==== TABLE OF CONTENTS ==== Numerical-Semiampleness-of-Nef-Adjoint-Divisors//completion.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//frobenius.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//geometry.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//introduction.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//jet-tools.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//main.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//positive.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//preliminaries.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//reduction.tex Numerical-Semiampleness-of-Nef-Adjoint-Divisors//references.tex =========================== ==== BEGIN Numerical-Semiampleness-of-Nef-Adjoint-Divisors//completion.tex ==== \section{Completion of the full-dimension induction} \label{sec:completion} Theorem~\ref{thm:terminal-big} treats varieties with terminal singularities and linearly trivial canonical divisor. We now pass to klt Calabi--Yau pairs and complete the full-dimension assertion. The boundary perturbation below follows the reduction in \cite[Section~7]{LP}: an effective representative of a small boundary perturbation plus the nef divisor allows ordinary log abundance to be applied to a multiple of that nef divisor. \begin{proposition}\label{prop:full-all} Let $n\geq 2$, and assume $(P_j)$ for every $j0$, \[ (T+aM)\cdot C>0 \] for every integral curve $C$ through a very general point of $X$. Then $T+cM$ is big for every rational $c>0$. \end{proposition} \begin{proof} Proposition~\ref{prop:full-positive} proves the assertion when $\kappa(T)>0$. Proposition~\ref{prop:reduction-cy} reduces the remaining case to the following claim in dimension $n$: \begin{equation}\label{eq:completion-cy-claim} \begin{gathered} (X,B)\text{ projective klt},\qquad K_X+B\equiv 0,\\ M\text{ nef and $\Q$-Cartier of full nef dimension} \quad\Longrightarrow\quad M\text{ big}. \end{gathered} \end{equation} We prove this claim first with $B=0$ and then with arbitrary boundary. We may take a small projective $\Q$-factorialization throughout: the adjoint and $M$ pull back, full nef dimension is preserved on the isomorphism locus, and bigness of the pullback implies bigness downstairs. \medskip \noindent\emph{The case of zero boundary.} Suppose that $X$ is klt and $K_X\equiv0$. Theorem~\ref{input:abundance} makes $K_X$ semiample, hence $K_X\sim_{\Q}0$. Indeed, a semiample numerically trivial divisor defines a morphism with zero-dimensional image, so a sufficiently divisible multiple is linearly trivial. Take the global index-one cover \[ \pi\colon X^{\mathrm{ind}}\longrightarrow X \] associated with a trivialization of the smallest positive multiple of $K_X$ that is linearly trivial. This is a finite quasi-\'etale cover, and \[ K_{X^{\mathrm{ind}}}=\pi^*K_X\sim0. \] The cover is klt. Since its canonical divisor is Cartier, its log discrepancies are positive integers, hence at least one; it therefore has canonical singularities. These are the standard index-one cover properties for klt varieties; see~\cite[Definition~5.19 and Proposition~5.20]{KM}. By the crepant terminalization theorem \cite[Corollary~1.4.3]{BCHM}, there is a projective birational morphism \[ h\colon \widetilde X\longrightarrow X^{\mathrm{ind}} \] with $\widetilde X$ terminal and $\Q$-factorial and $K_{\widetilde X}=h^*K_{X^{\mathrm{ind}}}$. In particular, $K_{\widetilde X}$ is Cartier and linearly trivial, as required by Theorem~\ref{thm:terminal-big}. Set $\widetilde M=(\pi\circ h)^*M$. This divisor is nef and has full nef dimension. To verify the latter assertion, choose a very general point upstairs outside the exceptional locus of $h$ and above the very general locus used for $M$. Every curve through this point has a curve as its image in $X$, and the projection formula gives strictly positive $\widetilde M$-degree. A positive Cartier multiple of $\widetilde M$ is therefore big by Theorem~\ref{thm:terminal-big}. Bigness is detected by a dominant generically finite pullback, so $M$ is big on $X$. \medskip \noindent\emph{Introducing the boundary.} Now let $(X,B)$ satisfy the hypotheses of \eqref{eq:completion-cy-claim}, with $B\ne0$. Theorem~\ref{input:abundance} gives \[ K_X+B\sim_{\Q}0. \] Choose a rational $\alpha>0$ sufficiently small that $(X,(1+\alpha)B)$ remains klt, and put \[ T_*=K_X+(1+\alpha)B\sim_{\Q}\alpha B. \] Thus $T_*$ is pseudo-effective and $\kappa(T_*)\geq0$. The full condition holds for $T_*$ and $M$: for a curve $C$ through a point outside $\Supp B$, we have $T_*\cdot C=\alpha B\cdot C\geq0$, whereas $M\cdot C>0$ if that point is very general. We will show that $T_*+M$ is big without using the conclusion of the present proposition. If $\kappa(T_*)>0$, this follows from Proposition~\ref{prop:full-positive}. Suppose instead that $\kappa(T_*)=0$, and apply Lemma~\ref{lemma:two-nef-model} to $(X,(1+\alpha)B)$ and $M$. Write $Y$ for its final model, $B_Y$ and $N$ for the transforms of $B$ and $M$, and \[ U=K_Y+(1+\alpha)B_Y. \] The output has $U\sim_{\Q}0$ and, for some rational $s>0$, the divisor $U+sN$ is nef of full nef dimension. The maps in the construction extract no divisors, so pushing forward the rational linear equivalence $K_X+B\sim_{\Q}0$ also gives \[ K_Y+B_Y\sim_{\Q}0. \] Subtracting these two equivalences shows that $\alpha B_Y\sim_{\Q}0$. Since $B_Y$ is effective on a projective variety, intersection with a sufficiently ample complete-intersection curve forces $B_Y=0$. The ordinary pair on $Y$ is therefore klt with zero boundary and $K_Y\sim_{\Q}0$. Moreover, $N$ is nef of full nef dimension, because $U+sN\equiv sN$. The zero-boundary case just proved makes $N$, and hence $U+sN$, big. On a common smooth resolution, with maps $p$ to $X$ and $q$ to $Y$, the effective exceptional comparison from the reduction is \[ p^*(T_*+sM)=q^*(U+sN)+E, \qquad E\geq0. \] It follows that $T_*+sM$ is big. Choose a positive rational $\lambda$ with $\lambda<\min\{1,1/s\}$. Then \[ T_*+M =\lambda(T_*+sM)+(1-\lambda)T_*+(1-\lambda s)M \] is big, since the first summand is big and the other two are pseudo-effective. We have thus established bigness of $T_*+M$ in both cases. Choose an effective $\Q$-divisor $G$ with \[ G\sim_{\Q}T_*+M\sim_{\Q}\alpha B+M. \] For a sufficiently small rational $u>0$, the divisor \[ \Delta_u=(1-u\alpha)B+uG \] is effective and $(X,\Delta_u)$ is klt. To see this explicitly, take $u\alpha<1$ and choose $u$ small enough that $(X,B+uG)$ is klt on a fixed log resolution; then $\Delta_u\leq B+uG$. Its adjoint satisfies \[ K_X+\Delta_u \sim_{\Q} K_X+B+u(G-\alpha B) \sim_{\Q}uM. \] This adjoint is nef, so Theorem~\ref{input:abundance} makes it semiample. Consequently $M$ itself is semiample. The morphism given by a free multiple of $M$ has zero-dimensional general fiber: a positive-dimensional fiber through a very general point would contain a curve of $M$-degree zero, contrary to full nef dimension. That morphism is therefore generically finite onto its image, and $M$ is big. This proves \eqref{eq:completion-cy-claim} for all klt Calabi--Yau pairs. Proposition~\ref{prop:reduction-cy} now supplies the case $\kappa(T)=0$, while Proposition~\ref{prop:full-positive} supplies $\kappa(T)>0$. These exhaust the possibilities furnished by the minimal-model and abundance inputs for a pseudo-effective klt adjoint, and establish the proposition. \end{proof} ==== END Numerical-Semiampleness-of-Nef-Adjoint-Divisors//completion.tex ==== ==== BEGIN Numerical-Semiampleness-of-Nef-Adjoint-Divisors//frobenius.tex ==== \section{Frobenius comparison} \label{sec:frobenius} The scalar estimate of Lemma~\ref{lemma:scalar-bound} and the projective-bundle estimate of Proposition~\ref{prop:large-seshadri} will now give the contradiction needed for Theorem~\ref{thm:terminal-big}. Their incompatibility is expressed by the following theorem about fixed complex data. It is the finite-data Frobenius theorem of \cite[Theorem~9.1]{LA}. We give its proof, including the bounds that must be uniform when the residual characteristic varies. The argument compares two orders of vanishing of one determinant. Ordinary jets make the determinant nonzero after reduction to positive characteristic. A small polarization bounds the rank on the diagonal, forcing high vanishing at each diagonal point. A fixed movable curve on a point blowup bounds the vanishing of every section of each determinant factor and gives the opposite inequality. \subsection{The finite-data statement} For a vector bundle on a smooth projective curve, its slope is degree divided by rank. The least and greatest Harder--Narasimhan slopes are denoted by $\mu_{\min}$ and $\mu_{\max}$. For a line bundle on a smooth variety, to generate jets through order $a$ at a point means surjectivity to its stalk modulo the $(a+1)$-st power of the maximal ideal. \begin{theorem}[Finite-data Frobenius incompatibility] \label{thm:finite-data} Let $W$ be a smooth connected projective complex variety of dimension $n\ge2$. Let $D,H$ be rational Cartier divisors, with $H$ ample, let $S$ be an integral Cartier divisor, and fix an integer $q>0$ and real numbers $r>1$, $\epsilon>0$. Suppose \begin{equation}\label{fr:intersection-bounds} \begin{gathered} H^n\le(2\epsilon)^n,\qquad K_WH^{n-1}\le2H^n/r,\qquad (2D+qS)H^{n-1}\le4H^n,\\ (14\epsilon)^n<\tfrac14,\qquad 80\epsilon<\tfrac34. \end{gathered} \end{equation} Assume that the following fixed data exist. \begin{enumerate} \renewcommand{\labelenumi}{\textup{(\roman{enumi})}} \item A smooth integral complete-intersection flag from $W$ to a curve $C$, whose successive divisor classes are $h_i=l_iH$ for fixed positive integers $l_i$ for which $l_iH$ is Cartier, with $\mu_{\min}(\Omega_W^1|_C)\ge0$. \item On the projective bundle \[ Z=\PP\bigl(\mathcal O_W(S)_1\oplus \mathcal O_W(S)_2\bigr)\longrightarrow W\times W, \qquad \xi=c_1(\mathcal O_Z(1)),\qquad M=D_1+D_2+q\xi, \] use the convention that the direct image of $\mathcal O_Z(a\xi)$ is the $a$-th symmetric power of the displayed sum for $a\ge0$; subscripts indicate pullback from the corresponding factor of $W\times W$. The two axes are the sections defined by the two summands. A smooth point $z_*$ in their complement and finitely many sections of $k_0M$, for some integer $k_0>0$ with $k_0D$ Cartier, generate jets through order $(2n+2)k_0$ at $z_*$. \item A point $x\in W$, its blowup $\pi_x:\widehat W\to W$ with exceptional divisor $J$, and a curve class \[ \gamma=f_*(A_1\cdots A_{n-1}), \] where $f:V\to\widehat W$ is one fixed birational morphism, $V$ is smooth integral projective, and the $A_i$ are ample integral Cartier divisors, such that $J\cdot\gamma>0$ and \begin{equation}\label{fr:movable-input} \pi_x^*(D+K_W/2+\lambda S)\cdot\gamma \le40\epsilon J\cdot\gamma\qquad(0\le\lambda\le q). \end{equation} It suffices to check the two endpoints in this affine inequality. \end{enumerate} These data cannot coexist. \end{theorem} The point $x$ is independent of the two base coordinates of $z_*$. No nefness is required of $D,S$, or $K_W$, and the theorem does not assume that $K_W$ is pseudo-effective. Its hypotheses specify the flag and the movable curve class themselves. All these data will be fixed before the residual characteristic tends to infinity. \subsection{Reduction and Frobenius jets} \label{fr:reduction} We begin the proof of Theorem~\ref{thm:finite-data} by spreading its hypotheses and converting the fixed ordinary jets into jets of order proportional to the residual characteristic. Put $\Lambda=\prod_{i=1}^{n-1}l_i$. The flag satisfies \begin{equation}\label{fr:flag-degrees} [C]=\Lambda H^{n-1},\qquad h_i\cdot C=l_i\Lambda H^n. \end{equation} Choose one sufficiently ample integral Cartier divisor $T_0$ so that every $T_0+aS$, for integers $0\le a\le(k_0-1)q$, has a section nonvanishing at each of the two base coordinates of $z_*$. Fix these finitely many sections. All models, morphisms, divisors, points, the flag, the complete-intersection witness for $\gamma$, and the jet and filler sections spread over an integral finitely generated $\Z$-algebra of characteristic zero. Shrink its spectrum so that the fibers and flag are smooth projective and geometrically integral, the birational morphisms remain birational, the fixed ample bundles remain ample, and the finite jet surjection and nonvanishings persist. Birationality is preserved by spreading an isomorphism between dense opens. All displayed intersection numbers are constant. We also preserve $\mu_{\min}(\Omega_W^1|_C)\ge0$. One way is to spread its characteristic-zero Harder--Narasimhan filtration, make all its factors locally free, and use openness of their semistability on curves; their degrees and ranks are fixed. Equivalently, a single relative ample twist globally generates the curve bundle. Every quotient of rank $a$ then has degree bounded below by $-ad$ for a fixed integer $d$. A negative-degree quotient therefore has one of finitely many ranks and degrees. The proper relative Quot schemes for those Hilbert polynomials have images missing the generic point: after removal of torsion, a negative-degree coherent quotient there would give a negative-degree vector-bundle quotient. Removing those finitely many images proves the same openness assertion directly. This base has closed points in arbitrarily large characteristics. Take algebraic closures of their residue fields and exclude denominator primes. Retain the notation for the reductions. The curve $\gamma$ continues to pair nonnegatively with every effective divisor: pull that divisor back under the fixed birational morphism and intersect with its fixed ample divisors. This tests effective divisors on the reduction itself. No specialization of a characteristic-zero effective cone is used. \begin{lemma}\label{fr:positive-characteristic-jets} For a sufficiently large residual characteristic $p$, set \[ N_p=\lfloor p/k_0\rfloor,\qquad B_p=k_0N_pD+T_0. \] The sections of $pq\xi+B_{p,1}+B_{p,2}$ generate jets through order $(2n+2)k_0N_p$ at $z_*$. In particular, they surject onto the quotient of its local ring by the $p$-th powers of all regular parameters. \end{lemma} \begin{proof} Multiply $N_p$ copies of the fixed jet system. Every monomial of total degree at most $(2n+2)k_0N_p$ is a product of $N_p$ monomials of degree at most $(2n+2)k_0$. Taking sections with those leading monomials gives a triangular system, ordered by total degree. Starting at degree zero and removing the error in each successive degree proves the required jet surjection. This argument takes place in the local ring and never divides by factorials. Put $d_p=(p-k_0N_p)q$, so $0\le d_p\le(k_0-1)q$. Among the sections fixed before reduction, choose \[ u_0\in H^0(W,\OO_W(T_0)),\qquad v_{d_p}\in H^0(W,\OO_W(T_0+d_pS)) \] nonzero at the first and second base coordinates of $z_*$, respectively. Their tensor product belongs to the summand of first-slot weight zero in the projective-bundle formula; its fiber monomial is the $d_p$-th power of the second coordinate. It defines a section of $d_p\xi+T_{0,1}+T_{0,2}$ nonzero at $z_*$, since $z_*$ lies off both axes. Multiplying the product jet system by this section changes its divisor from $N_pk_0M$ to $pq\xi+B_{p,1}+B_{p,2}$. Multiplication is invertible on the local jet algebra. Only the finitely many previously fixed values of $d_p$ occur. Since $\dim Z=2n+1$, the quotient by the $p$-th powers of regular parameters has top total degree $(2n+1)(p-1)$. For large $p$ this is at most $(2n+2)k_0N_p$. The underlying ordinary/Frobenius ideal comparison is also recorded in \cite[proof of Proposition~2.12, Equation~(2.8)]{MustataSchwede}. \end{proof} \subsection{Full rank from the two-slot jets} \label{fr:full-rank} Let $F:W\to W'$ be relative Frobenius to the base-field twist. It is finite flat because $W$ is smooth. Write $S'$ for the twist of $S$, so $F^*S'=pS$, and define \[ \mathcal E=F_*\OO_W(B_p),\qquad s=\rk\mathcal E=p^n, \qquad R=s^2. \] For integers $0\le a\le q$, put $U_a=H^0(W,\OO_W(B_p+paS))$. Projection formula gives an evaluation map on $W'\times W'$: \begin{equation}\label{fr:two-slot-evaluation} \bigoplus_{\substack{a+b=q\\a,b\ge0}} U_a\otimes U_b\otimes \OO(-aS'_1-bS'_2) \longrightarrow \mathcal E\boxtimes\mathcal E. \end{equation} \begin{lemma}\label{fr:generic-rank} The map in \eqref{fr:two-slot-evaluation} has generic rank $R$. \end{lemma} \begin{proof} Trivialize the two summands defining $Z$ near the selected base points. Let $z$ be the torus coordinate, whose value at $z_*$ is $z_0\ne0$. By the projective-bundle formula, the sections in Lemma~\ref{fr:positive-characteristic-jets} are weight polynomials in $z$ with weight-$j$ coefficients in \[ H^0(W,B_p+jS)\otimes H^0(W,B_p+(pq-j)S),\qquad 0\le j\le pq. \] Let $A_*$ be the tensor product of the local algebras of the two base Frobenius fibers. The local quotient in Lemma~\ref{fr:positive-characteristic-jets} is \[ A_*[z]/(z^p-z_0^p), \] free over $A_*$ with basis $1,z,\ldots,z^{p-1}$. Project to the coefficient of $1$ in this basis. Exactly the weights $j=pa$ survive, with multipliers $z_0^{pa}$. Their coefficients are the values of the corresponding summands of \eqref{fr:two-slot-evaluation}, in the chosen frames and the induced twisted frames. As the full jet map is surjective, these coefficients span $A_*$, of dimension $p^{2n}=R$. Thus the evaluation has full rank at this pair of base points, and hence generically. The calculation uses a basis over the possibly nonreduced algebra $A_*$, so reducedness of the Frobenius fibers is unnecessary. \end{proof} The jet calculation at $z_*$ proves generic full rank; the diagonal estimate below will be tested at $(x',x')$, obtained from the separately fixed point $x$. \subsection{Sections on the Frobenius diagonal} \label{fr:filtration} The Frobenius fiber product has the cartesian square \begin{equation}\label{fr:frobenius-square} \begin{tikzcd} \Delta_F=W\times_{W'}W \arrow[r,"\mathrm{pr}_2"] \arrow[d,"\mathrm{pr}_1"'] & W \arrow[d,"F"]\\ W \arrow[r,"F"'] & W'. \end{tikzcd} \end{equation} Its reduced subscheme is the diagonal $W$. Put \[ \mathcal L_p= \OO_{\Delta_F}(B_{p,1}+B_{p,2}+pqS_1). \] Write $h=F\circ\mathrm{pr}_1=F\circ\mathrm{pr}_2: \Delta_F\to W'$. On this fiber product, \[ \OO(pS_1)\simeq h^*\OO_{W'}(S')\simeq\OO(pS_2). \] Consequently every pair of weights $a+b=q$ gives the same line bundle: \[ \OO_{\Delta_F}(B_{p,1}+B_{p,2}+paS_1+pbS_2) \simeq\mathcal L_p. \] On the diagonal of $W'\times W'$, every source twist of \eqref{fr:two-slot-evaluation} becomes $-qS'$. Finite base change and projection formula identify the twisted target as \[ (\mathcal E\otimes\mathcal E)(qS')\simeq h_*\mathcal L_p. \] The product sections defining the columns restrict to global sections of this one bundle $\mathcal L_p$. Their values at a diagonal point therefore lie in the image of $H^0(\Delta_F,\mathcal L_p)$ in the corresponding fiber of $h_*\mathcal L_p$. This proves that the rank at every diagonal point is at most \begin{equation}\label{fr:diagonal-rank-upper} h^0(\Delta_F,\mathcal L_p). \end{equation} For a rank-$n$ bundle $V$ in characteristic $p$, denote by $A_j(V)$ the degree-$j$ part of its symmetric algebra modulo the $p$-th powers of local generators. This construction is independent of frame: the $p$-th power of a linear combination is the sum of the $p$-th powers in characteristic $p$. Each $A_j(V)$ is locally free and a quotient of $V^{\otimes j}$. Set \[ u=n(p-1),\qquad e_j=\rk A_j(V),\qquad \sum_{j=0}^u e_j=p^n. \] Filtering $\mathcal L_p$ by powers of the ideal of the reduced diagonal in $\Delta_F$ gives the graded bundles \begin{equation}\label{fr:diagonal-graded} \mathcal G_j=\OO_W(2B_p+pqS)\otimes A_j(\Omega_W^1), \qquad 0\le j\le u. \end{equation} Indeed, in smooth local coordinates the algebra is generated by parameter differences with their $p$-th powers zero, and its degree-one conormal is $\Omega_W^1$. This coordinate calculation can be made étale locally or in completions and identifies the global associated graded. This is the diagonal-ideal form of the canonical Frobenius filtration \cite[Corollary~3.5]{KS}. The same calculation, using only the bundle from the second factor, filters $F^*\mathcal E$ with pieces \cite[Theorem~3.7]{Sun} \begin{equation}\label{fr:Frob-pullback-graded} \OO_W(B_p)\otimes A_j(\Omega_W^1). \end{equation} Complementary multiplication is a perfect pairing, giving \begin{equation}\label{fr:socle-pairing} A_{u-j}(V)\simeq A_j(V)^\vee\otimes(\det V)^{p-1}. \end{equation} On monomials, each exponent vector pairs with its complement to $(p-1,\ldots,p-1)$. The top line transforms by the character $(\det V)^{p-1}$: this is immediate on diagonal matrices and hence identifies the character of $\mathrm{GL}_n$ on that line. \subsection{A uniform bound for the diagonal rank} \label{fr:rank-bound} We bound the sections of every graded bundle in \eqref{fr:diagonal-graded} by its rank times the same scalar polynomial in $p$. This uniformity lets us sum the ranks of the graded pieces without an additional factor for their number. We begin with a slope bound on the fixed curve; here slope means ordinary degree divided by rank. \begin{lemma}\label{fr:truncated-slope} There is a constant $c\ge0$, independent of the sufficiently large residual characteristic $p$ and of $j$, such that for $V=\Omega_W^1|_C$ one has \[ \mu_{\max}(A_j(V))\le(p-1)K_W\cdot C+nc. \] \end{lemma} \begin{proof} Write $F_C$ for absolute Frobenius of $C$. Langer's instability estimate \cite[Corollary~2.5]{Langer} gives \begin{equation}\label{fr:Langer-min} L_{\min}(V):= \lim_{e\to\infty}p^{-e}\mu_{\min}(F_C^{e*}V) \ge-\frac{c}{p-1}. \end{equation} To make the uniformity explicit, let $g$ be the fixed genus of $C$. One can choose a nef line bundle $A_C$ of degree at most $d_g=\max\{4g-2,0\}$ such that $T_C\otimes A_C$ is globally generated; for $g\le1$, take $A_C=\OO_C$. Langer's bound is \[ \mu_{\min}(V)-L_{\min}(V) \le\frac{(\rk V-1)\deg A_C}{p-1}. \] Since $\rk V=n$ and $\mu_{\min}(V)\ge0$, we may take $c=(n-1)d_g$, independently of the reduction. We also need, for every integer $i\ge0$, \begin{equation}\label{fr:tensor-min} \mu_{\min}(V^{\otimes i})\ge iL_{\min}(V). \end{equation} Fix $p$ and $i$. Choose a line bundle $A_e$ of degree $-\mu_{\min}(F_C^{e*}V)+O(1)$, rounded up with a fixed genus-dependent additive constant, so that $F_C^{e*}V\otimes A_e$ is globally generated. This follows from Serre duality: after subtracting any point, its minimum slope can be made greater than $2g(C)-2$, which annihilates $H^1$ and makes evaluation at that point surjective. For every vector-bundle quotient $Q$ of $V^{\otimes i}$, the bundle $F_C^{e*}Q\otimes A_e^{\otimes i}$ is then globally generated. Consequently \[ p^e\mu(Q)+i\deg A_e\ge0. \] Divide by $p^e$ and let $e\to\infty$ to prove \eqref{fr:tensor-min}. This limit is taken for each fixed $p,i$; there is no interchange of Frobenius-iteration and characteristic limits. Since $A_{u-j}(V)$ is a quotient of $V^{\otimes(u-j)}$, the perfect pairing \eqref{fr:socle-pairing} yields \begin{align*} \mu_{\max}(A_j(V)) &= (p-1)\deg\det V-\mu_{\min}(A_{u-j}(V))\\ &\le(p-1)K_W\cdot C+\frac{(u-j)c}{p-1}\\ &\le(p-1)K_W\cdot C+nc. \end{align*} \end{proof} The divisors $B_p-pD=T_0-(p-k_0N_p)D$ range over a fixed finite list. Combining~\eqref{fr:intersection-bounds}, \eqref{fr:flag-degrees}, and Lemma~\ref{fr:truncated-slope} therefore gives, uniformly for $0\le j\le u$, \begin{align} \mu_{\max}(\mathcal G_j|_C) &\le p(4+2/r)\Lambda H^n+O(1)\notag\\ &\le M_p:=7p\Lambda H^n+c_0,\label{fr:common-slope-bound} \end{align} where $c_0\ge0$ is fixed. The coefficient $7$ provides harmless room in this common bound. \begin{lemma}\label{fr:diagonal-sections} For all sufficiently large $p$, \[ h^0(\Delta_F,\mathcal L_p) \le R\bigl(7^nH^n+O(1/p)\bigr)M_p$, because then its maximum slope is negative. Each nonzero summand has at most $e_j(M_p+1)$ sections. Indeed, evaluation at $\lfloor M_p\rfloor+1$ distinct points is injective; the kernel has negative maximum slope after subtracting those points. By \eqref{fr:flag-degrees} it follows that \begin{align} h^0(W,\mathcal G_j) &\le e_j(M_p+1) \prod_{i=1}^{n-1} \left(1+\frac{M_p}{l_i\Lambda H^n}\right)\notag\\ &=e_jp^n\bigl(7^nH^n+O(1/p)\bigr). \label{fr:common-flag-polynomial} \end{align} The leading coefficient follows from $\Lambda=\prod_i l_i$. More importantly, the whole scalar polynomial on the first line is the same for every $j$. Sum over the filtration in \eqref{fr:diagonal-graded}. The ranks satisfy $\sum_j e_j=p^n$, so \eqref{fr:common-flag-polynomial} gives the asserted bound with $R=p^{2n}$. There is no extra factor for the number of graded pieces. Finally \[ 7^nH^n\le(14\epsilon)^n<1/4 \] by \eqref{fr:intersection-bounds}, proving the strict inequality for all large $p$. \end{proof} \subsection{The two orders of the determinant} \label{fr:determinant} Choose $R$ generically independent columns of \eqref{fr:two-slot-evaluation} from bases of its source vector spaces. Their determinant is a nonzero section \[ 0\ne\sigma\in H^0(W'\times W',\mathcal Q_1\boxtimes\mathcal Q_2) \] of an actual external-product line bundle. To specify its factors, let $m_i$ be the sum of the weights of the $R$ chosen columns in slot $i$. The target determinant is $(\det\mathcal E)^s\boxtimes(\det\mathcal E)^s$, and the chosen source determinant is $\OO(-m_1S')\boxtimes\OO(-m_2S')$. Thus \[ \mathcal Q_i=(\det\mathcal E)^{\otimes s} \otimes\OO_{W'}(m_iS'),\qquad \lambda_i=m_i/R. \] Every column weight lies between $0$ and $q$, so $0\le\lambda_i\le q$ independently of the columns chosen. Under the numerical identification with the base-field twist, \begin{equation}\label{fr:slot-determinant-class} \frac{c_1(\mathcal Q_i)}R =\frac{c_1(\mathcal E)}s+\lambda_iS =\frac{B_p}p+\frac{p-1}{2p}K_W+\lambda_iS, \qquad 0\le\lambda_i\le q. \end{equation} The Frobenius first-Chern-class identity is also \cite[Lemma~4.2]{Sun}; it is an identity of rational divisor classes. For the second equality, use \eqref{fr:Frob-pullback-graded}. Complementary degrees in \eqref{fr:socle-pairing} have total first Chern class $e_j(p-1)K_W$. Summing gives \[ \sum_jc_1(A_j(\Omega_W^1))=\frac{s(p-1)}2K_W. \] Thus $c_1(F^*\mathcal E)=sB_p+s(p-1)K_W/2$, and Frobenius pullback of twisted divisor classes multiplies by $p$, as required. For $0\le\lambda\le q$, the fixed movable inequality gives \[ (D+K_W/2+\lambda S)\cdot\gamma\le40\epsilon J\cdot\gamma, \] where pullback by $\pi_x$ is understood. This concerns only intersections of fixed divisors and the fixed complete-intersection witness, so it remains true on the reduction. The class in \eqref{fr:slot-determinant-class} differs from its left-hand divisor at $\lambda=\lambda_i$ by \[ \frac1p\bigl(B_p-pD-K_W/2\bigr). \] Its numerator belongs to the fixed finite list $T_0-aD-K_W/2$, $a\in\{0,\ldots,k_0-1\}$. The resulting error after intersection with $\gamma$ is therefore $O(p^{-1})$, uniformly over the selected determinant columns. Let $x'$ be the twisted point and let $0\ne\tau_i\in H^0(W',\mathcal Q_i)$. The strict transform of its effective divisor on the blowup at $x'$ pairs nonnegatively with the twist of $\gamma$. Since $J\cdot\gamma>0$, it follows that \begin{equation}\label{fr:slot-order} \frac{\ord_{x'}(\tau_i)}R\le40\epsilon+O(1/p). \end{equation} The constant is uniform in the columns and in $\tau_i$. This argument tests sections on the reduction itself; it does not require them to lift to characteristic zero. K\"unneth identifies the space containing $\sigma$ with $H^0(W',\mathcal Q_1)\otimes H^0(W',\mathcal Q_2)$. Choose bases adapted to the orders of vanishing at $x'$ in each factor. In each degree their leading terms are linearly independent. Tensor products of these leading terms remain independent in each bidegree of the two disjoint sets of parameters; terms of different bidegrees cannot cancel. Every nonzero tensor consequently has order at most the sum of the two maximum basis orders. Applying \eqref{fr:slot-order}, we obtain \begin{equation}\label{fr:determinant-order-upper} \ord_{(x',x')}\sigma\le R\bigl(80\epsilon+O(1/p)\bigr). \end{equation} On the other hand, \eqref{fr:diagonal-rank-upper} and Lemma~\ref{fr:diagonal-sections} bound the matrix rank at $(x',x')$ by a number strictly smaller than $R/4$. Trivialize its line bundles near that point. Invertible constant row operations make more than $3R/4$ rows vanish at the point. Each entry of those rows belongs to the maximal ideal, so every term of the determinant has order at least $3R/4$. Therefore \begin{equation}\label{fr:determinant-order-lower} \ord_{(x',x')}\sigma\ge3R/4. \end{equation} Equations \eqref{fr:determinant-order-upper} and \eqref{fr:determinant-order-lower} contradict $80\epsilon<3/4$ for sufficiently large $p$. This contradiction proves Theorem~\ref{thm:finite-data}. The full-rank calculation used $z_*$, while the diagonal estimate holds at every point and was tested at the separately chosen $x$. The errors were controlled by one fixed finite list of divisor classes and one fixed flag polynomial. Thus their constants do not depend on $p$, the filtration index, or the chosen columns. \subsection{Application to the terminal case} \label{fr:application} We return to Theorem~\ref{thm:terminal-big}. Thus $X$ is a projective terminal $\Q$-factorial variety of dimension $n\ge2$ with $K_X\sim0$, and $L$ is a nef Cartier divisor of full nef dimension. Assume that $L$ is not big. Recall the ample rational divisors $P=r(L+tA)$, where $A$ is a fixed ample Cartier divisor and $t>0$ tends to zero, with \[ r\longrightarrow\infty,\qquad P^n\longrightarrow\epsilon^n, \qquad \rho=(P^n)^{1/n}. \] The positive rational number $\epsilon$ was chosen so that $(14\epsilon)^n<1/4$ and $80\epsilon<3/4$. Choose the integer $q$ supplied by Proposition~\ref{prop:large-seshadri}, and then choose $t$ sufficiently small that this proposition and Lemma~\ref{lemma:scalar-bound} apply, that $r>q+1$, and that $P^n<(2\epsilon)^n$. We now fix $t$, and therefore also $P,r,\rho$ and the projective bundle in~\eqref{eq:bundle}. Every remaining choice is made over $\C$ before reducing to positive characteristic. \paragraph{The divisors and the flag.} Choose a smooth projective resolution $w:W\to X$ and set \[ D=w^*P,\qquad S=w^*L,\qquad H_0=w^*P. \] The divisor $S$ is integral Cartier, and $H_0$ is nef and big. Since $X$ is terminal and $K_X\sim0$, there is an integral effective $w$-exceptional divisor $E$ with $K_W\sim E$. The projection formula and the nefness of $P-rL$ give \[ K_WH_0^{n-1}=0,\qquad (2D+qS)H_0^{n-1}\le(2+q/r)H_0^n. \] At $H_0$, the inequalities $K_WH_0^{n-1}<2H_0^n/r$ and $(2D+qS)H_0^{n-1}<4H_0^n$ are strict. Thus, by continuity of intersection products, a sufficiently small rational ample perturbation $H=H_0+\eta A_W$, with $A_W$ an ample Cartier divisor on $W$ and $\eta>0$, satisfies \[ H^n\le(2\epsilon)^n,\qquad K_WH^{n-1}\le2H^n/r,\qquad (2D+qS)H^{n-1}\le4H^n. \] This gives~\eqref{fr:intersection-bounds}. The canonical divisor of $W$ is pseudo-effective. Generic semipositivity of the cotangent bundle, originating in \cite{Miyaoka} and used here in the form \cite[Theorem~2.1]{CP}, together with restriction to sufficiently general complete-intersection curves \cite[Theorem~6.1]{MR}, gives a smooth integral flag with successive divisor classes $l_iH$, for sufficiently large divisible integers $l_i$, such that \[ \mu_{\min}(\Omega_W^1|_C)\ge0. \] This is the flag in Hypothesis~\textup{(i)} of Theorem~\ref{thm:finite-data}. \paragraph{The finite jet system.} On the bundle of~\eqref{eq:bundle}, write $R=P_1+P_2+q\xi$. The divisor $R$ is ample: $\xi$ is nef and relatively ample, and $P_1+P_2$ is the pullback of an ample divisor on $X\times X$. Proposition~\ref{prop:large-seshadri} supplies a smooth point $z_*$ in the complement of the two axes with \[ \varepsilon(R;z_*)>2n+2. \] The point can be chosen over the open set where $w$ is an isomorphism. Choose a rational number $a$ strictly between $2n+2$ and $\varepsilon(R;z_*)$. On the blowup $b:\widetilde Z\to Z$ at $z_*$, with exceptional divisor $G$, the rational divisor $b^*R-aG$ is ample. For sufficiently large divisible $k_0$, Serre vanishing gives \[ H^1\bigl(\widetilde Z, \OO_{\widetilde Z}(k_0b^*R-k_0aG)\bigr)=0. \] Since $z_*$ is smooth, the sections on the thickened exceptional divisor $k_0aG$ identify with the jet quotient at $z_*$ modulo $\mathfrak m_{z_*}^{k_0a}$, in a local frame for $k_0R$. The restriction sequence therefore shows that sections of $k_0R$ generate jets through order $k_0a-1$, and hence through order $(2n+2)k_0$. Increase the divisibility of $k_0$ to make $k_0P$ Cartier. Pulling these sections to \[ Z_W=\PP\bigl(\OO_W(S)_1\oplus\OO_W(S)_2\bigr) \longrightarrow W\times W \] preserves the jets at the corresponding point, because the induced map $Z_W\to Z$ is an isomorphism near that point. The pulled-back divisor is $D_1+D_2+q\xi$. We have obtained the finite system required in Hypothesis~\textup{(ii)}. \paragraph{The movable curve class.} Choose a very general point $x\in W$ outside the exceptional locus, with image in the locus where Lemma~\ref{lemma:scalar-bound} holds. Let $\pi_x:\widehat W\to W$ be its blowup and $J$ its exceptional divisor. The rational divisor \[ D^+=2D+E \] is big. For every sufficiently divisible positive integer $m$, normality of $X$ and exceptionality of $E$ give $w_*\OO_W(mE)=\OO_X$. Thus every section of $mD^+$ comes from a section of $2mP$, multiplied by the section of the fixed divisor $mE$. Since $x\notin\Supp E$, Lemma~\ref{lemma:scalar-bound} implies \begin{equation}\label{fr:application-order} \ord_x(s)/m\le8\rho<16\epsilon \qquad\bigl(0\ne s\in H^0(W,\OO_W(mD^+))\bigr). \end{equation} It follows that \begin{equation}\label{fr:application-not-pe} \pi_x^*D^+-40\epsilon J\quad\text{is not pseudo-effective}. \end{equation} Indeed, if it were pseudo-effective, a convex combination with the big divisor $\pi_x^*D^+$ would make $\pi_x^*D^+-cJ$ big for any rational $c$ with $16\epsilon0$. For every $0\le\lambda\le q$, \[ D^+-(D+K_W/2+\lambda S) \sim_{\R}w^*(P-\lambda L)+E/2 \] is pseudo-effective: $P-\lambda L=(P-rL)+(r-\lambda)L$ is nef and $E$ is effective. Pulling back to $\widehat W$ and pairing with $\gamma$ therefore gives \[ \pi_x^*(D+K_W/2+\lambda S)\cdot\gamma \le(\pi_x^*D^+)\cdot\gamma <40\epsilon J\cdot\gamma. \] This proves Hypothesis~\textup{(iii)}. All the hypotheses of Theorem~\ref{thm:finite-data} now hold for one fixed collection of complex data, a contradiction. Hence $L$ is big, completing the proof of Theorem~\ref{thm:terminal-big}. ==== END Numerical-Semiampleness-of-Nef-Adjoint-Divisors//frobenius.tex ==== ==== BEGIN Numerical-Semiampleness-of-Nef-Adjoint-Divisors//geometry.tex ==== \section{Jets on a terminal variety with trivial canonical class} \label{sec:geometry} The reduction to a Calabi--Yau pair leaves a positivity problem for a nef divisor: full nef dimension must imply bigness. We first treat the case with trivial canonical bundle and terminal singularities. The argument follows the scalar and two-slot jet constructions of \cite[Sections~7--8]{LA}. Its geometric exclusions differ from the nonvanishing argument there: lower-dimensional instances of $(P_j)$ and the polarized effective birationality theorem replace the exclusions for a canonical nonvanishing counterexample. \begin{theorem}\label{thm:terminal-big} Let $n\ge2$, and assume $(P_j)$ for $j0$ tend to zero along a sequence of rational numbers. Since $L$ is nef and not big, $L^n=0$. We may therefore choose positive integers $r=r(t)$ satisfying \begin{equation}\label{eq:scaling} P=P_t=r(L+tA),\qquad r\longrightarrow\infty,\qquad P^n\longrightarrow\epsilon^n. \end{equation} For example, round $\epsilon/((L+tA)^n)^{1/n}$ to the nearest positive integer. The divisor $P$ is ample and rational Cartier, and $P-rL$ is nef. After discarding finitely many terms, we have \begin{equation}\label{j:volume-window} \frac{\epsilon^n}{2}\le P^n\le2\epsilon^n. \end{equation} Every curve through a very general point of $X$ has positive integral $L$-degree, hence $P$-degree at least $r$. Thus a covering family of curves of bounded $P$-degree will give a contradiction. We specify how the same obstruction applies to subvarieties and covers. A resolution $W\to X$ has an effective canonical divisor: pull back a nowhere-zero canonical form on the smooth locus of $X$ and use terminality. Hence $W$, and therefore $X$, is not uniruled \cite{BDPP}. By Lemma~\ref{j:very-general-images}, if a smooth projective variety $Y$ maps generically finitely to a positive-dimensional subvariety containing a sufficiently very general point of $X$, then $Y$ is not uniruled and the pulled-back Cartier divisor $L_Y$ has full nef dimension. In particular, $K_Y$ is pseudo-effective. If $f=\dim Y2n+2. \end{equation} More generally, for any prescribed dense open subsets $U_1,U_2$ of the smooth locus of $X$, the point may be chosen over $U_1\times U_2$. In particular, these opens may be the isomorphism loci of fixed resolutions of $X$. \end{proposition} We prove the proposition in the remainder of this section. Choose a positive rational gap $\delta$ such that \begin{equation}\label{j:slice-gap} (2/\delta)^nC_{\mathrm{sl},n}\epsilon^n<1. \end{equation} Next choose $q$ so large that $d+1$ increasing levels of gap $\delta$, with lowest level strictly greater than $2n+2$, lie strictly below $(c_nq\epsilon^{2n})^{1/d}$. These choices precede the limit $t\to0$. Fix the prescribed opens $U_1,U_2$, if any. Suppose that \eqref{j:large-seshadri} fails throughout their smooth torus preimage for arbitrarily small $t$. For any such $t$, asymptotic Riemann--Roch and the elementary jet count at a smooth point show that all the full kernels at the chosen levels are nonzero for sufficiently large divisible $k$. Indeed the highest level has $d$-th power smaller than $R^d$. Failure of \eqref{j:large-seshadri} supplies, through each very general point, a curve with degree divided by multiplicity smaller than the lowest level. Every section in the lowest kernel vanishes identically on that curve: otherwise its intersection with the curve is at least its vanishing order times the curve's multiplicity, contradicting the degree inequality. Thus the lowest base locus has a positive-dimensional component through the mark. Take $k\delta\ge4$ and apply Lemma~\ref{j:moving-kernel}. We obtain a family of proper integral components $V\subset Z$ whose incidence dominates $Z$. For its general members, of dimension $f$, the estimates are \begin{equation}\label{j:total-component-bounds} \begin{split} 00$. Resolve $W_0$ to $W_0^*$ and take as a resolution of $F$ the induced projective bundle $p_0:F^*\to W_0^*$. Its relative canonical divisor is $-2\xi+p_0^*L$, exactly the restriction of $K_{Z_{\mathrm{sl}}}$. Cancelling this term in \eqref{j:slice-bounds} gives \begin{equation}\label{j:saturated-canonical} p_0^*K_{W_0^*}R_{\mathrm{sl}}^w \le cR_{\mathrm{sl}}^{w+1}\cdot F. \end{equation} The divisor $K_{W_0^*}$ is pseudo-effective by the very general point condition. On expanding the left side and pushing down, its term with exactly one $\xi$ is $wqK_{W_0^*}P^{w-1}$; every other nonzero term is nonnegative, since it pairs the pseudo-effective canonical divisor with nef classes. On the right, the same expansion used for \eqref{j:bundle-volumes} gives \begin{equation}\label{j:saturated-volume} (w+1)qP^w\cdot W_0 \le R_{\mathrm{sl}}^{w+1}\cdot F \le C'_nqP^w\cdot W_0, \end{equation} using $r\ge q$. The slice degree bound and the first inequality in \eqref{j:saturated-volume} bound $00$. Intersect $F$ with each of the two Cartier axes and push the resulting cycles down to $X$. This gives effective integral Weil divisors $D_1,D_2$. The axes do not contain $F$, and their classes differ by $L$; projection of the associated rational section, or equivalently its norm, gives \begin{equation}\label{j:signed-L} D_1-D_2\sim\pm eL. \end{equation} For either axis $S$, the classes $R_{\mathrm{sl}}-qS$ and $R_{\mathrm{sl}}-P$ are nef. Intersecting with the effective cycle $S\cdot F$ and then using nefness on $F$ yields \begin{equation}\label{j:axis-degrees} qD_iP^{n-1}\le qS\cdot F\cdot R_{\mathrm{sl}}^{n-1} \le R_{\mathrm{sl}}^n\cdot F. \end{equation} Thus both axis-image degrees are uniformly bounded. Set $G=D_1+D_2$. It is $\Q$-Cartier because $X$ is $\Q$-factorial. Choose $b>0$ rational sufficiently small that $(X,bG)$ is klt. We claim \begin{equation}\label{j:positive-boundary-kappa} \kappa(K_X+bG)>0. \end{equation} The Iitaka dimension is at least zero, since $K_X+bG\sim_\Q bG$. If it were zero, Theorems~\ref{input:models} and \ref{input:abundance} would give a finite ordinary MMP to a good model $Y$ with transformed adjoint rationally linearly trivial. The transform of $K_X$ is linearly trivial as well. Consequently the effective transform of $G$ is zero: all its components have been contracted. On a common resolution $a:U\to X$, $b_Y:U\to Y$, relation \eqref{j:signed-L} then expresses $a^*L$ up to rational linear equivalence as a signed $b_Y$-exceptional divisor $E$. Divisors exceptional over $X$ are also exceptional over $Y$, since the MMP extracts none. Nefness of $a^*L$ and the negativity lemma imply $E\le0$. Its class is also pseudo-effective. A nonzero anti-effective divisor has negative intersection with a sufficiently high power of an ample divisor, so $E=0$. This contradicts full nef dimension of $L$, and proves \eqref{j:positive-boundary-kappa}. The already proved positive-Iitaka-dimension case (Proposition~\ref{prop:full-positive}) now gives \[ K_X+bG+L\ \text{big}. \] Its full-positivity hypothesis holds because $L$ has full nef dimension and $K_X+bG$ is pseudo-effective. Take a log resolution $a:Y\to X$ of $G$ and let $D_0$ be the reduced simple normal crossings divisor consisting of the strict support of $G$ and every exceptional divisor. Decrease $b$ if necessary; Proposition~\ref{prop:full-positive} still applies. Terminality and the support of $a^*G$ then give \[ K_Y+D_0+a^*L\ \ge\ a^*(K_X+bG+L) \] up to rational linear equivalence and an effective difference. Indeed the finitely many exceptional discrepancies are positive, and decreasing $b$ makes each coefficient of $b\,a^*G$ no larger than the corresponding coefficient of $K_Y-a^*K_X+D_0$. Hence the left side is big. Test on $Y$ with $a^*P$. Exceptional divisors contribute zero to intersection with $(a^*P)^{n-1}$, and $K_X\sim0$. The degree of the reduced strict support is no larger than $GP^{n-1}$, bounded by \eqref{j:axis-degrees}; also $LP^{n-1}\le P^n/r$. Since \eqref{j:volume-window} bounds $P^n$ away from zero, we obtain a uniform constant $C$ such that \[ (K_Y+D_0+a^*L)(a^*P)^{n-1}\le C(a^*P)^n. \] Lemma~\ref{lemma:bounded-curves}, now with its reduced boundary $D_0$, gives the final contradiction. This proves \eqref{j:no-intermediate-fibre}. Thus a general moving component $V$ has fibre dimension either zero or $n+1$ over its image in either factor. Fibre dimension $n+1$ would make $V$ the full inverse image of its image in that factor. Since $V$ is proper, the image has dimension at most $n-1$, so the other projection would have fibre dimension between $1$ and $n$, already excluded. Both projections of $V$ are therefore generically finite onto their images, and $\dim V\le n$. If $\dim V0$. Here nef reduction contracts the curves of degree zero through very general points; its precise properties are recalled in Section~\ref{sec:preliminaries}. On surfaces, Fontanari \cite[Theorem~1.4]{Fontanari} proved actual semiampleness when the nef summand is Cartier and its numerical class is not proportional to that of the ordinary adjoint. The birational reductions below follow \cite[Sections~4--7]{LP}. The companion results \cite{LA,MM} supply ordinary log abundance and the existence of the particular terminating programs required in those reductions. The further ingredient proved here is that a nef divisor of full nef dimension on a klt Calabi--Yau pair is big. Full nef dimension means that every curve through a very general point has positive degree. Bigness is stronger: it requires the volume of the divisor to be positive. The gap between these two conditions is the geometric issue addressed by the proof. Our treatment adapts the moving-jet and Frobenius arguments of \cite[Sections~7--9]{LA}. The moving-point method has antecedents in the local-positivity work of Ein--K\"uchle--Lazarsfeld \cite{EKL}. The finite-data comparison uses Frobenius jets, as studied by Musta\c t\u a--Schwede \cite{MustataSchwede}, the canonical filtrations of Sun and Kitadai--Sumihiro \cite{Sun,KS}, and Langer's control of Frobenius slopes \cite{Langer}. We reproduce the comparison proof with its uniformity conditions. The adaptation to an arbitrary nef summand occurs in the geometric exclusions preceding it: the polarized birationality theorem of Birkar--Zhang gives curves of bounded degree, contradicting a growing integral lower bound for their degrees. We give explicit proofs of those exclusions, including the case of correspondences between finite covers. \subsection{Structure of the proof} There are two simultaneous induction statements. The first says that $K_X+B+cM$ is big for every rational $c>0$ whenever one positive combination of $K_X+B$ and $M$ has positive degree on all curves through a very general point. The second says that the positive part of $K_X+B+M$ on a suitable smooth birational model is rational and numerically semiample. The positive part is the divisor left after removing the asymptotically fixed prime divisors. When the original adjoint is nef, it has no such negative part, and the second assertion descends to Theorem~\ref{thm:main}. Section~\ref{sec:reduction} sets up this induction. A pair of compatible minimal-model programs retains both a semiample ordinary adjoint and a nearby nef generalized adjoint. Positive Iitaka dimension then permits induction on the fibres. The remaining full-dimension case is a klt Calabi--Yau pair. Sections~\ref{sec:jet-tools} and~\ref{sec:geometry} treat the essential terminal case with trivial canonical bundle. If a nef Cartier divisor $L$ of full nef dimension were not big, its ample approximations could be rescaled to bounded volume while their degrees on very general curves tend to infinity. The bounded-degree obstruction controls the base loci of moving jet kernels. It produces a small vanishing-order bound on the variety itself and a large jet system on a projective bundle over its square. Section~\ref{sec:frobenius} shows that these two conclusions are incompatible. The comparison is made on fixed complex data before reduction to positive characteristic. Section~\ref{sec:completion} passes through an index-one cover and a boundary perturbation to obtain the full-dimension assertion for all klt Calabi--Yau pairs. Section~\ref{sec:positive} proves the positive-part assertion by nef reduction and the canonical bundle formula, then descends numerical semiampleness to the original variety and to any algebraically closed field of characteristic zero. ==== END Numerical-Semiampleness-of-Nef-Adjoint-Divisors//introduction.tex ==== ==== BEGIN Numerical-Semiampleness-of-Nef-Adjoint-Divisors//jet-tools.tex ==== \section{Bounded-degree curves and moving jet kernels} \label{sec:jet-tools} The geometric argument will produce subvarieties with bounded degree and bounded canonical intersection. We first explain how these bounds contradict a growing lower bound for the degrees of curves. We then record the moving-center results from \cite{LA} that supply the required subvarieties. All varieties in this section are over $\C$. \subsection{The bounded-degree obstruction} \begin{lemma}[Bounded-degree curves] \label{lemma:bounded-curves} For every integer $f\geq1$ there is an integer $m_f>0$ with the following property. Let $Y$ be a smooth integral projective variety of dimension $f$, let $D_0$ be a reduced simple normal crossing divisor or the zero divisor, and let $L_0$ be a nef integral Cartier divisor of full nef dimension on $Y$. Suppose that $P_0$ is a nef rational Cartier divisor and that $r,C_0,C_1$ are positive real numbers satisfying \[ 00$. Suppose that $(Z,\Theta)$ is log canonical at the generic point of $V$ and has a divisor of log discrepancy zero with center $V$. If $P$ is a nef rational Cartier divisor on $V$ and $\rho:V^*\to V$ is a smooth projective resolution factoring through its normalization, then \[ K_{V^*}\cdot(\rho^*P)^{f-1} \leq (K_Z+\Theta)|_V\cdot P^{f-1}. \] \end{lemma} Only generic log canonicity is required. The proof in \cite{LA} uses the dlt modification of \cite[Theorem~2.10]{FH}, whose truncated boundary leaves an effective surplus supported over the non-lc locus. On a minimal stratum over $V$, adjunction gives a pair that is klt over the generic point of its connected-fiber base. This suffices for the klt-trivial canonical bundle formula and its b-nef moduli divisor \cite[Definition~3.1 and Theorem~3.5]{FG}. The effective discriminant, the moduli divisor, and finite ramification have nonnegative intersections with the pulled-back nef class. The comparison uses canonical Weil cycles, so it requires no $\Q$-Gorenstein assumption on the normalization of $V$. \begin{lemma}[A base component of high multiplicity, \cite{LA}, Lemma~7.2] \label{j:base-component} Let $Z$ be a projective $\Q$-factorial klt variety of dimension $d$, let $R$ be a nef rational Cartier divisor, and let $k>0$ be an integer such that $kR$ is Cartier. Let $0\ne\mathcal H\subset H^0(Z,\OO_Z(kR))$ be a linear subspace with proper base locus and base ideal $\mathfrak b$. Suppose that $V$ is an integral base-locus component of dimension $f0$, there are a rational number $00), \end{split} \end{equation} where $V^*$ is a smooth projective resolution. The degree bound also holds for $f=0$. \end{lemma} The degree estimate is the local multiplicity bound $h^{d-f}$ combined with successive general cuts by the given linear series. The boundary is obtained by averaging general members with total coefficient equal to the generic log canonical threshold of $\mathfrak b$. Thus it is an actual divisor to which Lemma~\ref{j:subadjunction} applies. \subsection{Full jet kernels and restriction to a fiber} \begin{lemma}[Moving multiplicity, \cite{LA}, Lemma~7.3] \label{j:moving-kernel} Let $Z$ be an integral projective variety of dimension $d$, let $R$ be a nef, big, semiample rational Cartier divisor, and fix an integer $k>0$ such that $kR$ is Cartier. Choose positive real numbers $\tau_0<\cdots<\tau_d$ with consecutive gap $\delta>0$. For $x\in Z_{\mathrm{sm}}$, put \[ \mathcal H_i(x)=H^0\bigl(Z,\OO_Z(kR) \otimes\mathfrak m_x^{\lceil k\tau_i\rceil}\bigr). \] Suppose that for very general $x$ all these spaces are nonzero and the base locus of $\mathcal H_0(x)$ has a positive-dimensional component through $x$. If $k\delta\geq4$, then, after an algebraic parameter extension and restriction to nonempty opens, there are an integral parameter variety $T$ with a dominant marked-point map $x:T\to Z_{\mathrm{sm}}$, a fixed index $i\in\{0,\ldots,d-1\}$, and a family of integral subvarieties $V_t$ containing $x(t)$ and common to the base loci of $\mathcal H_i(x(t))$ and $\mathcal H_{i+1}(x(t))$. In particular, the incidence \[ \Gamma=\{(t,z):z\in V_t\}\subset T\times Z \] dominates $Z$. For general $t$, the dimension $f$ of $V_t$ lies in $[1,d-1]$, $V_t$ is an isolated component of the higher base locus at its generic point, and the entire higher-kernel base ideal is contained there in the ordinary power $\mathcal I_{V_t}^h$, where \[ h=\lceil k\delta/2\rceil,\qquad k/h\leq2/\delta. \] The incidence base ideal is likewise contained in $\mathcal I_\Gamma^h$ on a dense open of $\Gamma$. These containments are taken in the smooth ambient open. If $Z$ is also $\Q$-factorial and klt, then \begin{equation} \label{j:moving-bounds} \begin{split} R^f\cdot V_t&\leq(2/\delta)^{d-f}R^d,\\ K_{V_t^*}\cdot R^{f-1} &\leq(K_Z+cR)|_{V_t}\cdot R^{f-1}, \qquad 00$, integrality gives \[ (T+mL)\cdot C\geq-2n+m>0, \] which is impossible. Hence $L\cdot C=0$. The Cartier descent statement in the ordinary cone and contraction theorem gives a Cartier divisor on the contraction base whose pullback is $L$. It is nef: lift any curve on the base to a curve dominating it and use the projection formula. In a flip its pullback to the flipped variety is therefore again Cartier and nef. The relative signs of $T+mL$ are those of $T$, so the flip is also an ordinary flip for the klt pair. Thus the inductive hypotheses persist. The terminating program supplied by Theorem~\ref{input:models} has all the stated properties. \end{proof} Apply the lemma after a small $\Q$-factorialization, with $L=m_0M$ for an integer $m_0>0$ making $L$ Cartier. The program is $M$-trivial, so on a common resolution its ordinary adjoint comparison also gives \[ p^*(T+M)=q^*(T'+M')+E, \] where $E$ is effective and exceptional over the output. The transformed ordinary adjoint $T'$ remains pseudo-effective, and $M'$ is nef. By \eqref{eq:positive-transfer}, it suffices to prove $(Z_n)$ on that output. We may consequently retain the notation $X,B,T,M,L$ and assume in addition that \begin{equation}\label{eq:large-adjoint-nef} T+mL\text{ is nef}. \end{equation} \subsection{The case of full nef dimension} Suppose first that $T+mL$ has full nef dimension. Then \eqref{eq:full-condition} holds, and $(P_n)$ shows that $T+M$ is big. Write, by Kodaira's lemma, \[ T+M\sim_\Q A_1+G_1, \] where $A_1$ is ample and $G_1\geq0$ is rational. Choose a rational $v>0$ sufficiently small that $(X,B+vG_1)$ is klt. Since $M+vA_1$ is ample, a general effective rational representative $A'$ of this divisor can be chosen with sufficiently small coefficients that $(X,B+vG_1+A')$ is klt. Its adjoint satisfies \begin{equation}\label{eq:big-ordinary-boundary} K_X+B+vG_1+A'\sim_\Q(1+v)(T+M). \end{equation} It is big and has a good minimal model: one may use the big-case theorem of \cite{BCHM}, or combine Theorems~\ref{input:models} and \ref{input:abundance}. On a common resolution, the left side of \eqref{eq:big-ordinary-boundary} is the pullback of a semiample divisor plus an effective exceptional divisor. Its positive part is therefore the semiample pullback. Dividing by $1+v$ proves $(Z_n)$ in this case. \subsection{Descent along a nef reduction} We treat the remaining case, where the nef dimension of $T+mL$ is strictly smaller than $n$. Let \[ X\dashrightarrow Z \] be its nef reduction, with $Z$ normal and projective. A very general compact fibre $F$ is contained in the morphism locus and has positive dimension. On this fibre, \[ (T+mL)|_F\equiv0. \] The restriction $T|_F$ is pseudo-effective. To see this, choose effective rational approximations to $T$ after adding positive ample perturbations tending to zero, and choose $F$ outside the countably many exceptional choices for these approximations. Their restrictions are effective, and their classes approach $T|_F$. After a resolution of $F$ the same argument applies to the pullbacks. As $L|_F$ is nef and $T|_F\equiv-mL|_F$, both divisors are numerically trivial. Indeed, intersect this equality with an ample complete-intersection curve on a smooth resolution of $F$. Pseudo-effectivity makes the left side nonnegative and nefness makes the right side nonpositive. Their common value is zero, and a nef divisor with zero intersection against an ample complete-intersection curve is numerically trivial. Restriction to a sufficiently general fibre gives a klt pair $(F,B|_F)$ with adjoint $T|_F$. By Theorem~\ref{input:abundance}, its numerically trivial adjoint is semiample, hence \begin{equation}\label{eq:fibre-adjoint-trivial} K_F+B|_F\sim_\Q0. \end{equation} These restrictions are made where the almost holomorphic nef reduction is a morphism. Normality and the klt restriction statement hold after shrinking the base; alternatively they follow by restricting a log resolution over a general smooth open. Numerical triviality, initially obtained for very general fibres, also follows on a general fibre by the constancy of the relevant intersection numbers in this family. We next use the birational descent of nef divisors \cite[Lemma 3.1]{LP}. After resolving the nef reduction and modifying its base birationally, it supplies a diagram \[ X\xleftarrow{p}X_1\xrightarrow{f}Z_1 \] in which $X_1,Z_1$ are smooth and projective, $p$ is birational, $f$ is a contraction, and there is a rational divisor $M_1$ on $Z_1$ with \begin{equation}\label{eq:nef-descent} p^*M\equiv f^*M_1. \end{equation} The divisor $M_1$ is nef, since its pullback is numerically equivalent to the nef divisor $p^*M$ and nefness descends under a surjective morphism. We may take a further log resolution of the pair on top without affecting \eqref{eq:nef-descent}. Write the crepant boundary as $B_1^+-B_1^-$, with effective rational divisors of disjoint support, so that \[ K_{X_1}+B_1^+-B_1^-=p^*(K_X+B). \] Set $B_1=B_1^+$ and $E=B_1^-$. Since the original boundary is effective, $E$ is $p$-exceptional. Klt singularities make all coefficients of $B_1$ less than one, and its support is simple normal crossings. Thus $(X_1,B_1)$ is klt and \begin{equation}\label{eq:resolved-boundary} K_{X_1}+B_1=p^*T+E,\qquad E\geq0. \end{equation} In particular its adjoint is pseudo-effective. If $F_1$ is a general fibre of $f$, it maps birationally onto the corresponding compact fibre $F$ of the nef reduction. Every component of $E$ that meets $F_1$ restricts to an exceptional divisor over $F$: for a centre dominating the base, codimension at least two is preserved on the general fibre, and the other centres are avoided. Equations~\eqref{eq:fibre-adjoint-trivial} and \eqref{eq:resolved-boundary} therefore show that \begin{equation}\label{eq:fibre-kappa-zero} \kappa\bigl(F_1,(K_{X_1}+B_1)|_{F_1}\bigr)=0. \end{equation} The restriction is an effective exceptional divisor up to rational linear equivalence. Equivalently, its section spaces are those of the trivial divisor on the normal fibre $F$. In particular the general fibre pair has pseudo-effective adjoint and has a good minimal model by Theorems~\ref{input:models} and \ref{input:abundance}. \subsection{A relative good model and a lower-dimensional adjoint} At this point the nef summand comes from $Z_1$, while the ordinary adjoint has Iitaka dimension zero on its general fibres. We use a relative good model to express that ordinary adjoint as a pullback from a variety of dimension less than $n$. The relative log canonical ring of the rational klt pair $(X_1,B_1)$ over $Z_1$ is finitely generated. More precisely, for a positive integer $r$ making $r(K_{X_1}+B_1)$ Cartier, its Veronese algebra \[ \bigoplus_{\ell\geq0} f_*\OO_{X_1}\bigl(\ell r(K_{X_1}+B_1)\bigr) \] is a finitely generated $\OO_{Z_1}$-algebra. This is the relative finite-generation theorem for rational klt pairs, in the form \cite[Theorem 1.4]{FujinoFG}, obtained from the finite-generation theorem of \cite{BCHM} and the canonical-bundle reduction of Fujino--Mori \cite[Theorem~5.2]{FM}. Together with the good minimal models of the very general closed fibres established above, this verifies both hypotheses of \cite[Theorem 2.12]{HX}. That theorem gives a good minimal model over $Z_1$; the termination statement for an MMP with scaling once a good model exists, \cite[Corollary 2.9]{HX}, supplies a relative program \[ (X_1,B_1)\dashrightarrow(X_2,B_2). \] Thus $X_2$ is projective and $\Q$-factorial, $(X_2,B_2)$ is klt, and $K_{X_2}+B_2$ is semiample over $Z_1$. This application uses very general closed fibres over $\C$ and requires no change of the ground field to a geometric generic field. By \eqref{eq:nef-descent}, we may replace the nef term numerically by $f^*M_1$ before making the positive-part comparisons. This pullback is trivial on every step of the relative program and is carried unchanged from the base. The ordinary adjoint remains pseudo-effective by numerical pushforward. Let \[ X_2\xrightarrow{g}Z_2\xrightarrow{j}Z_1 \] be the relative semiample fibration of $K_{X_2}+B_2$, with $g$ a contraction. Its construction gives a $j$-ample $\Q$-Cartier divisor $A_2$ on $Z_2$ such that \begin{equation}\label{eq:relative-iitaka} K_{X_2}+B_2\sim_\Q g^*A_2. \end{equation} The relative program preserves Iitaka dimension on the general fibres, by its effective exceptional pullback comparisons. From \eqref{eq:fibre-kappa-zero}, the relative Iitaka dimension is zero. Hence \[ \dim Z_2=\dim Z_10$ and a line bundle $P_K\in\Pic^0(X_K)$ such that \[ \OO_{X_K}(rD_K)\otimes P_K \] is globally generated. Choose a $k_0$-point of $X_0$ and rigidify line bundles there. The finite-type scheme $\Pic^0(X_0)$ then has a Poincar\'e family. Tensor this family by the fixed line bundle $\OO_{X_0}(rD_0)$. The set of parameters for which the resulting line bundle is globally generated is constructible. Indeed, stratify the parameter scheme so that the zeroth direct image is locally free and cohomology and base change hold. On each stratum use the evaluation map; the image in the parameter space of the support of its cokernel is closed, because projection from $X_0$ is proper. Its complement is the globally generated locus on that stratum. Formation of the Picard scheme and of this constructible condition commutes with the field extension. The locus is nonempty after extension to $K$, by the chosen $P_K$, and is consequently nonempty over $k_0$. As $k_0$ is algebraically closed, it has a $k_0$-point. The corresponding globally generated line bundle, represented by a Cartier divisor and divided by $r$, is the desired semiample rational divisor numerically equivalent to $D_0$. \end{proof} \begin{proof}[Completion of the proof of Theorem~\ref{thm:main}] Let $k$ be the given algebraically closed field of characteristic zero. Descend $X,B,M$, the required rational Cartier structures, and a projective embedding to a finitely generated subfield of $k$. Let $k_0\subset k$ be its algebraic closure inside $k$, and choose an embedding $k_0\hookrightarrow\C$. The hypotheses persist under the resulting algebraically closed field extensions. For klt singularities this follows from a log resolution, which can be included among the descended data. Nefness is invariant under field extension, since it can be tested by ampleness after arbitrarily small rational ample perturbations. Pseudo-effectivity is invariant for the same reason with bigness in place of ampleness: a rational divisor is pseudo-effective exactly when all its positive rational ample perturbations are big, and bigness is preserved and reflected by extension of the section spaces. These tests show both that the descended data satisfy the hypotheses and that their complex extension does. The result over $\C$ supplies a semiample numerical representative of $K_X+B+M$ after this extension. Lemma~\ref{lemma:field-transfer} descends it to $k_0$. Extending that representative to $k$ preserves global generation of a suitable multiple and numerical equivalence. It is the required semiample $\Q$-Cartier divisor on the original variety. \end{proof} ==== END Numerical-Semiampleness-of-Nef-Adjoint-Divisors//positive.tex ==== ==== BEGIN Numerical-Semiampleness-of-Nef-Adjoint-Divisors//preliminaries.tex ==== \section{Conventions and birational inputs} \label{sec:preliminaries} Varieties are integral and projective over an algebraically closed field of characteristic zero unless stated otherwise. A $\Q$-pair $(X,B)$ consists of a normal variety, an effective rational boundary $B$, and a rational Cartier divisor $K_X+B$. We use log discrepancies: on a resolution $h:Y\to X$, the equality $K_Y+B_Y=h^*(K_X+B)$ assigns log discrepancy $1-b_E$ to a prime divisor with coefficient $b_E$ in $B_Y$. The pair is klt if all log discrepancies are positive, and log canonical if they are nonnegative. We use the standard cone, contraction, negativity, and resolution theorems as in \cite{KM,BCHM}. The symbols $\sim_\Q$ and $\equiv$ denote rational linear and numerical equivalence. A divisor is pseudo-effective if its numerical class is in the closure of the cone generated by effective divisors, and big if its class lies in the interior of that cone. Its Iitaka dimension is denoted by $\kappa$. Over $\C$, a very general point means a point outside a specified countable union of proper closed subsets. For a nef divisor $N$, its nef reduction is an almost holomorphic dominant rational map with connected compact general fibres, on which $N$ is numerically trivial, such that every curve through a very general point not contracted by the map has positive $N$-degree \cite{NefReduction}. Its base dimension is the \emph{nef dimension} $n(X,N)$. Thus $n(X,N)=\dim X$ precisely when every curve through a very general point has positive $N$-degree. This numerical condition does not assert bigness. For a pseudo-effective real divisor $D$ on a smooth variety and a prime divisor $\Gamma$, fix an ample divisor $A$ and set \[ \sigma_\Gamma(D)=\lim_{\delta\downarrow0} \inf\{\mult_\Gamma D':D'\ge0,\ D'\sim_\R D+\delta A\}. \] These numbers are independent of $A$ and depend only on the numerical class of $D$. The divisorial Zariski decomposition is \[ N_\sigma(D)=\sum_\Gamma\sigma_\Gamma(D)\Gamma, \qquad P_\sigma(D)=D-N_\sigma(D). \] The sum has finite support. A nef divisor has $N_\sigma(D)=0$. Further properties from \cite{Nakayama,LP} are recorded at their point of use in Section~\ref{sec:positive}. \begin{lemma}\label{lemma:pseudo-vg} Let $D$ be a pseudo-effective rational Cartier divisor on a projective complex variety $X$. Every integral curve through a sufficiently very general point has nonnegative $D$-degree. \end{lemma} \begin{proof} Fix an ample Cartier divisor $A$ and positive rational numbers $\delta_i\to0$. Each $D+\delta_iA$ is big, so there is an effective rational divisor $E_i\sim_\Q D+\delta_iA$. Choose a point outside $\bigcup_i\Supp E_i$. A curve through it is contained in none of these supports; hence $(D+\delta_iA)\cdot C\ge0$ for every $i$. Let $i\to\infty$. \end{proof} We state explicitly the companion results used throughout. We require existence of a terminating choice of program, rather than termination of every sequence of flips. \begin{theorem}[Log abundance, {\cite[Theorem~1.1]{LA}}] \label{input:abundance} Let $(X,B)$ be a projective log canonical $\Q$-pair over an algebraically closed field of characteristic zero. If $K_X+B$ is nef, then it is semiample. \end{theorem} For the next input a nef rational Cartier b-divisor is specified by a nef rational Cartier divisor on a projective birational model; it is pulled back unchanged on all higher models. Its trace on a lower model is its pushforward and need not be nef. Generalized discrepancies are defined by pulling back the adjoint while keeping this fixed nef data. \begin{theorem}[Minimal-model programs, {\cite[Theorem~1.1 and Section~3.3]{MM}}] \label{input:models} Let $(X,B+\mathbf M)$ be a projective generalized klt $\Q$-pair over $\C$, with $X$ $\Q$-factorial and with fixed nef rational Cartier b-divisor data. If $K_X+B+M_X$ is pseudo-effective, there is a terminating choice of its minimal-model program, with $\Q$-factorial generalized klt models and a nef endpoint. The program extracts no divisors and generalized log discrepancies do not decrease. On a common resolution of its initial and final models, the pullback of the initial adjoint is the pullback of the final adjoint plus an effective divisor exceptional over the final model. \end{theorem} The ordinary klt case follows by taking the nef data to be zero. In that case Theorem~\ref{input:abundance} makes the endpoint a good minimal model. The moving-center and Frobenius results used from \cite{LA} are stated separately in Sections~\ref{sec:jet-tools} and~\ref{sec:frobenius}; neither theorem above asserts abundance for a nonzero generalized nef summand. ==== END Numerical-Semiampleness-of-Nef-Adjoint-Divisors//preliminaries.tex ==== ==== BEGIN Numerical-Semiampleness-of-Nef-Adjoint-Divisors//reduction.tex ==== \section{Reduction of full nef dimension to Calabi--Yau pairs} \label{sec:reduction} We work over $\C$ until the final descent argument. The reductions in this section follow the strategy of Lazi\'c--Peternell \cite[Sections 4 and 5]{LP}. We use the terminating programs in Theorem~\ref{input:models}, together with the abundance input Theorem~\ref{input:abundance}, to make the precise reductions needed below. The essential point is to retain control of two adjoints: an ordinary semiample adjoint and a nearby generalized nef adjoint. \subsection{The induction statements} For a projective klt $\Q$-pair $(X,B)$ and a nef $\Q$-Cartier divisor $M$ on $X$, write \[ T=K_X+B. \] Suppose that $T$ is pseudo-effective. We will use the condition that, for some $a\in\Q_{>0}$, \begin{equation}\label{eq:full-condition} (T+aM)\cdot C>0 \quad\text{for every integral curve $C$ through a very general point of $X$.} \end{equation} The divisor $T+aM$ need not be nef. By Lemma~\ref{lemma:pseudo-vg}, Condition~\eqref{eq:full-condition} holds whenever $M$ has full nef dimension. When $T+aM$ is nef, the condition says exactly that its nef dimension is $\dim X$. We prove the following two assertions together, by induction on $n$. The notation $P_\sigma$ denotes the positive part of the divisorial Zariski decomposition; its properties used in the proof are recalled in Section~\ref{sec:positive}. \begin{description} \item[$(P_n)$] If $(X,B)$ is a projective klt $\Q$-pair of dimension $n$, $T=K_X+B$ is pseudo-effective, $M$ is a nef $\Q$-Cartier divisor on $X$, and \eqref{eq:full-condition} holds, then $T+cM$ is big for every $c\in\Q_{>0}$. \item[$(Z_n)$] If $(X,B)$ and $M$ have the same hypotheses, without \eqref{eq:full-condition}, then there is a projective birational morphism $w\colon W\to X$ from a smooth projective variety such that \[ P_\sigma\bigl(w^*(T+M)\bigr) \] is a rational divisor numerically equivalent to a semiample $\Q$-divisor on $W$. \end{description} Both assertions hold in dimensions zero and one. In dimension one, pseudo-effectivity of $T$ and nefness of $M$ mean that their degrees are nonnegative. Condition~\eqref{eq:full-condition} makes every positive combination have positive degree, hence be ample. A divisor of degree zero is numerically equivalent to zero, and a divisor of positive degree is ample. These observations also give $(Z_1)$, since a nef divisor has zero negative part. On a point the assertions use the usual conventions. Fix $n\geq2$, and assume $(P_j)$ and $(Z_j)$ for $j0$ rational and small enough that $G+\eta A$ is still negative on the ray and \[ -(G+\eta A)\cdot\gamma \geq -\tfrac12G\cdot\gamma \] for its nonzero classes $\gamma$. We may represent $G+\eta A$ as an ordinary klt adjoint. Here is the reason this remains valid when the nef b-divisor does not descend to $V$. Choose a projective log resolution $h\colon V'\to V$ carrying its nef part $N'$ and write \[ K_{V'}+\Delta'+N'=h^*G. \] The coefficients of $\Delta'$ are strictly less than one. The resolution can be chosen with an effective exceptional divisor $F$ such that $-F$ is $h$-ample. For sufficiently small rational $\varepsilon>0$, the divisor $N'+\eta h^*A-\varepsilon F$ is ample. A general effective rational representative of this ample class can be chosen with sufficiently small coefficients that, after adding $\varepsilon F$, it preserves the sub-klt condition of $\Delta'$. Its pushforward, added to $\Delta$, is effective and defines a klt boundary $\Theta$ on $V$ with \[ K_V+\Theta\sim_\Q G+\eta A. \] The crepant equality can be checked upstairs: its difference is exceptional and relatively numerically trivial, hence zero by negativity. The ordinary cone theorem now gives a rational curve generator $C$ with $0<-(G+\eta A)\cdot C\leq2n$. The preceding comparison yields $-G\cdot C\leq4n$. The contraction $c\colon V\to S$ is the contraction of the same ray for this ordinary klt pair. Since its relative Picard number is one, there is a rational $\lambda>0$ such that $G-\lambda(K_V+\Theta)$ is numerically trivial over $S$. The Cartier descent statement for an ordinary klt extremal contraction, after clearing denominators, gives a rational Cartier divisor $D_S$ with \[ G-\lambda(K_V+\Theta)\sim_\Q c^*D_S. \] In the small case, let $c^+\colon V^+\to S$ be the ordinary flip. Taking strict transforms gives \[ G^+\sim_\Q \lambda(K_{V^+}+\Theta^+)+(c^+)^*D_S, \] which is ample over $S$. Thus the ordinary flip is also the flip of $G$. The ordinary contraction and flip preserve $\Q$-factoriality on the birational output. Their effective exceptional adjoint comparison, multiplied by $\lambda$, gives the asserted comparison for $G$, since the terms pulled back from $S$ cancel. With the nef b-divisor fixed, this comparison is also the improvement of generalized discrepancies used below. \end{proof} \begin{lemma}\label{lemma:two-nef-model} Let $(X,B)$ be a projective $\Q$-factorial klt $\Q$-pair of dimension $n$, let $T=K_X+B$ be pseudo-effective, and let $M$ be a nef $\Q$-Cartier divisor on $X$. Suppose \eqref{eq:full-condition} holds. Then there are a birational contraction $X\dashrightarrow Y$, a number $s_0\in\Q_{>0}$, and a common smooth projective resolution \[ X\xleftarrow{p}W\xrightarrow{q}Y \] with the following properties. If $B_Y$ and $N$ denote the transforms of $B$ and $M$, and $U=K_Y+B_Y$, then \begin{enumerate} \item $(Y,B_Y)$ is klt and $\Q$-factorial, $U$ is semiample, and $\kappa(Y,U)=\kappa(X,T)\geq0$; \item $N$ is pseudo-effective, and $U+s_0N$ is nef of full nef dimension; \item there are effective $q$-exceptional rational divisors $E_0,E_1$ on $W$ such that \begin{equation}\label{eq:two-comparisons} p^*T=q^*U+E_0, \qquad p^*(T+s_0M)=q^*(U+s_0N)+E_1. \end{equation} \end{enumerate} \end{lemma} \begin{proof} By Theorem~\ref{input:models}, there is a terminating $T$-MMP, ending on a $\Q$-factorial klt pair $(Y_0,B_0)$ with nef adjoint $U_0=K_{Y_0}+B_0$. Theorem~\ref{input:abundance} makes $U_0$ semiample. The effective exceptional comparisons for the program give \[ \kappa(X,T)=\kappa(Y_0,U_0)\geq0. \] Let $N_0$ be the transform of $M$. It is the trace on $Y_0$ of the fixed nef b-divisor determined by $M$ on $X$. Because this first program has finitely many steps, there is a rational $s>0$ such that every one of them remains negative for the transform of $T+uM$, for every rational $u\in(0,s]$. Choose $s$ also small enough to retain the strict sign on the flipped side of each flip. These are therefore programs for the corresponding generalized klt pairs: they start generalized klt on $X$, and their discrepancies improve at each step. In particular, $(Y_0,B_0+s\mathbf M)$ is generalized klt. Pushforward to a $\Q$-factorial model preserves pseudo-effectivity, so $N_0$ is pseudo-effective. For completeness, this last numerical assertion applies also to birational contractions with flips. On a common resolution, numerical pushforward to a $\Q$-factorial target is well defined: if a divisor upstairs is numerically trivial, its difference from the pullback of its pushforward is exceptional and relatively numerically trivial, and the negativity lemma makes that difference zero. Effective approximation then proves preservation of pseudo-effectivity. Choose $d>0$ so that $dU_0$ is Cartier and globally generated. Choose a rational $s_0\in(0,s)$ such that \begin{equation}\label{eq:small-nef-coefficient} \frac{1-s_0/s}{d}>4n\frac{s_0}{s}. \end{equation} The generalized adjoint $U_0+s_0N_0$ is pseudo-effective. Apply Theorem~\ref{input:models} to obtain a terminating program for it. We claim that every step is $U_0$-trivial and is also negative for the generalized adjoint with coefficient $s$, where throughout this claim the symbols denote their transforms. Suppose the claim holds up to the next step, and denote the current transforms by $U_i,N_i$. Then $dU_i$ is still Cartier and free, and the pair with nef b-divisor coefficient $s$ is generalized klt. For a $(U_i+s_0N_i)$-negative ray, the equality \[ U_i+s_0N_i=(1-s_0/s)U_i+(s_0/s)(U_i+sN_i) \] and nefness of $U_i$ show that the ray is $(U_i+sN_i)$-negative. Choose its rational curve generator $C$ by Lemma~\ref{lemma:generalized-ray-bound}. If $U_i\cdot C>0$, then integrality of $dU_i$ gives $U_i\cdot C\geq1/d$, whence \[ (U_i+s_0N_i)\cdot C \geq\frac{1-s_0/s}{d}-4n\frac{s_0}{s}>0, \] a contradiction. Thus $U_i$ is trivial on the ray. The morphism defined by $dU_i$ is constant on every fibre of the extremal contraction and factors through it. Its descended line bundle is globally generated; pulling it to the flipped model, when there is a flip, gives the transform of $dU_i$. Hence that transform is again Cartier and free, with equal pullbacks on a common resolution. Since the step is also negative for coefficient $s$, generalized klt singularities with that coefficient persist. This proves the claim by induction through the finite program. Let $Y$ be its output. The ordinary adjoint $U$ is crepant throughout the second program, so $(Y,B_Y)$ is klt, $U$ is semiample, and its Iitaka dimension equals that of $T$. The generalized adjoint $U+s_0N$ is nef. Combining the pullback comparisons of the two programs gives \eqref{eq:two-comparisons} with effective $q$-exceptional divisors. The transform $N$ remains pseudo-effective. It remains to prove full nef dimension. Suppose instead that the nef reduction of $U+s_0N$ has a positive-dimensional very general compact fibre $F$. Shrink the base of the almost holomorphic map so that these compact fibres lie entirely in its morphism locus. The image in $Y$ of the exceptional locus of $q$ has codimension at least two. A general fibre $F$ meets each of its components in codimension at least two, or not at all: discard the images of components that do not dominate the nef-reduction base, and use the fibre dimension theorem for the others. We may therefore choose a curve $C\subset F$ through a very general point, avoiding this exceptional image. If $\dim F>1$, take sufficiently general complete intersections in $F$ through that point; if $\dim F=1$, take $F$ itself. Choose the point also in the common isomorphism locus of the birational contraction and with the very general conditions of \eqref{eq:full-condition} transported from $X$. Since $U$ is nef and $N$ is pseudo-effective, Lemma~\ref{lemma:pseudo-vg} gives \[ U\cdot C=N\cdot C=0, \] because their positive combination has degree zero on $F$. The strict lift $\widetilde C\subset W$ avoids $E_0$ and $E_1$. Both equalities in \eqref{eq:two-comparisons} then give \[ p^*T\cdot\widetilde C =p^*M\cdot\widetilde C=0. \] The curve $p(\widetilde C)$ passes through the chosen very general point of $X$ and contradicts \eqref{eq:full-condition}. \end{proof} \subsection{Positive Iitaka dimension and the remaining case} The preceding construction leaves only one obstruction to bigness: the ordinary semiample adjoint may define a map to a point. If it defines a positive-dimensional base, the induction hypothesis handles its fibres. \begin{proposition}\label{prop:full-positive} Assume $(P_j)$ for $j0$, then $T+cM$ is big for every $c\in\Q_{>0}$. \end{proposition} \begin{proof} After a small $\Q$-factorialization, apply Lemma~\ref{lemma:two-nef-model}. Let $h\colon Y\to S$ be the contraction defined by the semiample divisor $U$. There is an ample rational divisor $A_S$ on $S$ with $U\sim_\Q h^*A_S$, and $\dim S=\kappa(X,T)>0$. Put $Q=U+s_0N$. If a very general fibre $F$ has positive dimension, it has dimension strictly less than $n$. The restriction pair $(F,B_Y|_F)$ is klt and \[ K_F+B_Y|_F\sim_\Q U|_F\sim_\Q0. \] Moreover, $Q|_F$ is nef of full nef dimension. Indeed, a very general point of a very general $F$ can be chosen outside the countable union excluded in the full-nef-dimension assertion for $Q$ on $Y$. Since $U|_F\equiv0$, the divisor $N|_F$ is nef. Apply $(P_{\dim F})$ to this restriction pair and $s_0N|_F$ to conclude that $Q|_F$ is big. When the fibre dimension is zero, relative bigness is automatic. Thus $Q$ is big over $S$. To justify the passage to the generic fibre, fix an ample Cartier divisor $H$ on $Y$ and shrink $S$ so that $h$ is flat. For every positive integer $\ell$ clearing the denominators of $Q$, upper semicontinuity makes the locus \[ \bigl\{s\in S: H^0(F_s,\OO_{F_s}((\ell Q-H)|_{F_s}))\ne0\bigr\} \] closed. Bigness on a very general fibre puts its base point in one of these countably many loci. Choose that point outside every locus that is proper; the locus containing it must then be all of $S$. On the generic fibre, $\ell Q$ is consequently the sum of the ample divisor $H$ and an effective divisor, which proves the asserted relative bigness. A relatively big divisor becomes big after adding a sufficiently large pullback of an ample divisor on the base; equivalently, apply the usual relative form of Kodaira's lemma \cite[Lemma 3.2.1]{BCHM}. It follows that \[ Q+bU=(1+b)U+s_0N \] is big for some rational $b>0$. Equation~\eqref{eq:two-comparisons} then shows that $(1+b)T+s_0M$ is big. Finally, fix $c>0$ rational and choose $\lambda>0$ rational with $\lambda(1+b)<1$ and $\lambda s_00$. \end{proof} Consequently, the remaining work for $(P_n)$ is the full-nef-dimension statement on klt Calabi--Yau pairs. Theorem~\ref{thm:terminal-big} establishes its terminal, zero-boundary form. 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