\) times the measure of the remaining coordinate range. On each compact subneighborhood these ranges, taken over closed base balls, are nested compact sets whose intersection is the image of the central fiber. That image has measure zero in the \(e+1\) coordinates, because the fiber has dimension \(e\). Continuity of finite measure from above shows that the range measures tend to zero. Thus the mass is \[ o\bigl(\delta^{2(b_0-1)}\bigr). \] This also covers \(b_0=1\), when it asserts absence of an atom. The pushforward current has zero Lelong numbers at every point, and Demailly regularization \cite[Theorem~1.1]{DemaillyRegularization1992} proves that its class \(B'_0\) is nef. Pushforward under \(p\) gives \(p_*B'_0=B_0\). For a modification between smooth compact Kähler manifolds, the kernel of pushforward on real Bott--Chern \((1,1)\)-classes is generated by the classes of its exceptional prime divisors. Since \(p_*p^*B_0=B_0\), it follows that \(B'_0-p^*B_0\) is an exceptional real divisor class. It is \(p\)-nef because \(B'_0\) is nef. Apply relative negativity to this exceptional real divisor. Locally over the base, the usual proof for a projective modification cuts by general hyperplanes to a surface and uses the negative definite intersection matrix of exceptional curves; it forces every coefficient of a relatively nef exceptional divisor to be nonpositive. Thus \[ B'_0=p^*B_0-\{D_0\},\qquad D_0\ge0\ \text{\(p\)-exceptional}. \] In particular \begin{equation}\label{nv:flattened-volume} \int_{Z'_0}(B'_0)^{b_0} =\vol(B'_0)\le\vol(p^*B_0)=\vol(B_0). \end{equation} Choose a Kähler class \(\omega'\) on \(Z'_0\) and put \(C=B'_0+\varepsilon\omega'\). For \(0\le k\le b_0\), set \[ I_k=\int_{U'}(\beta')^{e+k}(f'^*C)^{b_0-k}. \] These are mixed intersections of nef classes. The first two satisfy \[ I_0=w\int_{Z'_0}C^{b_0},\qquad I_1=(e+1)w\int_{Z'_0}B'_0C^{b_0-1} \le(e+1)w\int_{Z'_0}C^{b_0}. \] The mixed nef inequalities make the sequence \(I_k\) log-concave. Every \(I_k\) is positive: on the dense open where \(q\) is a local biholomorphism and \(f'\) is a submersion, \(\beta'\) is positive definite and \(f'^*C\) has rank \(b_0\), so the defining top form is strictly positive. The successive ratios are therefore at most \(I_1/I_0\le e+1\). Thus \[ \int_U\beta^{b_0+e}=I_{b_0} \le(e+1)^{b_0}w\int_{Z'_0}C^{b_0}. \] Letting \(\varepsilon\downarrow0\) and using Equation~\eqref{nv:flattened-volume} proves Equation~\eqref{nv:mixed-bound}. \end{proof} We will also use the determinant formula without its volume bound. \begin{corollary}\label{nv:slope-descent} Let \(Y\) be a smooth compact Kähler manifold and \(\mathscr E\) a holomorphic vector bundle of rank at least two. Write \(\pi_{\mathscr E}:\PP_Y(\mathscr E)\to Y\) and \(\zeta_{\mathscr E}=c_1(\OO_{\PP(\mathscr E)}(1))\). Suppose that a real class \(D\) on \(Y\) satisfies \[ c_1(\mathscr H)\le kD \quad\text{whenever}\quad 0\ne\bigl(\mathscr H\longrightarrow\mathscr E^{\otimes k}\bigr) \] for a line bundle \(\mathscr H\) and integer \(k\ge0\). For a real class \(A\) on \(Y\) and a real number \(b\ge0\), \begin{equation}\label{nv:slope-descent-equation} \pi_{\mathscr E}^*A+b\zeta_{\mathscr E}\ge0 \quad\Longrightarrow\quad A+bD\ge0 . \end{equation} \end{corollary} \begin{proof} Choose a real class \(H_0\) on \(Y\) so that \(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0\) is Kähler; relative ampleness of \(\OO(1)\) permits such a choice. Take numbers \(\delta\downarrow0\) with \(b+\delta>0\) rational. Adding \(\delta(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0)\) to the pseudo-effective class in Equation~\eqref{nv:slope-descent-equation} makes it big. A Fujita decomposition on a smooth projective modification \(h:U\to\PP_Y(\mathscr E)\), with the divisor coefficients rounded upwards as in Equation~\eqref{nv:rational-fujita}, has the form \[ \beta=f^*(A+\delta H_0)+c_1(\Lambda),\qquad \Lambda=(b+\delta)h^*\OO(1)-\OO_U(D') \ \text{in }\Pic(U)\otimes\Q, \] where \(f=\pi_{\mathscr E}h\), \(\beta\) is Kähler, and \(D'\ge0\) is rational. For divisible \(j\), \[ f_*\OO_U(j\Lambda)\ \subseteq\ \Sym^{j(b+\delta)}\mathscr E . \] Its determinant of rank \(R_j\) consequently maps into \(\mathscr E^{\otimes j(b+\delta)R_j}\). The assumed line inequality gives \[ \frac{c_1(\mathcal F_j)}{jR_j}\le(b+\delta)D. \] Lemma~\ref{nv:direct-image} makes \(A+\delta H_0+\lim c_1(\mathcal F_j)/(jR_j)\) pseudo-effective. Adding the preceding pseudo-effective difference shows that \(A+\delta H_0+(b+\delta)D\) is pseudo-effective. Let \(\delta\downarrow0\). This proves the corollary. In this argument only \(b+\delta\) and the coefficients of \(D'\) are rational; \(A\), \(H_0\), and \(D\) remain real classes. \end{proof} \subsection{Restriction to the diagonal} We now use the volume of \(M\) to obtain a high-rank subsheaf of a symmetric cotangent power on \(Z\). Distinguish the two copies of \(Z\) by bracketed indices and blow up their diagonal: \[ b:B=\Bl_{\Delta_Z}(Z\times Z)\longrightarrow Z\times Z,\qquad E=b^{-1}(\Delta_Z). \] Thus \(E=\PP_Z(\Omega_Z)\); its points are normal lines in \(T_Z\). The dimensions are \(\dim B=2d\) and \(\dim E=2d-1\), and \(E\to Z\) has relative dimension \(d-1\). Put \(\zeta=c_1(\OO_E(1))\), so that \(\{E\}|_E=-\zeta\). For \(s\ge0\) define \[ C_s=b^*(M_{[1]}+M_{[2]})-s\{E\},\qquad s_{\max}=\sup\{s\ge0:C_s\ge0\}. \] The class \(C_0\) is big, so \(s_{\max}>0\) and \(C_s\) is big for \(0\le s0\) such that, for every sufficiently large \(q\), one can choose a rational number \[ c_nA_q\le s\le C_nA_q \] and a Kähler current \(T\) in \(C_s\) with analytic singularities, not generically singular on \(E\), with the following property. On a smooth projective modification \(h:U\to E\), its restricted mass has a rational-divisor approximation \begin{equation}\label{nv:selected-decomposition} h^*(2M+s\zeta)=\beta+\{D'\}, \qquad \beta\ \text{Kähler},\quad D'\ge0\ \text{rational}, \end{equation} which retains all log-ideal orders of \(T|_E\). If \(f:U\to Z\) is the natural map and \[ w=\int_{U_z}\beta^{d-1} \] on a general fiber, then \begin{equation}\label{nv:selected-w} w\ge c_ns^{d-1}. \end{equation} \end{lemma} \begin{proof} First let \(s\) be any rational number in \((0,s_{\max})\) with \(v_E(s)>0\), and use the approximation in Equation~\eqref{nv:rational-fujita} for any current used to compute this restricted volume. In Lemma~\ref{nv:direct-image}, the data for Equation~\eqref{nv:restriction-class} are \[ b_0=d,\quad e=d-1,\quad G=2M,\quad \Lambda=s h^*\OO_E(1)-\OO_U(D') \ \text{in }\Pic(U)\otimes\Q. \] For large divisible \(j\), the associated sheaves satisfy \begin{equation}\label{nv:symmetric-inclusion} \mathcal F_j=f_*\OO_U(j\Lambda) \subseteq\Sym^{js}\Omega_Z,\qquad R_j=\rk\mathcal F_j,\qquad 00\). The contribution of \(0(1-\delta)m\). Summing instead over their internal degree gives \[ H_m(\delta) \le \binom{m+n}{n}\binom{\lceil\delta m\rceil+n}{n},\qquad \limsup_{m\to\infty}\frac{H_m(\delta)}{N_m} \le\binom{2n}{n}\delta^n. \] On the other hand, \(w^+\ge w/2\) and Equation~\eqref{nv:plus-rank} show that the approximation on \(E^+\) retains at least half the rank fraction in Equation~\eqref{nv:selected-rank-fraction}: \[ \lim_{j\to\infty}\frac{R_j^+}{N_{js}} =\frac{w^+}{s^{d-1}} \ge\frac{w}{2s^{d-1}}\ge c_n>0. \] Fix \(\delta>0\), depending only on \(n\), so small that, for each fixed \(q\) and its selected data, fewer than \(R_j^+/2\) monomials have transverse degree greater than \((1-\delta)js\) for all sufficiently large divisible \(j\). For each remaining monomial, Equation~\eqref{nv:coefficient-order} and Lemma~\ref{nv:incidence-pole} give \[ \ord_u(c_{\alpha,\gamma}) \ge j(s-C_n)-(1-\delta)js =j(\delta s-C_n)\ge\frac{\delta js}{2}. \] The last inequality holds once \(q\) is large, since \(s\ge c_nA_q\to\infty\). Over the same discrete valuation ring \(R\), the torsion-free sheaf \(\mathcal F_j^+\) becomes a free module of rank \(R_j^+\). Write its inclusion into the symmetric power as a matrix with the monomials as rows. Every maximal minor uses at least \(R_j^+/2\) of the rows whose coefficients have order at least \(\delta js/2\). All maximal minors therefore have order at least \(\delta jR_j^+s/4\). By the definition of \(h_j\), this proves Equation~\eqref{nv:determinant-order-equation}. Dividing the determinant map by \(\sigma_{J_+}^{h_j}\) gives a holomorphic map at every point of codimension one: the coefficients have the required order along \(J_+\), and its local equation is a unit away from \(J_+\). Hartogs' theorem extends the divided map across the remaining set of codimension at least two, giving the stated factorization through \(\det\mathcal F_j^+(h_jJ_+)\). \end{proof} \begin{proof}[Completion of the proof of Theorem~\ref{nv:meromorphic}] Fix \(q\) sufficiently large for the preceding lemmas, with its chosen \(P,s,T\) and modifications. Choose a very general point \(x\) in the first factor of \(X^2\). The slice of \(Z^+\) above \(x\) is \[ S=\PP_Y(\OO_Y\oplus a^*\mathscr L),\qquad a:Y=\Bl_xX\longrightarrow X, \] and \(J_+|_S\) is the pullback of \(F\). We choose \(x\) so that Equation~\eqref{nv:slope-blowup}, Equation~\eqref{nv:point-bound}, the restrictions of the currents below, and all determinant maps for divisible \(j\) can be tested on this slice. These exclude at most countably many proper analytic sets and the measure-zero exceptional sets for current restrictions. Write the restricted determinant line as \[ (\det\mathcal F_j^+)|_S =\pi_S^*\mathscr Q_j^+\otimes\OO_S(\ell_j^+),\qquad Q_j^+=c_1(\mathscr Q_j^+). \] Factoring the common zero of order \(h_j\) from the determinant map gives a nonzero map whose source on \(S\) is \[ \pi_S^*(\mathscr Q_j^+\otimes\OO_Y(h_jF)) \otimes\OO_S(\ell_j^+). \] The plus sign of \(h_jF\) follows because division of the map by the local equation of \(J_+^{h_j}\) enlarges its source from \(\det\mathcal F_j^+\) to \(\det\mathcal F_j^+\otimes\OO(h_jJ_+)\). Its target embeds into the \(jsR_j^+\)-fold tensor of the restriction of \(p_+^*\Omega_Z\). In particular we use the cotangent bundle of the original \(Z\), with its restricted filtration \[ 0\longrightarrow \pi_S^*(\OO_Y^{\oplus n}\oplus a^*\Omega_X) \longrightarrow (p_+^*\Omega_Z)|_S \longrightarrow \OO_S(-2)\otimes\pi_S^*a^*\mathscr L \longrightarrow0. \] The same homogeneous-monomial calculation as in Equation~\eqref{nv:determinant-bound-Z}, now applying Equation~\eqref{nv:slope-blowup} to the second factor of \(X^2\), gives \begin{equation}\label{nv:determinant-bound-slice} \ell_j^+\le0,\qquad Q_j^++h_j\{F\} \le(jsR_j^+-\ell_j^+)a^*L. \end{equation} More explicitly, if the chosen graded term has \(i\) relative factors, the pushed polynomial has degree \(-2i-\ell_j^+\ge0\). Its exponent in \(a^*\mathscr L\), together with the \(i\) relative factors and the cotangent order from the second factor of \(X^2\), is at most \(jsR_j^+-\ell_j^+\). The cotangent factors from the first factor of \(X^2\) are constant. This proves the displayed inequality using the tensors of \(a^*\Omega_X\), without introducing \(\Omega_Y\) or \(\Omega_{Z^+}\). Apply Lemma~\ref{nv:direct-image} to the decomposition in Equation~\eqref{nv:plus-decomposition}. Equation~\eqref{nv:GRR-base} shows that the limits of the restricted determinant components divided by \(jR_j^+\) exist; denote them by \(Q^+\) and \(\ell^+\). Divide Equation~\eqref{nv:determinant-bound-slice} by \(jR_j^+\), use Equation~\eqref{nv:determinant-order-equation}, and pass to the closed pseudo-effective cone. We obtain \begin{equation}\label{nv:slice-upper} \ell^+\le0,\qquad Q^++c_ns\{F\}\le(s-\ell^+)a^*L. \end{equation} The pseudo-effective class in Equation~\eqref{nv:GRR-base}, restricted to the very general slice \(S\), is \begin{equation}\label{nv:slice-psef} \pi_S^*(Q^++2a^*P)+(\ell^++2q)\xi\ge0. \end{equation} Testing on a general vertical line gives \(\ell^++2q\ge0\). Let \(A\subset S\) be the axis of \(\PP_Y(\OO_Y\oplus a^*\mathscr L)\) with divisor class \(\{A\}=\xi\). On \(A\simeq Y\), its normal class is \(\{A\}|_A=\xi|_A=a^*L\ge0\). Add an arbitrarily small Kähler class to the class in Equation~\eqref{nv:slice-psef}, and choose a Kähler current with analytic singularities in the resulting big class. Remove its generic divisorial pole along \(A\) and restrict the residual current to \(A\). Adding back the removed nonnegative multiple of \(a^*L\) preserves pseudo-effectivity. Passing to the limit gives \begin{equation}\label{nv:slice-lower} Q^++2a^*P+(\ell^++2q)a^*L\ge0. \end{equation} The slope bound in Equation~\eqref{nv:slice-upper} supplies the pseudo-effective class \((s-\ell^+)a^*L-Q^+-c_ns\{F\}\). Adding it to Equation~\eqref{nv:slice-lower} cancels the entire normalized determinant class restricted to the axis, \(Q^++\ell^+a^*L\), leaving \[ 2a^*P+(s+2q)a^*L-c_ns\{F\}\ge0. \] Since \(L\le P/r\), the point bound in Equation~\eqref{nv:point-bound} implies \begin{equation}\label{nv:contradiction} \frac{c_ns}{\,2+(s+2q)/r\,}\le C_n. \end{equation} The denominator is bounded independently of \(q\), because \(s\le C_nq^{1/d}\) and \(r\ge q^2\). The numerator tends to infinity, because \(s\ge c_nq^{1/d}\). Equation~\eqref{nv:contradiction} is impossible for arbitrarily large \(q\). For each \(q\), the choice of \(t(q)\), the rational \(s\), the current \(T\), and the modifications precedes the limit in \(j\). The monomial cutoff \(\delta\) depends only on \(n\), while the required lower bound for divisible \(j\) may depend on the fixed data. We choose the very general points after those data are fixed. Thus every simultaneous general-point test above involves at most countably many conditions, and no bound depends on the point. The contradiction disproves the contrary hypothesis and proves Theorem~\ref{nv:meromorphic}. \end{proof} ==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//06-simple-nonvanishing.tex ==== ==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//07-signed-rigidity.tex ==== \section{Signed rigidity on simple spaces} \label{sec:signed} The meromorphic section furnished by Theorem~\ref{nv:meromorphic} need not have an effective divisor. We therefore need a boundary argument that retains both its zeroes and its poles. The result below supplies that argument. \begin{theorem}[Signed rigidity]\label{signed:torsion} Let \((X,D)\) be a dlt pair on a normal irreducible compact Kähler space, and suppose that the pair has the resolution property of Definition~\ref{def:lc-strata-resolution}. Suppose also that \(X\) is globally \(\Q\)-factorial, that \(a(X)=0\), that \(X\) is simple, and that \(D=\sum_{i=1}^sD_i\) is reduced. Put \(L=K_X+D\), and assume the following. \begin{enumerate} \item The rational line \(L\) is analytically nef and has an actual rational linear equivalence \[ L\sim_{\Q}G=\sum_{i=1}^s a_iD_i,\qquad a_i\in\Q. \] \item For some real \(c>0\), the pullback of the class \(\{L-cD\}\) to a projective resolution with smooth compact Kähler source is pseudo-effective. \item The rational holomorphic line \(L|_D\) is semiample on the whole reduced analytic space \(D\): some Cartier multiple is generated at every point of \(D\). \end{enumerate} Then \(L\) is torsion. More precisely, for some positive integer \(q\), the holomorphic line \(\OO_X(qL)\) is isomorphic to \(\OO_X\). \end{theorem} To apply this theorem in the proof of Proposition~\ref{simple:case}, Theorem~\ref{nv:meromorphic} supplies a signed representative of the canonical bundle. After resolving, we enlarge the support of the transformed boundary and this signed canonical divisor to a reduced SNC divisor; the resulting adjoint remains pseudo-effective. Corollary~\ref{prog:signed-nef} gives an ordinary dlt nef model \((X_{\mathrm{nef}},D_{\mathrm{nef}})\) whose adjoint \(L_{\mathrm{nef}}=K_{X_{\mathrm{nef}}}+D_{\mathrm{nef}}\) has a signed representative supported on \(D_{\mathrm{nef}}\), supplying the first hypothesis. Theorem~\ref{floor:generation} supplies semiampleness on the whole reduced \(D_{\mathrm{nef}}\). On a common smooth resolution, simplicity excludes uniruledness, so Ou's criterion \cite[Theorem~1.1]{Ou2025} and exceptional translation for the canonical comparison give pseudo-effectivity of the pullback of \(K_{X_{\mathrm{nef}}}=L_{\mathrm{nef}}-D_{\mathrm{nef}}\). Thus the second hypothesis holds with \(c=1\). Once signed rigidity makes the actual line \(L_{\mathrm{nef}}\) torsion, the program comparison and Lemma~\ref{bir:unique-current} let us subtract the added boundary and give the required decomposition for the original adjoint. The final subsection verifies these steps. The signed construction in \cite[Proposition~3.3]{LI} is the source of the root and separation operations used below. The lifting method is that of \cite[Sections~10--12]{BL}. We will establish the analytic form of the signed construction and the change to the lifting argument caused by its residual poles. The numerical argument first singles out positive-dimensional fibers in the positive boundary. We then lift those fibers through every finite boundary neighborhood and deform them to compact subspaces outside the boundary. An argument using the cycle space and Baire's theorem turns these deformations into a covering family, which simplicity excludes. \subsection{The boundary detected by the nef class} If \(D=0\), the signed equivalence already gives \(L\sim_{\Q}0\). We may therefore assume in the constructions below that \(D\ne0\); in particular \(\dim X>0\). Fix the data of Theorem~\ref{signed:torsion}, and write \[ G=G_+-G_-,\qquad D_0=\sum_{a_i=0}D_i, \] where \(G_+\) and \(G_-\) are effective with disjoint prime supports. Choose a positive integer \(m\) so that \(mG_+\), \(mG_-\), \(mD_0\), and \(mL\) are Cartier, the given rational equivalence induces a fixed isomorphism \[ N_0:=\OO_X(mG)\simeq\OO_X(mL), \] and \(N_0|_D\) is generated. Its sections give a morphism and an actual line isomorphism \begin{equation}\label{signed:boundary-map} \phi:D\longrightarrow P=\PP^b,\qquad N_0|_D\simeq\phi^*\OO_P(1). \end{equation} Let \(n=\dim X\). Choose a projective resolution \(\mu:\widetilde X\to X\) as in the second hypothesis of the theorem, and a Kähler class \(\omega\) on \(\widetilde X\). The numerical dimension of the nef pullback is \[ v=\nu(L):=\max\bigl\{k\in\{0,\ldots,n\}: (\mu^*\{L\})^k\omega^{n-k}>0\bigr\}. \] The next argument is the Kähler form of \cite[Lemma~3.2]{LI}. \begin{lemma}\label{signed:boundary-geometry} If \(v=0\), then \(D=0\) and \(L\sim_{\Q}0\). If \(v=n>0\), then \(a(X)=n\). If \(00}\dim\phi(D_i)=r, \] and \begin{equation}\label{signed:small-intersections} \dim\phi\bigl(\Supp G_+\cap(\Supp G_-\cup D_0)\bigr)0\) such that \(C\omega-h\) is nef and \(h-\epsilon\omega\) is pseudo-effective. Pairing these two inequalities successively with products of the nef classes \(\mu^*\{L\}\), \(h\), and \(\omega\) shows that \[ (\mu^*\{L\})^k h^{n-k}>0 \quad\Longleftrightarrow\quad (\mu^*\{L\})^k\omega^{n-k}>0. \] We may therefore use \(\theta\) in all the tests defining \(v\). Suppose first that \(v0\) attains dimension \(r\). It remains to separate its intersections from a general fiber. On \(H^{1,1}_{\BC}(\widetilde X,\R)\) consider the form \[ q(\alpha,\beta)= \alpha\beta(\mu^*\{L\})^r h^{n-r-2},\qquad Q_{ij}=q(\mu^*\{D_i\},\mu^*\{D_j\}). \] The mixed Kähler Hodge index theorem, followed by approximation of the nef factors by Kähler classes, says that \(q\) has at most one positive direction; see \cite[Theorems~A and C]{DinhNguyen}. Put \(l=\mu^*\{L\}\). The definition of \(v\) and Equation~\eqref{signed:boundary-zero} give \[ q(l,l)=0,\qquad q(l,\mu^*\{D_i\})=0, \qquad q(l,h)=\{L\}^{v}\theta^{n-v}>0. \] Linear algebra now makes \(q\) negative semidefinite on \(l^\perp\): a positive vector in \(l^\perp\), together with a suitable vector in the span of \(l,h\), would give two positive directions. In particular \(Q\) is negative semidefinite. For \(i\ne j\), one has \(Q_{ij}\geq0\). This assertion uses the effective intersection cycle of the distinct \(\Q\)-Cartier divisors \(D_i,D_j\) on \(X\), and projection formula. It does not require their total transforms to intersect effectively. The two Cartier equations have a proper intersection: the dlt ambient space is klt, hence Cohen--Macaulay, after dropping the reduced boundary, and the distinct divisor equations form a regular sequence. The intersection cycle is consequently pure of codimension two, and each of its mixed nef degrees is nonnegative. Write the coefficient vector as \(\mathbf a=\mathbf a^+-\mathbf a^-\), with nonnegative vectors of disjoint supports. The signed equivalence and orthogonality to \(l\) give \(Q\mathbf a=0\). Thus \[ Q(\mathbf a^+,\mathbf a^+) =Q(\mathbf a^+,\mathbf a^-)\geq0. \] Negative semidefiniteness forces equality and \(Q\mathbf a^+=0\). For every \(j\) with \(a_j\leq0\), its row is a sum of nonnegative off-diagonal terms: \[ 0=\sum_{a_i>0}a_iQ_{ji}. \] Each term is zero. Applying the generated-line degree test to the effective intersection cycles proves Equation~\eqref{signed:small-intersections}. If \(r=0\), any nonempty intersection would have positive \(\theta^{n-2}\)-degree, so these intersections are empty. Finally, if \(v=n>0\), the nef volume criterion makes the rational line on \(\widetilde X\) big \cite[Theorem~0.5]{DemaillyPaun2004}. A big holomorphic line on a compact Kähler manifold has maximal Iitaka dimension. Hence \(a(\widetilde X)=n\), and birational invariance gives \(a(X)=n\). \end{proof} Thus, when \(00\). Equation~\eqref{signed:small-intersections} makes those fibers disjoint from \(\Supp G_-\cup D_0\). The next construction prepares their normal directions and adjunction for infinitesimal lifting. \subsection{Local roots and residue with residual poles} We work for the rest of the signed argument in the range \(01\) divisible by \(m\) and by every nonzero integer \(|ma_i|\), and put \(a=\ell/m\), a positive integer. \begin{lemma}[Analytic signed charts]\label{signed:root-charts} For each \(t\in P\), after shrinking an open neighborhood \(U\subset P\), choose a line \(R_U\) together with an isomorphism \(R_U^\ell\simeq\OO_P(1)|_U\). A comparison of two such roots on an overlap is power-compatible if its \(\ell\)-th power commutes with these isomorphisms. If \(t\notin\phi(\Supp G_+)\), one may take \(U\) disjoint from this closed image and set \(Z_U=S=T=\varnothing\); the assertions below are then vacuous for the unique maps from the empty spaces. If \(t\in\phi(\Supp G_+)\), there are a Hausdorff normal analytic space \(Z_U\) of pure dimension \(n\), considered near a reduced Cartier divisor \(S\) of pure dimension \(n-1\), a reduced Weil divisor \(T\) with no component in common with \(S\), and maps \[ \pi:Z_U\longrightarrow X,\qquad g:S\longrightarrow U \] with the following properties. \begin{enumerate} \item The map \(g\) is proper, and \(\pi|_S\) is finite over \(D_U=\phi^{-1}(U)\), with \(g=\phi\pi|_S\). Its image covers the positive components over \(U\), and \[ \OO_S(S)\simeq g^*R_U. \] The image of \(S\cap T\) lies in \(\phi(\Supp G_+\cap(\Supp G_-\cup D_0))\). Near \(S\), the underlying sets satisfy \[ \pi^{-1}(|D|)=|S|\cup|T|. \] \item The pair \((Z_U,S+T)\) is log canonical, and there is a fixed actual isomorphism of divisorial sheaves \begin{equation}\label{signed:adjoint-chart} \tau:\OO_{Z_U}(aS)\xrightarrow{\ \simeq\ }\omega_{Z_U}(S+T). \end{equation} The spaces \(Z_U\) and \(S\) are Cohen--Macaulay. The support of \(T\) is locally set-theoretically principal. Off \(T\), the space \(Z_U\) is klt and its canonical sheaf is invertible. The map \(\pi\) is quasi-finite near \(S\setminus T\). \item Projective log resolutions of the total divisor data can be chosen with Kähler forms on neighborhoods of the compact sets over each compact fiber of \(g\). Power-compatible changes of the boundary root extend to isomorphisms of ambient germs that preserve the full ideal \(\OO_{Z_U}(-S)\), the reduced divisor \(T\), and Equation~\eqref{signed:adjoint-chart}. Resolutions may be chosen functorially for these germ isomorphisms. \end{enumerate} Here the nonempty space \(Z_U\) is a neighborhood of the whole compact fiber under consideration; no extension of \(g\) to \(Z_U\) is asserted. \end{lemma} \begin{proof} The empty branch follows by shrinking away from the closed image \(\phi(\Supp G_+)\). We construct the nonempty charts in four stages: take roots and trivialize the remaining adjoint torsion, separate the positive and negative root divisors, retain the root line that controls the normal direction, and pass to an ordinary neighborhood of the compact fiber. First consider an analytic open set on which the Cartier lines of \(mG_+\), \(mG_-\), and \(mD_0\) are trivial, and let \(f_+,f_-,f_0\) be the functions representing their sections. Normalize the entire finite space \[ \{y_+^\ell=f_+,\qquad y_-^\ell=f_-,\qquad y_0^\ell=f_0\}, \] retaining all its components and the lifted action of \(\mu_\ell^3\) that multiplies the three root coordinates. On an overlap, the functions \(f_+,f_-,f_0\) change by holomorphic units. Local \(\ell\)-th roots of those units give comparisons of the finite spaces that lift uniquely to their normalizations; different choices differ by \(\mu_\ell^3\). The quotient stacks of these normalized charts by \(\mu_\ell^3\) glue to the simultaneous normalized root stack over \(X\), which we denote by \(\mathcal X_{\mathrm{rt}}\). The zero divisors of the three root sections are denoted by \(S_+,S_-,S_0\). At a generic prime of multiplicity \(b\), a normalized chart of \(y^\ell=x^b\) has ramification index \(\ell/b\) and \(\ord(y)=1\). Here \(b=|ma_i|\) or \(b=m\), so it divides \(\ell\). Thus each root divisor is generically reduced and Cartier. A Cartier divisor on a normal space satisfies Serre's condition \(S_1\), so it is reduced. Log ramification with the full reduced boundary gives a log canonical pair and the rational equivalence \[ K_{\mathcal X_{\mathrm{rt}}}+S_++S_-+S_0 \sim_{\Q}a(S_+-S_-). \] The ambient charts of \(\mathcal X_{\mathrm{rt}}\) are klt. Off the boundary this follows from the klt complement downstairs and the unramified charts. For a place centered in the Cartier boundary, dropping that boundary increases its log discrepancy by its strictly positive order; this also makes every zero-discrepancy place positive. To turn the rational adjoint equivalence into an actual isomorphism, consider the integral reflexive sheaf on \(\mathcal X_{\mathrm{rt}}\) \[ \mathcal F= \bigl(\omega_{\mathcal X_{\mathrm{rt}}}(S_++S_-+S_0)\otimes \OO_{\mathcal X_{\mathrm{rt}}}(-a(S_+-S_-))\bigr)^{**}. \] It is torsion. Choose a periodicity \(\mathcal F^{[q]}\simeq\OO\), take the relative spectrum of its reflexive-power algebra, and normalize; write \(\rho_{\mathrm{ind}}\) for this finite map, and retain \(S_+,S_-,S_0\) for the pulled-back root divisors on the cover. At codimension one this is a cover \(w^q=u\) for a unit \(u\), hence is unramified. Log canonicity, the klt ambient property, and reduced Cartier root divisors persist. The tautological evaluation of this algebra trivializes the reflexive pullback \((\rho_{\mathrm{ind}}^*\mathcal F)^{**}\): it does so in codimension one, and the identity then extends reflexively. The evaluation is a sheaf morphism on the cover stack itself, so the resulting actual divisorial isomorphism \[ \omega(S_++S_-+S_0)\simeq\OO\bigl(a(S_+-S_-)\bigr) \] on the cover is equivariant on every atlas. Next separate the positive and negative root divisors. Let \(\mathcal X_{\mathrm{sep}}\) be the normalization of the blowup of the ideal generated by their equations on the cover, and write \(p_1\) for its map. This blowup embeds in a relative \(\PP^1\); its normalization still has fibers of dimension at most one. Each exceptional prime therefore lies over a codimension-two positive--negative intersection. At its generic point the original dlt pair is a two-branch SNC pair. After extracting roots of units the normalized Kummer chart there has smooth coordinates \(x=u^{\ell/b_+}\), \(z=w^{\ell/b_-}\), where \(b_+\) and \(b_-\) are the two corresponding multiplicities. On this chart \(\mathcal F\) is a line and its periodicity cover is unramified. The two-coordinate blowup calculation consequently applies at every exceptional generic point. If \(E\) denotes its exceptional divisor, it gives reduced Cartier divisors \[ E,\qquad S'_+=p_1^*S_+-E,\qquad S'_-=p_1^*S_--E, \] with \(S'_+\cap S'_-=\varnothing\). No such two-branch stratum is contained in \(S_0\), so \(p_1^*S_0\) is reduced and has no exceptional component. The full boundary \(S'_++S'_-+E+p_1^*S_0\) is crepant and reduced. Its codimension-one equality with the pullback adjoint gives the discrepancy equality for every further valuation, hence log canonicity. The ambient charts after this blowup are still klt. Off the full boundary the blowup is an isomorphism to the old klt complement. A place of zero pair discrepancy centered in that boundary gains its strictly positive order when the effective Cartier boundary is dropped, while a place of positive pair discrepancy remains positive. The strict divisor \(S'_+\) is finite over \(S_+\): before normalization it lies in one section of the projective ratio coordinate, so its proper map has finite fibers, and normalization is finite. On \(\mathcal X_{\mathrm{sep}}\), put \(S=S'_+\) and, temporarily, \(T=S'_-+E+p_1^*S_0\). Write \(\pi_{\mathrm{sep}}:\mathcal X_{\mathrm{sep}}\to X\) for the composite map, and define the retained root line \[ \mathcal R:=\OO_{\mathcal X_{\mathrm{sep}}}(S'_+-S'_-), \qquad \mathcal R^\ell\simeq\pi_{\mathrm{sep}}^*N_0. \] Near \(S\), the negative strict root section is a unit. The fixed adjoint isomorphism becomes \(\omega_{\mathcal X_{\mathrm{sep}}}(S+T) \simeq\OO_{\mathcal X_{\mathrm{sep}}}(aS)\), and the positive root section divided by that negative unit section is a section of \(\mathcal R\) with zero divisor \(S\). This line and section determine the Cartier divisor \(S\) and its normal line; they must survive passage to an ordinary chart. The pair consisting of \(\mathcal R\) and its displayed power isomorphism defines a map to the gerbe \(\sqrt[\ell]{N_0}\), which parametrizes \(\ell\)-th roots of \(N_0\). Take the relative coarse space over this gerbe. On an atlas this quotients by the finite relative inertia kernel, the subgroup of each stabilizer acting trivially on \(\mathcal R\). Thus the quotient retains the root character of \(\mathcal R\). The deck transformations of the representable periodicity cover are not this inertia kernel and are not removed. Both \(\mathcal R\) and its section descend through the kernel, so the reduced image \(S\) remains Cartier and its full ideal descends. We retain \(S,T\) for the reduced images. A finite-group norm of a local equation of the upstairs \(T\) has precisely its image as zero set. Thus the reduced image \(T\) is locally set-theoretically principal; it need not be Cartier. All divisorial ramification lies on the full boundary. Log ramification identifies the invariant log dualizing sheaf in codimension one with the downstairs one. Taking reflexive hulls and invariants of the equivariant adjoint isomorphism gives an actual descended isomorphism \(\OO(aS)\simeq\omega(S+T)\). Its ordinary pullback below is Equation~\eqref{signed:adjoint-chart}. Outside \(S+T\), the local finite quotient is quasi-étale, so its klt upstairs complement gives a klt downstairs complement by finite discrepancy comparison. Away from \(T\), dropping the Cartier divisor \(S\) from the lc pair increases the log discrepancy of every place centered in \(S\) by its strictly positive order. Thus the quotient charts are klt away from \(T\), and the same isomorphism makes their canonical sheaves invertible there. Off \(T\) the separating blowup is an isomorphism, while the other operations are finite; hence the composite map to \(X\) is quasi-finite near \(S\setminus T\). The same quotient preserves Cohen--Macaulayness of both the ambient space and \(S\). Before the quotient the ambient space is klt and hence Cohen--Macaulay, and \(S\) is Cartier there. Locally its finite quotient ring, and the quotient ring of \(S\), are invariant direct summands of those finite Cohen--Macaulay rings. A system of parameters downstairs is one upstairs; the vanishing of lower local cohomology upstairs and the invariant projection give its vanishing for the downstairs direct summand. This proves the stated depth. Removal of the relative inertia kernel also makes the strict positive divisor finite and representable over \[ D\times_P\sqrt[\ell]{\OO_P(1)}. \] Its root line is \(\OO_S(S)\), and its image covers \(\Supp G_+\). The intersection \(S\cap T\) maps to the positive--negative or positive--zero intersections: the only new component is the separating exceptional divisor. Before the quotient, the inverse image of \(|D|\) is the support of \(S'_++S'_-+E+p_1^*S_0\). The finite quotient sends this full support to \(|S|\cup|T|\), and the equality of underlying sets persists under the ordinary base change below. The image of \(S\cap T\) in \(P\) consequently has dimension \(0\). This proves Equation~\eqref{signed:full-residual-comparison}. Projection formula gives the same comparison after adding \(D_S=p^*S\). The quotient sequence therefore identifies \[ \begin{aligned} R(p_D)_*\left( \frac{\omega_{\widehat Z}(D_S+C)}{\omega_{\widehat Z}(C)} \right) &\simeq \omega_Z(S+T)|_S[0]\\ &\simeq \OO_S(aS)[0], \end{aligned} \] where \(p_D:D_S\to S\) and the quotient is a sheaf on the possibly nonreduced divisor \(D_S\). To justify the displayed derived statement, first push it by the exact closed immersion \(S\hookrightarrow Z\). There it is the quotient of the two acyclic comparisons. Closed pushforward detects cohomology sheaves, and canonical truncation yields the statement on \(S\). There is a quotient morphism \begin{equation}\label{signed:quotient-map} \begin{aligned} \omega_H(A)&= \frac{\omega_{\widehat Z}(H+A)}{\omega_{\widehat Z}(A)} \\ &\longrightarrow \frac{\omega_{\widehat Z}(D_S+C)}{\omega_{\widehat Z}(C)}. \end{aligned} \end{equation} Indeed \(H+A\leq D_S+C\) and \(A\leq C\). This morphism need not be injective when \(C\) and \(H\) share a component. Compose its derived pushforward with the preceding isomorphism. The result retracts Equation~\eqref{signed:derived-insertion}: at each generic point of \(S\), its composite is the identity under the fixed residue convention. The composite is an endomorphism of the invertible sheaf \(\OO_S(aS)\) in degree zero; since \(S\) is reduced, agreement at every generic point implies agreement everywhere. Applying \(Rg_*\) proves Equation~\eqref{signed:residue-injection}. All maps use the fixed adjoint isomorphism, divisor ideals, and residue, so they commute with the germ comparisons. Relative canonical forms under products, followed by wedging with the canonical frame of the new factor, give the last assertion. \end{proof} \subsection{The localized Hodge argument} The target in Equation~\eqref{signed:residue-injection} contains the additional poles \(A\). The following version of the filtered SNC calculation incorporates them. In its application, we embed \(H\) by the graph of \(h\) in \(\widehat Z\times U\) and use the projection to \(U\) as the ambient map. This projection is submersive, while its restriction to the graph is the proper map \(h\); it requires no extension of \(h\) to \(\widehat Z\). We will use the same graph construction after a change of parameter. The parameter is projective only in the last assertion below; the closed-stratum maps are proper Kähler maps. \begin{proposition}\label{signed:localized-hodge} Let \(q:\mathscr V\to Y\) be a holomorphic map of complex manifolds, submersive near a reduced closed analytic subspace \(i:H\hookrightarrow\mathscr V\) of pure dimension \(d\) and codimension \(c_0>0\). Suppose \(h=q|_H\) is proper. Assume that locally \(H\) is an SNC divisor in a smooth submanifold of codimension \(c_0-1\), and that a reduced divisor \(A\) is simultaneously SNC and transverse to those support equations. Assume that the components \(H_i\) and \(A_j\) near \(H\) have finite global smooth indexing, and that every closed stratum \(H_I\cap A_J\), with \(I\ne\varnothing\), is smooth and proper over \(Y\) and carries a global relative Kähler form. Empty or disconnected strata are allowed. Here \(H_I=\bigcap_{i\in I}H_i\) and \(A_J=\bigcap_{j\in J}A_j\), with \(A_\varnothing=\mathscr V\). Use right \(\mathscr D\)-modules and finite-pole local cohomology, and put \[ \mathcal K_A=\mathcal H^{c_0}_{[H]}(\omega_{\mathscr V})(*A). \] Its order filtration is generated from \(F_0\mathcal K_A=i_*\omega_H(A)\), with \(F_p=0\) for \(p<0\). For \(M^j=\mathcal H^j q_+\mathcal K_A\), the filtered direct image is strict at every level, and \(M^j\) has a finite filtration by submodules, strict for \(F\), whose quotients are polarizable real pure Hodge modules. In particular, \begin{equation}\label{signed:lowest-step} F_0M^j=R^jh_*\omega_H(A)\lhook\joinrel\longrightarrow M^j, \qquad F_pM^j=0\quad(p<0). \end{equation} Define the symbol morphism \[ \sigma:F_0M^j\otimes T_Y\longrightarrow\gr^F_1M^j, \qquad \sigma(m\otimes\xi)=[m\xi]. \] If \(Y\) is smooth projective, then for every ample line \(N\) on \(Y\), \begin{equation}\label{signed:symbol-vanishing} \Hom_{\OO_Y}(N,\Ker\sigma)=0. \end{equation} This includes any coherent torsion in the kernel. \end{proposition} \begin{proof} We verify the residual localization in the proof of \cite[Proposition~11.1]{BL}. We first identify the lowest order step and a filtration by the number of branch poles. Its graded pieces will be dual constant modules on smooth strata, to which proper Kähler direct image applies. The resulting strictness gives the injection at the lowest filtration step, and the graded de Rham complex on a projective base gives the symbol vanishing. At a point of \(H\), choose simultaneous coordinates \[ (z_1,\ldots,z_{c_0-1},x_1,\ldots,x_t,y_1,\ldots,y_s,\xi), \quad \mathscr I_H=(z_1,\ldots,z_{c_0-1},x_1\cdots x_t), \quad A=(y_1\cdots y_s=0). \] Let \(\eta\) be the coordinate volume form. The localization Čech complex for the displayed regular support equations has cohomology only in degree \(c_0\). Localizing this cohomology also in the \(y\)'s gives \(\mathcal K_A\). Successive finite Taylor divisions give its unique additive polar normal forms: finite sums of \begin{equation}\label{signed:polar-form} f(x_{I^c},y_{J^c},\xi) \prod_{\nu=1}^{c_0-1}z_\nu^{-u_\nu} \prod_{i\in I}x_i^{-v_i} \prod_{j\in J}y_j^{-w_j}\eta, \quad I\ne\varnothing, \end{equation} where every displayed order is positive and \(J\) is arbitrary, including empty. The coefficient is independent of exactly the variables occurring negatively. The normal form is an additive statement, not an \(\OO_{\mathscr V}\)-linear decomposition. The simple fractions, with each displayed order equal to one, are exactly the image of \(\omega_H(A)\). More explicitly, their unreduced representatives are \[ \frac{f\eta} {z_1\cdots z_{c_0-1}(x_1\cdots x_t)(y_1\cdots y_s)}. \] The regular-equation residue identifies their numerators modulo \(\mathscr I_H\) with the absolute dualizing sheaf of \(H\), twisted by \(A\). This map is injective. If the fraction has no remaining \(x\)-pole, Taylor division says that its numerator modulo the \(z\)'s is divisible by \(x_1\cdots x_t\). Localization in the \(y\)'s does not enlarge this kernel, because each \(y_j\) is a nonzerodivisor modulo \(\mathscr I_H\). This also proves the identification under changes of coordinates, including the determinant of the normal conormal frame. Give a term of Equation~\eqref{signed:polar-form} the excess \[ e=\sum_\nu(u_\nu-1)+\sum_{i\in I}(v_i-1) +\sum_{j\in J}(w_j-1). \] The right canonical action of a coordinate derivative is minus differentiation of the coefficient of \(\eta\). A derivative in a denominator variable increases its order by one with a nonzero scalar. Because the coefficient in the normal form is independent of those variables, every term of excess \(e\) is obtained from a simple term by \(e\) such derivatives. Conversely, order \(p\) differentiations produce excess at most \(p\). Thus \(F_p\mathcal K_A\) consists exactly of the finite sums of excess at most \(p\). This is the good order filtration generated from \(i_*\omega_H(A)\). There is also an increasing filtration \(W\) by the total number of branch poles. Intrinsically, its step of index \(-d+k-1\) is the sum of the images of support modules for subunions of the \(H_i\), localized along sublists of the \(A_j\), for which the total number of selected branches is at most \(k\). In the normal form it imposes \(|I|+|J|\leq k\). The induced double filtration therefore has \begin{equation}\label{signed:localized-graded} \gr^W_{-d+k-1}(\mathcal K_A,F) \simeq \bigoplus_{\substack{|I|+|J|=k\\ I\ne\varnothing}} (i_{I,J})_+(\omega_{H_I\cap A_J},F^{(0)}). \end{equation} Here \(F^{(0)}_p\omega_{H_I\cap A_J}=0\) for \(p<0\) and is \(\omega_{H_I\cap A_J}\) for \(p\geq0\). Here \(A_\varnothing=\mathscr V\), and empty intersections contribute zero. To check this formula, retain the polar type using exactly \(I,J\). Its normal denominators are the \(c_0-1\) equations \(z_\nu\), the \(|I|\) selected \(x\)'s, and the \(|J|\) selected \(y\)'s. They are the regular equations of the smooth stratum \(H_I\cap A_J\), whose dimension is \(d-k+1\). Right closed transfer has no codimension shift in \(F\), and the excess counts exactly its normal derivative orders. Mayer--Vietoris boundary maps for the \(x\)-subunions and the localization boundary for each selected \(y\) identify these quotients intrinsically with the support module of that intersection. For the latter assertion, localization modulo the nonlocalized module in one transverse \(y\) direction is its first local cohomology in that direction. A fixed order of the component labels fixes the residue signs, so the identifications glue. The smooth- support graded modules are regular holonomic, and their finite extension \(\mathcal K_A\) is regular holonomic as well. The filtration \(W\) has a real realization. Tensor the finite-pole de Rham comparisons for the regular support coordinates with the open localization comparison for the \(y\) coordinates, whose one-coordinate factor is the cohomology of a punctured disk. These comparisons are natural for all sublists. The resulting real support and localization complexes complexify to the de Rham complexes just computed. The latter are perverse by regular holonomicity; exact and faithful complexification gives the same assertion for the real complexes. Their real perverse images then give \(W\). On a graded stratum of dimension \(d-k+1\), the real object is its dual constant object \(\R^H[d-k+1](d-k+1)\). Its first right filtration step is zero and its weight is \(-d+k-1\), in agreement with Equation~\eqref{signed:localized-graded}. Residue signs and constant normalizations preserve its real polarization. This is the real comparison in \cite[Section~11.2]{BL}, with the punctured-disk factors supplying the additional localization boundaries. The proper-support check also survives localization. If a holomorphic germ \(f\) vanishes on reduced \(H\), then \begin{equation}\label{signed:support-condition} fF_p\mathcal K_A\subset F_{p-1}\mathcal K_A. \end{equation} For a generator \(z_\nu\) of \(\mathscr I_H\), multiplication reduces its pole order or kills the class. Multiplication by \(x_1\cdots x_t\) also lowers the excess or kills the class, because the polar type always has \(I\ne\varnothing\). The \(y\)-poles do not affect this argument. In particular every graded stratum stays in the proper support \(H\), and properness is needed only there. We have now identified the order filtration, its real pure stratum pieces, and the proper support of each level. We next pass these data to the base. Apply Saito's constant-source Kähler direct-image theorem \cite[Theorem~1 and Remark~1.2]{SaitoKahler} to each smooth closed stratum in Equation~\eqref{signed:localized-graded}, using its stipulated relative Kähler form. The resulting degree-\(j\) direct image of the step indexed by \(\lambda\) is strict and polarizable real pure of weight \(\lambda+j\). In the submersion neighborhood, the level-\(F_p\) relative right de Rham complex has term \[ F_{p-e}\mathcal K_A\otimes\bigwedge^e T_{\mathscr V/Y} \quad\text{in degree }-e. \] The simultaneous excess and branch bounds show that passing to a \(W\)-quotient commutes with taking each such level. To prove strictness, use the finite \(W\) spectral sequence \[ E_1^{-\lambda,j+\lambda} =\mathcal H^j q_+\gr^W_\lambda\mathcal K_A \quad\Longrightarrow\quad \mathcal H^j q_+\mathcal K_A. \] Its first-page terms and their level filtrations are strict pure objects by the preceding application. The differentials are real and filtration preserving, because the support and localization comparisons are real and natural. The first differential preserves weight, hence is a strict morphism of pure Hodge modules; its kernels and cokernels remain pure. A higher differential sends \((\lambda,j)\) to \((\lambda-k,j+1)\) for \(k\geq2\) and strictly lowers weight. Such a real filtered map is zero. On distinct strict supports this follows from strict support; on the same dense smooth support a real map preserves both Hodge filtrations, and a source vector of type \((p,q)\) would have image in \(F^p\cap\overline F^q\) of a pure object of smaller weight, which is zero. Both the unfiltered and each level spectral sequence therefore degenerate at the second page. Their comparison is injective there. A least-\(W\)-step argument on the finite abutment proves injection at every \(F\) level and strictness of its induced \(W\) filtration. This is the strict-direct-image argument of \cite[Proposition~11.1]{BL}, now justified for every graded term with \(A\)-poles. For \(p<0\), the relative complex is zero. For \(p=0\), its only term is \(F_0\mathcal K_A=i_*\omega_H(A)\) in degree zero. The level injection gives Equation~\eqref{signed:lowest-step}. Suppose finally that \(Y\) is smooth projective. Each strict- support pure quotient just obtained is of exactly the type in \cite[Lemma~11.2]{BL}: a polarizable real pure direct-image piece arising from a dual constant on a smooth Kähler source. That lemma compares its actual right filtration with the Sabbah--Schnell pure component extending the same generic polarized real variation. Its proof uses uniqueness of the intermediate extension and the reconstruction of \(F\) from the canonical \(V\)-filtration, so it depends only on this pure piece, not on the divisor presentation of its source. It gives the negative-ample vanishing \[ \mathbb H^u\bigl(Y,N^{-1}\otimes\gr^F_p\mathrm{DR}_Y Q\bigr)=0 \quad(u<0) \] for each pure quotient \(Q\), by \cite[Theorem~16.3.10]{SabbahSchnell}. The finite strict filtration extends this vanishing to \(M^j\). Since its negative \(F\) levels vanish, its degree-one graded right de Rham complex is exactly \[ \gr^F_1\mathrm{DR}_Y M^j= \left[F_0M^j\otimes T_Y\xrightarrow{\sigma}\gr^F_1M^j\right] \quad\text{in degrees }-1,0. \] Its negative hypercohomology after tensoring by \(N^{-1}\) is \(H^0(Y,N^{-1}\otimes\Ker\sigma)\). It vanishes, which is Equation~\eqref{signed:symbol-vanishing}. This calculation does not assume that the kernel is locally free. \end{proof} \subsection{Lifting every finite boundary neighborhood} Return to the charts and resolution of Lemma~\ref{signed:root-charts} and Equation~\eqref{signed:resolved-divisors}. On an empty chart the sheaves below are zero. On a nonempty chart put \(I=\OO_Z(-S)\), an invertible ideal. For every integer \(j\) and positive integer \(k\), put \[ \mathcal A_j=I^j/I^{j+1}=\OO_S(-jS),\qquad \mathcal T_{j,k}=I^j/I^{j+k}. \] These are sheaves on the underlying topological space of \(S\). Pushforward by \(g\) has that meaning for \(\mathcal T_{j,k}\); it does not require a morphism from a thickening to \(U\). A local frame of \(R_U^{-1}\) gives a Laurent graded frame \(u^j\) of \(\mathcal A_j\) for every \(j\). It is a frame on the associated graded, not an extension of a frame to \(Z\). \begin{proposition}[All finite lifting orders]\label{signed:all-orders} For every root chart, every \(j\in\Z\), and every \(k\geq1\), the map of sheaves of complex vector spaces on \(U\) \begin{equation}\label{signed:all-order-surjection} g_*\mathcal T_{j,k+1}\longrightarrow g_*\mathcal T_{j,k} \end{equation} is surjective. The assertion holds on every parameter germ, including germs supported at special parameters. \end{proposition} \begin{proof} The assertion is immediate for an empty chart. We induct on \(k\), simultaneously for all integer \(j\) and all nonempty charts. The layer algebra and its connecting maps are the ones in \cite[Lemma~12.1]{BL}; we recall their construction to specify exactly which obstruction is to be killed. Assume all smaller orders. The connecting map for \[ 0\longrightarrow\mathcal A_{j+k} \longrightarrow\mathcal T_{j,k+1} \longrightarrow\mathcal T_{j,k}\longrightarrow0 \] factors uniquely through a map \begin{equation}\label{signed:obstruction} \delta_k:g_*\mathcal A_j\longrightarrow R^1g_*\mathcal A_{j+k}. \end{equation} Indeed the smaller orders make \(g_*\mathcal T_{j,k}\to g_*\mathcal A_j\) surjective. For \(k>1\), the kernel comes from \(g_*\mathcal T_{j+1,k-1}\), by the leading-layer sequence. The order \(k-1\) in degree \(j+1\) lifts this kernel into \(g_*\mathcal T_{j,k+1}\), so the current connecting map kills it. For \(k=1\) the leading map is the identity. Vanishing of all maps in Equation~\eqref{signed:obstruction} is equivalent to the surjectivity at the current order. The maps \(\delta_k\) form a degree-\(k\) derivation on the Laurent graded algebra \(\bigoplus_jg_*\mathcal A_j\), with values in its first-cohomology module. On a parameter germ, choose a lift of a leading section \(x\) to \(g_*\mathcal T_{j,k}\), and represent that lift locally by \(x_\alpha\in I^j\). Then \(x_\beta-x_\alpha\in I^{j+k}\). Their differences modulo \(I^{j+k+1}\) represent \(\delta_k(x)\). For another degree-\(l\) section \(z\), the product difference is \[ x_\beta z_\beta-x_\alpha z_\alpha =x_\alpha(z_\beta-z_\alpha)+z_\alpha(x_\beta-x_\alpha) +(x_\beta-x_\alpha)(z_\beta-z_\alpha). \] The last term lies in \(I^{j+l+2k}\subset I^{j+l+k+1}\). This proves the derivation rule. If \(t_1,\ldots,t_b\) are coordinates on \(U\), choose simultaneous representatives \(G_{i,\alpha}\) of their lifts to \(g_*\mathcal T_{0,k}\); their differences lie in \(I^k\). Taylor's formula modulo their squared differences gives, for any holomorphic \(F\), \begin{equation}\label{signed:chain-rule} \delta_k(F(t))=\sum_i F_{t_i}(t)\delta_k(t_i). \end{equation} Both formulas hold on germs, without removing torsion in the target, and commute with the prescribed germ comparisons. Put \(m_k=a+k>0\). An order-\(k\) obstruction raises degree by \(k\), whereas the residue injection accepts degree \(-a\). We therefore test leading sections \(x\) of degree \(-m_k=-(a+k)\). Apply the split residue injection to define \[ c(x)=\operatorname{res}_*\delta_k(x),\qquad e_i(x)=\operatorname{res}_*\bigl(x\delta_k(t_i)\bigr) \quad\text{in }R^1h_*\omega_H(A). \] Both arguments have degree \(-a\), the degree of \(\OO_S(aS)\) in Equation~\eqref{signed:residue-injection}. The next construction places these classes in a filtered direct-image module. For the identity graph, the calculation will give the undivided relation \(c(x)+\sum_i e_i(x)\partial_{t_i}=0\). Taking its degree-one symbol gives \(\sigma(\sum_i e_i(x)\otimes\partial_{t_i})=0\). We will globalize that symbol on a projective coordinate-power cover, where the boundary root is an ample line. The adjugate formula expresses the symbol across the ramification of this cover without dividing by its Jacobian determinant. The symbol vanishing will then kill the obstructions of the parameter coordinates on every stalk; the undivided relation will kill \(\delta_k(x)\), and the derivation rule will recover the other degrees. Let \(\psi:U'\to U\) be a holomorphic map between coordinate opens of dimension \(b\), with coordinates \(w\) on \(U'\), transverse to every smooth closed stratum \(H_I\cap A_J\). The identity map is allowed. In \(\widehat Z\times U'\), the graph support \(H^\flat=H\times_U U'\) is an SNC divisor in a smooth complete intersection of codimension \(b\). The residual divisor remains transverse. After shrinking about a compact fiber, the finite-component Lemma~10.3 of \cite{BL} gives finite smooth indexing of the graph strata, and their relative Kähler forms restrict from the product neighborhood. Complete-intersection adjunction gives the canonical factor \[ \omega_{U'}\otimes\psi^*\omega_U^{-1}, \] whose coordinate frame we denote by \(dw/dt\). This is a ratio of canonical frames, with no inverse Jacobian. Pull the Čech representatives of \(c(x),e_i(x)\) to this graph and tensor by that frame. Proposition~\ref{signed:localized-hodge} places the resulting classes \(\widetilde c,\widetilde e_i\) in \(F_0M^\flat\), where \(M^\flat\) is the degree-one direct-image module of the localized graph support. Write \(J=(\partial\psi_i/\partial w_j)\) and \(J^\#=\operatorname{adj}(J)\). Because the adjugate is polynomial in \(J\), the following identity remains meaningful where \(J\) is singular: \begin{equation}\label{signed:adjugate} \widetilde c\det J+ \sum_{i,j}(\widetilde e_i\partial_{w_j})J^\#_{ji}=0 \quad\text{in }M^\flat. \end{equation} We verify the change from \cite[Lemma~12.2]{BL} explicitly. Choose simultaneous representatives of the lifts to length \(k\) \[ x_\alpha\in I^{-a-k},\quad x_\beta-x_\alpha\in I^{-a}, \qquad G_{i,\beta}-G_{i,\alpha}\in I^k. \] After pulling to \(\widehat Z\), put \(\eta_\alpha=x_\alpha\tau\), using the meromorphic form defined by Equation~\eqref{signed:adjoint-chart}, and put \(Q_{i,\alpha}=\psi_i(w)-G_{i,\alpha}\), \(P_\alpha=\prod_iQ_{i,\alpha}\). On an overlap write \(\Delta_i=G_{i,\beta}-G_{i,\alpha}\). The insertion and \(D_S\geq H\) give the precise pole bounds \begin{align} \eta_\alpha&\in\omega_{\widehat Z}(H+A+kD_S),\nonumber\\ \eta_\beta-\eta_\alpha,\quad \eta_\alpha\Delta_i &\in\omega_{\widehat Z}(H+A),\label{signed:pole-bounds}\\ \eta_\alpha\Delta_i\Delta_l,\quad (\eta_\beta-\eta_\alpha)\Delta_i &\in\omega_{\widehat Z}(H+A-kD_S) \subset\omega_{\widehat Z}(A).\nonumber \end{align} Thus every quadratic or cross difference has no \(H\)-pole, although it may retain a pole on \(A\). Such a term is zero in the support module localized along \(A\). In that localized graph support module consider the finite generalized fractions \[ S_\alpha=\left[\frac{\eta_\alpha\wedge dw}{P_\alpha}\right]. \] The \(H\)-pole is already in the numerator. Each polar class is killed by a power of a reduced equation of \(H\), and each \(\Delta_i\) is a multiple of that equation. Changing \(Q_{i,\alpha}\) to \(Q_{i,\beta}\) therefore has a finite geometric expansion on each class. The last line of Equation~\eqref{signed:pole-bounds} kills all terms beyond the linear terms, and gives the exact equality \[ S_\beta-S_\alpha= \left[\frac{(\eta_\beta-\eta_\alpha)\wedge dw}{P_\alpha}\right] +\sum_i\left[ \frac{\eta_\alpha\Delta_i\wedge dw} {Q_{i,\alpha}P_\alpha}\right]. \] Let \(C_\bullet\) be the first cochain. Let \(E_{i,\bullet}\) be the cochain with the numerator of the \(i\)-th summand and only denominator \(P_\alpha\), and let \(E_{i,\bullet}^{(l)}\) have its additional denominator \(Q_{l,\alpha}\). The simple cochains \(C_\bullet,E_{i,\bullet}\) are cocycles representing \(\widetilde c,\widetilde e_i\); their possible triple-overlap errors are precisely the cross terms killed above. The normal residue frame in this identification is \(dw/dt\). The numerator of \(E_{i,\bullet}\), including its \(A\)-poles, is independent of \(w\). Right differentiation therefore gives \[ E_{i,\bullet}\partial_{w_j}=\sum_lJ_{lj}E_{i,\bullet}^{(l)}. \] The preceding fraction equality is \(\check dS_\bullet=C_\bullet+\sum_iE_{i,\bullet}^{(i)}\). Multiplying by \(\det J\) and using \(JJ^\#=(\det J)\id\) gives at the cochain level \[ \check d(S_\bullet\det J)=C_\bullet\det J+ \sum_{i,j}(E_{i,\bullet}\partial_{w_j})J^\#_{ji}. \] These cochains are in the end term of the relative right de Rham complex, so a Čech coboundary there is a total coboundary. Passing to the direct image proves Equation~\eqref{signed:adjugate}, with \(J^\#_{ji}\) placed after the right derivative as displayed. Passing to \(\gr^F_1\) yields \begin{equation}\label{signed:local-symbol} \sigma\left(\sum_{j,i}J^\#_{ji}\widetilde e_i \otimes\partial_{w_j}\right)=0. \end{equation} For \(\psi=\id\) the undivided identity is \begin{equation}\label{signed:undivided} c(x)+\sum_i e_i(x)\partial_{t_i}=0. \end{equation} For \(b=0\) there are no graph equations, and the same undivided cochain computation gives \(c(x)=0\). We now prove that the obstructions of the parameter coordinates vanish on every stalk, including an obstruction supported only at a special parameter. Suppose \(b>0\), and fix \(t_*\in P\). We will use one compact graph over a projective parameter space, obtained by a parameter change unbranched above \(t_*\). The vanishing of \(\Hom\) in Equation~\eqref{signed:symbol-vanishing} applies to the entire coherent symbol kernel, and the local isomorphism above \(t_*\) will detect vanishing on the full germ there. The construction of \cite[Proposition~10.5]{BL} gives a coordinate- power map, here of exponent \(\ell\), \[ \begin{gathered} \psi:P'=\PP^b\longrightarrow P,\qquad [w_0:\cdots:w_b]\longmapsto[w_0^\ell:\cdots:w_b^\ell],\\ R=\OO_{P'}(1),\qquad R^\ell\simeq\psi^*\OO_P(1), \end{gathered} \] in suitably chosen coordinates, unbranched over \(t_*\). We spell out why its geometry also accommodates \(A\). Choose finitely many relatively compact parameter opens covering \(\phi(D)\), with the chart resolutions defined on slightly larger opens. Empty charts contribute no strata. Properness of the nonempty supports leaves only finitely many closed smooth strata \(H_I\cap A_J\) meeting the corresponding compact sets. A general tuple of coordinate hyperplanes is transverse to the maps of every such stratum, for every subtuple, and avoids \(t_*\). This follows by Sard's theorem applied to the incidence with variable hyperplanes; varying each hyperplane supplies its normal direction. In these coordinates \(d\psi\) has image the tangent space to the intersection of its vanishing coordinate hyperplanes. The transversality condition is consequently \[ dh(T_z(H_I\cap A_J))+d\psi(T_wP')=T_{h(z)}P \quad\text{whenever }h(z)=\psi(w). \] It gives the required simultaneous regular graph equations and SNC transverse residual divisors. To obtain a single compact graph support, first form the compact graph \(\Xi=D\times_P P'\subset X\times P'\). On it the pullback of \(N_0\) has the specified root \(R\). Lemma~10.1 of \cite{BL} extends that root to a neighborhood of \(\Xi\). Perform the full construction of Lemma~\ref{signed:root-charts} on this neighborhood, including the relative quotient and its ordinary pullback, and use the canonical sheaf relative to \(P'\) in the periodicity algebra; the absolute canonical factor is added only when taking graph residues. Resolve the same ordered total divisor data before imposing the graph equations. Near a compact graph slice, the root comparison lemma identifies this construction with the ordinary product of a local chart with a parameter open, as an ambient germ preserving the ideal and the adjoint form. Uniqueness of the comparison as a germ and properness of the graph allow a base shrink on which the identification holds on the entire slice. Normalization thus occurs before ramified graph base change. Smooth-functorial resolution preserves the product identification. The graph cut \(H'\) in this resolved ambient space is compact, has pure dimension \(n-1\), and is locally an SNC divisor in a smooth complete intersection of codimension \(b\). Its ambient projection to \(P'\) is submersive near \(H'\). Let \(A'\) be the reduced ambient resolved residual divisor, restricted to a neighborhood of \(H'\). Under each product-germ identification it is \(A\times U'\); its restriction to the graph complete intersection is the transverse residual cut. The chosen transversality makes \(A'\) simultaneously SNC with the graph support equations. All closed strata \(H'_I\cap A'_J\) are compact and smooth; splitting their connected components leaves finitely many indices. The relative metric construction in Lemma~\ref{signed:root-charts} gives one Kähler form near \(H'\), and hence the required forms on every stratum. Thus Proposition~\ref{signed:localized-hodge} applies to this single global graph. Write \(M'\) for its degree-one module. Take \(x=u^{-m_k}\). On a graph chart choose a frame \(\vartheta\) of \(R\) compatible with the downstairs frame of \(\OO_P(1)\). The local rule \begin{equation}\label{signed:global-symbol} \vartheta^{m_k}\longmapsto \sum_{j,i}J^\#_{ji}\widetilde e_i(u^{-m_k}) \otimes\partial_{w_j} \end{equation} defines a global morphism \(R^{m_k}\to F_0M'\otimes T_{P'}\). Here is the transition check, including the ramification locus. For changes of coordinates let \(A_t=\partial t'/\partial t\) and \(B_w=\partial w'/\partial w\), and let the root frame change by \(\vartheta'=\lambda\vartheta\). Locally \(\lambda\) descends from a holomorphic unit downstairs: its \(\ell\)-th power does, and after choosing a local \(\ell\)-th root of that unit the remaining ratio is a locally constant element of \(\mu_\ell\). Root germ compatibility and Equation~\eqref{signed:chain-rule} transform the downstairs column of \(e_i\)'s by \(\lambda^{m_k}A_t\). No derivative of \(\lambda\) appears, since \(x\) multiplies outside \(\delta_k(t_i)\). The absolute canonical factor changes by \(\det B_w/\det A_t\). Hence \[ \begin{aligned} \widetilde e'&=\frac{\det B_w}{\det A_t}\lambda^{m_k}A_t\widetilde e, &J'&=A_tJB_w^{-1},\\ (J')^\#\widetilde e'&=\lambda^{m_k}B_wJ^\#\widetilde e. \end{aligned} \] The adjugate equality is polynomial in \(J\), so holds also when \(J\) is singular. The last formula, together with the inverse change of the vector basis, is exactly the transition of Equation~\eqref{signed:global-symbol}. The graph classes here are natural Čech pullbacks through ambient germs; no ramified base-change isomorphism for \(R^1h_*\) is used. Equation~\eqref{signed:local-symbol} puts the image of this morphism in the first-symbol kernel. Since \(R^{m_k}=\OO_{\PP^b}(a+k)\) is ample, Equation~\eqref{signed:symbol-vanishing} makes the morphism zero. At a point above \(t_*\), the map \(\psi\) is locally biholomorphic and \(J^\#\) is invertible. The ambient germ comparison identifies the graph classes with the downstairs classes there. Thus \(e_i(u^{-m_k})=0\) as germs at \(t_*\), including any class supported there. The point was arbitrary, so these classes vanish on every chart. The split injection in Lemma~\ref{signed:residue} and the invertible Laurent frame now give \(\delta_k(t_i)=0\). For every leading section \(x\) of degree \(-m_k\), the classes \(e_i(x)\) vanish. Equation~\eqref{signed:undivided}, the injection in Equation~\eqref{signed:lowest-step}, and then the split residue injection give \(\delta_k(x)=0\). When \(b=0\), the direct identity \(c(x)=0\) and the same two injections give this conclusion without a graph test. In particular, the derivation rule and characteristic zero give \[ 0=\delta_k(u^{-m_k})=-m_k u^{-m_k-1}\delta_k(u), \qquad \delta_k(u)=0. \] For any local \(F\in g_*\OO_S\), the degree-\(-m_k\) section \(Fu^{-m_k}\) then gives \(0=\delta_k(Fu^{-m_k})=u^{-m_k}\delta_k(F)\). Every graded section is \(Fu^j\), so \(\delta_k\) vanishes in every degree. This proves the current order and completes the induction. \end{proof} \subsection{Compact deformations and the signed theorem} We now turn the infinitesimal lifting into actual compact subspaces. The role of all the integer layers in Proposition~\ref{signed:all-orders} is that both the equations of a boundary fiber and a generator of its normal direction can be lifted compatibly. This is the analytic counterpart of the compact-family step in \cite[Proposition~5.1]{LI}. \begin{lemma}\label{signed:escaping-fibers} Assume \(00\). On the selected component the locus where it is singular or where \(g\) has rank less than \(r\) is proper. Removing its image, along with the preceding proper bad subsets, under the finite map leaves a nonempty open \(D_i^\circ\subset D_i\). Its points have preimages at smooth points of these fiber components. Fix \(x\in D_i^\circ\) and a point above it in the compact fiber \(F=g^{-1}(t)\), for \(t\in Y\). Work on the single ordinary chart neighborhood of that fiber from Lemma~\ref{signed:root-charts}. Take regular coordinates \(t_1,\ldots,t_r\) on \(Y\) centered at \(t\), extended to local coordinates of its smooth embedding in \(P\). Their pullbacks form a regular sequence on \(S\) along \(F\), by flatness. In addition, \(S\) is Cohen--Macaulay, so \(F\) is Cohen--Macaulay of pure dimension \(d\). Choose once and for all a Hausdorff open neighborhood \(W\subset Z\) of \(F\), disjoint from \(T\), on which \(\pi\) is quasi-finite. Set \(I=\OO_Z(-S)\). Proposition~\ref{signed:all-orders} lets us lift the finitely many \(t_i\)'s to compatible germs of sections of \(\OO_Z/I^q\), \(q\geq1\), along \(F\). It also lifts the conormal frame \(u\in I/I^2\) to compatible germs \(\widetilde y_q\in I/I^{q+1}\) along \(F\); write \(\widetilde y\) for this system. At each order choose simultaneous representatives on a neighborhood of \(F\) contained in \(W\). These neighborhoods may shrink with the order. The leading coefficient of \(\widetilde y\) is a unit frame, so it generates \(I\) on each finite thickening. For \(R_q=\C[s]/(s^q)\), cut the lifted \(r\) equations in the Cartier thickening \(qS\), and map \(s\) to \(\widetilde y\). Denote the resulting subspace by \(F_q\). It is flat over the Artin analytic point \(\Spec R_q\), with closed fiber \(F\). Before cutting, multiplication by the Cartier generator identifies each successive \(s\)-layer with \(\OO_S\); this is the flatness criterion over \(R_q\). Quotienting by lifts of the closed-fiber regular sequence preserves flatness, by the local flatness criterion. Compatibility of the lifted equations gives compatible reductions of all \(F_q\). Each of these is an embedded compact deformation in the same fixed neighborhood \(W\). To see why the shrinking representative neighborhoods cause no difficulty, the support of \(F_q\) is the fixed compact set \(F\). Its ideal on its representative neighborhood glues with the unit ideal on \(W\setminus F\), since the deformation is empty on their overlap away from \(F\). This realizes it as a closed subspace of \(W\times\Spec R_q\). The ideal is coherent locally on both opens, and the family is proper over the Artin point because its support is compact. The Douady space of compact subspaces of \(W\) represents proper flat embedded families \cite[Section~9]{Douady1966}. The compatible \(F_q\)'s thus define a formal arc in its analytic germ at \([F]\). Embed that germ in a finite dimensional analytic coordinate space. The arc is a formal solution of its convergent defining equations. Analytic Artin approximation \cite[Theorem~1.2]{Artin1968} gives a convergent arc agreeing modulo \(s^2\). Pull back the universal family and shrink its disk to obtain \(\mathscr F\subset W\times\Delta\), proper and flat over \(\Delta\), with closed fiber \(F\) and the prescribed first normal displacement. The family may be taken to have pure \(d\)-dimensional fibers throughout the disk. Here is a local justification. The central fiber is Cohen--Macaulay of dimension \(d\). Flatness over the regular one-dimensional base makes its parameter a nonzerodivisor and gives the depth and dimension of the total local ring as \(d+1\) along that fiber. Cohen--Macaulayness is open for analytic local rings, and properness allows a disk shrink for which it holds on the whole family. Every total component then meets the central fiber after shrinking, and flatness makes it dominate the disk; the dimension formula gives it dimension \(d+1\). Quotient by the nonzerodivisor at each parameter gives Cohen--Macaulay fibers of dimension \(d\). The analytic dimension formula and unmixedness of these local rings give the asserted purity. Let \(f\) be a local equation of \(S\). On \(\mathscr F\) it vanishes on the central fiber, so flatness writes it as \(f=s h\). Agreement modulo \(s^2\) with the formal deformation says that \(h|_F\) is a unit: this is precisely the lifted conormal generator. Finitely many such neighborhoods cover the compact \(F\). Properness of \(\mathscr F\to\Delta\) lets us shrink the disk so that they cover the entire family and all their \(h\)'s are units; remove the closed image of the complement. Thus a nonzero fiber misses \(S\). The neighborhood \(W\) already misses \(T\), and the set equality in Lemma~\ref{signed:root-charts} gives \(\pi^{-1}(|D|)\cap W=|S|\cap W\). Hence the nonzero fiber images miss \(D\). The induced map \(\Pi:\mathscr F\to X\times\Delta\) is proper: the source is proper over \(\Delta\) and the target is Hausdorff over it, so the graph is closed and the projection is proper. It is quasi-finite by the choice of \(W\), hence finite. It therefore preserves the dimension of every fiber component. Every total component through a central point dominates the disk by flatness, and its punctured part is dense. Thus every central point, including the point chosen above \(x\), is a limit of points of nonzero fibers of dimension \(d\). Let \(\mathscr Z\subset X\times\Delta\) be the schematic image of this finite map. Its algebra is the coherent image of \(\OO_{X\times\Delta}\to\Pi_*\OO_{\mathscr F}\). Multiplication by a nonzero germ from the disk is injective on \(\OO_{\mathscr F}\), by flatness, and remains injective after the exact finite pushforward. It is therefore injective on the image algebra. That algebra is torsion-free, hence flat, over each local discrete valuation ring of the disk. Thus \(\mathscr Z\) is a proper flat family of subspaces of \(X\). The finite map to its schematic image is surjective, also on each fiber as a map of supports. Each \(\mathscr Z_s\) consequently has pure dimension \(d\). The Douady-to-cycle morphism, in the form recorded in \cite[Section~3.3]{Fujiki1978}, makes their fundamental cycles an analytic family in the cycle space of \(X\). The restriction to a smaller closed disk is compact. The volumes of these cycles against a Kähler form are bounded by continuity, proving the last assertion. \end{proof} \begin{proof}[Proof of Theorem~\ref{signed:torsion}] The assertion is immediate in dimension zero, and the case \(D=0\) follows directly from the signed equivalence. By Lemma~\ref{signed:boundary-geometry}, the case \(v=0\) already gives \(L\sim_{\Q}0\), and \(v=n>0\) contradicts \(a(X)=0\). Suppose, therefore, that \(0