==== TABLE OF CONTENTS ====
Bounded-klt-complements-for-Fano-contractions//bounded-axes.tex
Bounded-klt-complements-for-Fano-contractions//cheap-collapse.tex
Bounded-klt-complements-for-Fano-contractions//compatible-filtrations.tex
Bounded-klt-complements-for-Fano-contractions//constrained-minimum.tex
Bounded-klt-complements-for-Fano-contractions//discrepancy-push.tex
Bounded-klt-complements-for-Fano-contractions//essential-system.tex
Bounded-klt-complements-for-Fano-contractions//initial-cone.tex
Bounded-klt-complements-for-Fano-contractions//introduction.tex
Bounded-klt-complements-for-Fano-contractions//main.tex
Bounded-klt-complements-for-Fano-contractions//preamble.tex
Bounded-klt-complements-for-Fano-contractions//rank-growth.tex
Bounded-klt-complements-for-Fano-contractions//reduction.tex
Bounded-klt-complements-for-Fano-contractions//scale-tower.tex
Bounded-klt-complements-for-Fano-contractions//trace-lifting.tex
Bounded-klt-complements-for-Fano-contractions//trace-survival.tex
Bounded-klt-complements-for-Fano-contractions//valuation-optimization.tex
Bounded-klt-complements-for-Fano-contractions//width-estimates.tex
Bounded-klt-complements-for-Fano-contractions//references.bib
===========================
==== BEGIN Bounded-klt-complements-for-Fano-contractions//bounded-axes.tex ====
\section{Bounded axes and specialization of effectivity}
\label{b:bounded-axes}
The width estimates still allow torus widths to approach zero, and their
upper counts involve the cost as well as the physical degree. This section
removes both difficulties in the range needed for the first optimization
step. Its key obstruction is a degree-zero section of bounded positive
cost, which would contradict concentration of the entire base maximal
ideal. To construct such a section from a hypothetical obstruction, we
must retain effectivity, not only numerical divisor classes.
Pass to a subsequence on which each width is either bounded above or tends
to infinity. Call the former axes \emph{cheap}, and let \(J\) be the set of
cheap torus indices. The cheap residual axes are precisely the \(p_f\) axes
from \(K_f\); there are \(r=|J|+p_f\) cheap axes in total.
Put \(F=Y_f\); this is \(S_f\) up to the small
\(\Q\)-factorialization in the last step of the tower. We use \(H_f\) for
its pullback to \(F\) as well. Its positive strict coefficients are bounded
below by a fixed positive constant, including after the final normalization
\(s_f=1\). All constants in this section are uniform along the retained
sequence; the finite covers and open restrictions introduced below may
depend on a subsequence.
For each \(j\in J\), the opposite probes constructed in
\eqref{eq:C5} and after \eqref{eq:C8} give
\[
\bar h^{m_j/n}y_j^+,\ \bar h^{m_j/n}y_j^-\in R,
\qquad \operatorname{wt}(y_j^\pm)=\pm d_j\beta_j.
\]
Here \(m_j\in n\Z_{\geq0}\) and the positive integers \(d_j\) are uniformly
bounded. Their boundedness follows from the upper bounds on the cheap
widths. The functions \(y_j^\pm\) are angular functions and have physical
degree zero.
Here is the estimate we will prove. For a homogeneous section of
\(R\), its physical degree is its \(D\)-degree \(m\), and its cost
\(b\) is its \(\tilde v\)-value. On an actual angular lift
\(F_0\in V_m\), the same cost is \(b=m+v(F_0)\).
\begin{proposition}[The bounded axes estimates]\label{b:bounded-axes-estimates}
All widths \(W_j\) are bounded below by a uniform positive constant.
There are cutoffs \(S_0\to\infty\) and a uniform constant \(C_4\) such
that
\begin{equation}\label{eq:D3}
\begin{gathered}
b=\tilde v(\text{weight})\leq C_4m
\qquad(0<b\leq S_0),\\
\dim\bigl(\text{initial spaces at degree }m
\text{ through cost }S_0\bigr)
\leq C_4(1+m)^r,
\qquad r=|J|+p_f.
\end{gathered}
\end{equation}
The independent probes after \eqref{eq:C8} give matching box lower bounds,
up to uniform factors, for all the widths. There are \(r\) algebraically independent probes
of bounded degree and cost: the positive cheap torus probes and the
\(p_f\) residual probes from \(K_f\).
\end{proposition}
The proof is completed at the end of the section. Its main step is to
bound the cheap torus charges in terms of physical degree, as long as
both grow slowly compared with the collapsing scales. A failure will
produce a regular eigenfunction of physical degree zero and bounded
positive cost. We first reduce its regularity to a finite problem on
the terminal field \(K_f\).
\begin{lemma}[Descent of slow pure sections]\label{b:pure-descent}
Suppose, along the sequence, that
\[
g\in K_1^*,\qquad \operatorname{div}_{S_1}(g)+MH_1\geq0,
\qquad M\in\Q_{\geq0},\qquad Ms_{\exp}\longrightarrow0.
\]
Then, for all sufficiently late members,
\[
g\in K_f^*,\qquad \operatorname{div}_{F}(g)+CMH_f\geq0
\]
for a uniform constant \(C\).
\end{lemma}
\begin{proof}
Maintain the inequality
\(\operatorname{div}(g)\geq-CM\Lambda_i\) on all divisorial valuations
of \(K_i\). It holds initially by effectiveness and pullback of the given
rational Cartier divisor. At a collapsing level, the pivot on the proper normal
arithmetic generic fibre has \(\Lambda_i\)-value \(s_i\). Every other strict
prime of that fibre comes from a strict prime on \(S_i\) with
\(H_i\)-coefficient zero. The effective trace boundary of \(Q_i\), when
present, gives \(\Lambda_i\leq\lambda_i\) there. Since
\(\lambda_i\lesssim s_i\) and \(Ms_i\to0\), every possible pole order of
\(g\) on this fibre is an integer bounded in absolute value by a quantity
tending to zero. Thus \(g\) has no poles on the fibre. Properness, normality
and the contraction property imply \(g\in K_{i+1}\).
For a divisorial valuation downstairs, choose a lift attaining the minimum
for \(C_i\) in \eqref{eq:B1}. Use
\(\Lambda_i\lesssim\lambda_{i+1}C_i\) and
\(\lambda_{i+1}\pi_\#C_i=\Lambda_{i+1}\). Dividing by the restriction
multiplicity gives the same maintained inequality at level \(i+1\).
There are at most dimension-many steps, so the resulting constant is
uniform. At the last level \(\lambda_f=O(s_{\exp})\), with the usual zero
convention when the tower has no collapsing step. Off \(\Supp H_f\), the
same integral-order argument excludes poles. On \(\Supp H_f\), the strict
coefficient gap and the effective trace of \(Q_f\) give
\(\Lambda_f\lesssim H_f\). This proves the claimed divisor inequality.
\end{proof}
Applying \eqref{eq:A9} and Lemma~\ref{b:pure-descent} to the products of
opposite probes gives
\[
g_j:=y_j^+y_j^-\in K_f^*,\qquad
\operatorname{div}_{F}(g_j)+CH_f\geq0.
\]
Let \(\Sigma\) be the union of \(\Supp H_f\) and the supports of the zero
divisors of the \(g_j\). The degrees of its components, and the degrees of
the zero and pole divisors of every \(g_j\), are bounded. Indeed the last
displayed inequality bounds the pole degrees; principality bounds the zero
degrees by the same quantity. The degree of \(H_f\) is bounded by its
moduli reserve and the effective Calabi--Yau trace, as in \eqref{eq:C3};
its strict coefficient gap bounds the degree of its support. Divisor data
are omitted when \(\dim F=0\).
For an invariant prime divisor \(E\) of \(\Spec R\), write
\(E|_{K_f}=cP\), with \(P\) normalized when the restriction is nontrivial,
and put \(\ell_j=E(y_j^+)\). At the trivial restriction put \(c=0\) and
interpret all restricted orders as zero. The last step of the tower and
regularity of the two probes give
\begin{equation}\label{b:finite-row-bounds}
cC_f(P)\leq1-h_E/2\quad(c>0),\qquad
-m_jh_E\leq\ell_j\leq cP(g_j)+m_jh_E.
\end{equation}
Consequently the rows \((c,h_E,(\ell_j)_{j\in J})\) belong to a fixed
finite set. To see this, \(c\) is integral and bounded because
\(C_f(P)\) has a uniform positive lower bound; also
\(h_E\in\frac1n\Z\cap[0,1]\) by \eqref{eq:A10}--\eqref{eq:A11}.
Applying \eqref{eq:C1} to the zero and pole divisors bounds
\(c|P(g_j)|\), using \eqref{b:finite-row-bounds}; finally each \(\ell_j\)
is integral. If \(h_E=0\), and either \(c=0\) or the centre of \(P\) is
not contained in \(\Sigma\), then all \(\ell_j=0\).
\subsection{A finite marked regularity problem}
Consider an occurring weight supported on the cheap torus axes,
\[
m\chi+\sum_{j\in J}\alpha_j\beta_j,\qquad
\left(m+\sum_{j\in J}|\alpha_j|\right)s_{\exp}\longrightarrow0.
\]
Raise a nonzero section of this weight to a bounded positive common power
clearing \(n\) and the \(d_j\). It becomes
\[
\bar h^{m'/n}\prod_{j\in J}(y_j^+)^{a_j}g,
\qquad n\mid m',\quad a_j\in\Z,\quad g\in K_1^*.
\]
Here \(m'\) is that common multiple of \(m\), and \(a_j\) is the same
multiple of \(\alpha_j/d_j\). Multiplying this expression by the opposite
regular probe to the power \(a_j\) when \(a_j>0\), and by the positive
regular probe to the power \(-a_j\) when \(a_j<0\), produces a pure section.
Its degree is \(m'+\sum|a_j|m_j\) and its angular coefficient is
\(g\prod_{a_j>0}g_j^{a_j}\). Equation~\eqref{eq:A9} and
Lemma~\ref{b:pure-descent} imply that \(g\in K_f\) and that its poles on
\(F\) lie in \(\Sigma\). Regularity of the original eigenfunction is
exactly the family of inequalities
\begin{equation}\label{eq:D1}
cP(g)+m'h_E+a\cdot\ell\geq0.
\end{equation}
These tests suffice because a pole divisor of a rational eigenfunction is
invariant, and \(R\) is normal.
We shall exclude an obstruction sequence with
\(\sum|a_j|/m'\to\infty\), counting \(m'=0\), \(a\ne0\) as infinite.
Suppose such a sequence exists and first assume \(\dim F>0\).
Our aim is a single rational polyhedral test for the possible degrees
and charges. The finitely many row types do not yet give such a test:
the projected prime valuations \(P\) and the existence of an effective
representative for the required pole divisor still depend on the
member. We next place the relevant primes on bounded marked models.
\begin{lemma}[A bounded guard model]\label{b:guard-model}
After passing to a subsequence, there are bounded projective
\(\Q\)-factorial varieties of Fano type \(F'\to F\) and labelled prime
divisors on \(F'\), of bounded number and degree, with the following
properties. Every nontrivial projection in \eqref{eq:D1} for which
\(h_E>0\), or whose centre is contained in \(\Sigma\), is one of the
marked prime valuations. Every component of \(\Sigma\) has marked strict
transform, and every exceptional prime of \(F'\to F\) is marked. The
pullback of \(H_f\) has bounded degree on \(F'\).
\end{lemma}
\begin{proof}
The effective Calabi--Yau datum \(C_f\) has a uniform positive discrepancy
bound, a boundary dominating a fixed positive multiple of \(H_f\), and a
big nef b-part with a fixed positive \(\mathbf H_f\)-reserve. Choose a
fixed rational \(C_0\) such that
\[
\mathbf T_j=\operatorname{div}(g_j)+C_0\mathbf H_f
\]
is the b-Cartier closure of an effective divisor on \(F\). For sufficiently
small fixed rational \(\varepsilon>0\), define guard data by
\[
C^{\mathrm g}=C_f-\varepsilon\sum_{j\in J}\mathbf T_j.
\]
Subtract the same terms from the moduli presentation. Equation~\eqref{eq:C1}
bounds \(\mathbf T_j(P)\) by a uniform multiple of \(C_f(P)\), so this
retains a uniform positive discrepancy bound. Principal terms do not affect
nefness, and the reserve absorbs \(\varepsilon C_0|J|\mathbf H_f\).
Thus the new moduli remain big and nef, with a positive
\(\mathbf H_f\)-reserve, and the trace boundary remains effective.
There is a fixed \(e_0>0\) such that all the prime valuations specified in
the statement satisfy \(C^{\mathrm g}(P)\leq1-e_0\). For a projection
with \(h_E>0\), this follows from \eqref{b:finite-row-bounds}, since
\(h_E\geq1/n\) and \(c\geq1\). If \(h_E=0\), Equation~\eqref{eq:B2}
gives \(P(H_f)=0\). A centre contained in \(\Sigma\) therefore lies
generically outside \(\Supp H_f\) and inside a zero divisor of some
\(g_j\); its integral order \(P(g_j)\) is at least one. Since
\(C_f(P)\leq1\), subtracting \(\varepsilon\mathbf T_j\) gives the
required gap. A strict prime of \(F\) in \(\Supp H_f\) has a boundary
coefficient bounded below by domination and the strict coefficient gap.
A remaining strict prime of \(\Sigma\) has the same positive zero order
just used. This also proves the assertion for all components of \(\Sigma\).
Apply Lemmas~\ref{b:ordinary-approximation} and
\ref{b:controlled-extraction}. The monomial calculation in the proof of
Lemma~\ref{b:scale-bad-primes} gives only finitely many exceptional
prime valuations with \(C^{\mathrm g}(P)\leq1-e_0<1\), including
equality at the upper endpoint: their positive integral normal weights
satisfy a weighted-sum bound below one, and each coefficient in that
sum is positive.
Ordinary approximation with a tiny ample reserve does not increase their
discrepancies. The extraction construction gives a projective
\(\Q\)-factorial model \(F'\to F\) extracting exactly these primes.
The guard trace is effective on \(F'\), retains its uniform positive
log discrepancy bound, and has big nef moduli. Ordinary approximation
therefore gives an effective big boundary with rationally trivial adjoint
and a uniform positive discrepancy bound. By
\cite[Corollary~1.4]{BirkarBAB}, the varieties \(F'\) are bounded; they
are also of Fano type by the ample-reserve version of the construction.
Mark every strict prime of \(F'\) whose guard discrepancy is at most
\(1-e_0\). Each has guard boundary coefficient at least \(e_0\).
The trace degree of the nef b-part is nonnegative by the perturbation
argument in \eqref{eq:C2}. Thus the degree of the effective guard boundary
is at most the degree of \(-K_{F'}\), which is uniformly bounded for the
bounded very ample polarizations by the cone-theorem calculation preceding
\eqref{eq:C1}. This bounds both the number and the degrees of the marks.
The surviving \(\mathbf H_f\)-reserve gives the asserted bound on the
degree of the pullback of \(H_f\) in the same way.
\end{proof}
Label the marks along a subsequence, preserving which are exceptional and
which are strict over components of \(\Sigma\). Stabilize the finite sets
of rows in \eqref{eq:D1} projecting to each mark and the rows with trivial
projection. Introduce a rational signed pole allowance \(r_P\) at each
mark: a function \(g\) is allowed when \(P(g)\ge-r_P\), so a
negative allowance requires a zero. Consider the conditions
\begin{equation}\label{eq:D2}
\begin{gathered}
m'\geq0,\qquad
cr_P\leq m'h_E+a\cdot\ell
\quad\text{for every row projecting to a mark }P,\\
0\leq m'h_E+a\cdot\ell
\quad\text{for every row with }c=0,\\
r_P\leq0
\quad\text{at marks strict over primes of }F\text{ outside }\Sigma,\\
\sum_{P\ \mathrm{marked}}r_P[P]\in
\overline{\operatorname{Eff}}(F').
\end{gathered}
\end{equation}
An actual candidate satisfies these conditions with \(r_P=-P(g)\):
\(\operatorname{div}_{F'}g+\sum r_PP\) is effective. Conversely, suppose
a rational solution has \(\sum r_PP\) rational-linearly equivalent to an
effective divisor. Choose a positive integer \(q\) and a function
\(f\in K_f^*\) with
\[
\operatorname{div}_{F'}f+q\sum r_PP\ge0,
\qquad qm'\in n\Z,\qquad qa\in\Z^J.
\]
The integer \(q\) clears the divisor coefficients and the degree and
charge denominators. For a marked row, effectivity and \eqref{eq:D2}
give
\[
cP(f)+q(m'h_E+a\cdot\ell)
\ge q\bigl(m'h_E+a\cdot\ell-cr_P\bigr)\ge0.
\]
The rows with trivial projection also remain valid after multiplication
by \(q\). The poles of \(f\) downstairs on \(F\) are confined to
\(\Sigma\): the sign condition excludes other strict marked primes,
and effectivity excludes all unmarked strict primes. Any remaining
nontrivial projection has \(h_E=0\), \(\ell=0\), and centre not
contained in \(\Sigma\), by Lemma~\ref{b:guard-model}. The function
is regular at the generic point of that centre, so its order is
nonnegative. Thus all tests \eqref{eq:D1} hold, and
\(\bar h^{qm'/n}\prod_j(y_j^+)^{qa_j}f\) is an actual regular
eigenfunction.
\subsection{A marked family with a Fano type generic fibre}
We now construct a common cone test for \eqref{eq:D2}. We need two
implications. Every actual candidate must satisfy the cone inequalities
of one geometric generic marked fibre, even when its coefficients
\(r_P\) are unbounded. Conversely, one fixed rational solution of
those inequalities must give an effective representative, and hence
an actual eigenfunction, on all sufficiently late retained members.
The first implication will use sweeping curves; the second will
spread a line bundle with a nonzero section.
Both arguments require the geometric generic marked fibre to be
\(\Q\)-factorial and of Fano type. We establish these properties by
retaining an ordinary klt Calabi--Yau boundary with a fixed denominator
that dominates a big marked divisor. The member-dependent generalized
guard data do not yet give this finite set of transferable divisor
coefficients and adjoint relations. The next lemma constructs it.
\begin{lemma}[A bounded auxiliary Calabi--Yau boundary]
\label{b:bounded-guard-boundary}
The models of Lemma~\ref{b:guard-model} admit effective integral divisors
\(P'\) of bounded degree, supported on the marks, such that \(|P'|\) is
birational. They also admit klt boundaries \(\Theta^+\), with bounded
degrees and coefficients in a fixed finite rational set, satisfying
\[
\Theta^+\geq e'P',\qquad N(K_{F'}+\Theta^+)\sim0
\]
for a fixed positive rational \(e'\) and a bounded positive integer \(N\).
The marks, \(P'\), and \(\Theta^+\) together form a bounded family of
labelled divisor data.
\end{lemma}
\begin{proof}
Use the bounded \(k\) at the final level in \eqref{eq:B3} and set
\(P'=\lfloor kH_f\rfloor\) on \(F'\), using total pullback before
rounding. This is effective integral of bounded degree. Its support lies
in the marks, since the strict support of \(H_f\) and all exceptional
primes of \(F'\to F\) are marked. The functions in \eqref{eq:B3} still
belong to its complete system: their integral orders and positivity of
pullback allow the indicated rounding. Thus this system remains
birational, and \(P'\) is big.
Choose a small fixed rational \(e'>0\) so that
\[
\Delta_0=e'\sum_{P\ \mathrm{marked}}P+e'P'
\]
is at most the guard boundary. Such a uniform choice is possible because
the mark coefficients have a fixed lower bound and the coefficients of
\(P'\) are bounded by its degree. By ordinary approximation of the big
nef b-part, obtain an effective uniformly positive-lc boundary \(\Theta\)
with
\[
K_{F'}+\Theta\sim_{\Q}0,\qquad \Theta\geq\Delta_0,
\qquad \Theta-\Delta_0\ \text{big}.
\]
Here the added effective representative of the big b-part has big trace:
sections on a descent remain sections after pushforward. It supplies the
last bigness assertion.
Set \(D'=-(K_{F'}+\Delta_0)\), which is big. Since \(F'\) is a
\(\Q\)-factorial variety of Fano type, it is a Mori dream space by
\cite[Corollary~1.3.1]{BCHM}.
Run a \(D'\)-MMP to a \(\Q\)-factorial model \(Y\) with nef and big
transform \(D_Y\). The program extracts no divisors. The rationally
trivial log adjoint \(K+\Theta\) is crepant through the program, so
\((Y,\Theta_Y)\), and hence \((Y,(\Delta_0)_Y)\), retain the uniform
positive discrepancy bound. Moreover
\(D_Y=-(K_Y+(\Delta_0)_Y)\) is nef and big. Thus \(Y\) is bounded by
\cite[Theorem~1.1]{BirkarBAB}. It is of Fano type: writing
\(D_Y\sim_{\Q}A+E\) with \(A\) ample and \(E\geq0\), a sufficiently
small positive multiple of \(E\) can be added to \((\Delta_0)_Y\) while
keeping klt, and the resulting negative log adjoint is ample.
The coefficients of \((\Delta_0)_Y\) have a fixed common denominator.
Lemma~\ref{b:class-lattices}, applied to the bounded \(\Q\)-factorial
varieties \(Y\), gives a bounded integer \(q>0\) with \(qD_Y\) Cartier.
Apply \cite[Theorem~1.1 and Remark~1.2]{FujinoBPF} to the klt pair
\((Y,(\Delta_0)_Y)\), with \(L=qD_Y\) and \(a=1\). The divisor
\[
L-(K_Y+(\Delta_0)_Y)=(q+1)D_Y
\]
is nef and big; for a klt pair there are no additional lc-centre
conditions in log bigness. The theorem gives a bounded integer \(N>1\)
with \(ND_Y\) Cartier and free. Increase the free multiple if necessary.
For a general member
\(G_Y=\operatorname{div}_Y\psi+ND_Y\), Bertini on a log resolution
shows that
\((Y,(\Delta_0)_Y+G_Y/N)\) is klt.
Use the same rational canonical form on all models and define
\[
\Theta^+=-K_{F'}+\frac1N\operatorname{div}_{F'}\psi.
\]
On a common resolution with maps \(p\) to \(F'\) and \(q_Y\) to \(Y\),
the MMP negativity comparison reads
\(p^*D'=q_Y^*D_Y+E'\), with \(E'\geq0\) exceptional over \(Y\).
Consequently
\[
p^*(\Theta^+-\Delta_0)=q_Y^*(G_Y/N)+E'\geq0.
\]
Thus \(\Theta^+\geq\Delta_0\). Its log adjoint is crepant to that of
the klt pair just constructed on \(Y\), so it too is klt. Moreover
\(N(K_{F'}+\Theta^+)=\operatorname{div}_{F'}\psi\).
Its coefficients belong to \(\frac1N\Z\cap[0,1)\), and its degree is
the bounded degree of \(-K_{F'}\). In bounded projective embeddings,
the supports are parametrized by finitely many fixed-dimension,
fixed-degree Chow spaces \cite[Definition~3.21]{KollarChow2017}. Their
universal cycles may be taken over normal strata. The finite coefficient
set bounds the possible multiplicities, and the degree bound bounds the
number of prime components. Taking finite products and labelling their
components therefore bounds these data together with the marks and
\(P'\).
\end{proof}
Parametrize the labelled data of Lemma~\ref{b:bounded-guard-boundary} in
finite-type projective families over \(\C\), and pass to a subsequence
with fixed coefficient types and fixed \(N\). Choose a minimal closed
parameter subset containing infinitely many selected indices, then an
irreducible component \(T\) containing infinitely many of them. Every
proper closed subset of \(T\) contains only finitely many selected indices:
otherwise it would contradict minimality. Hence every dense open
restriction retains all but finitely many of those indices. Repetitions
of a parameter value cause no difficulty; in that case a minimal subset
may be a point. Denote the resulting marked family by
\(X^0\to T\), with fibres \(X^0_t\), marked primes, integral divisor
\(P'_t\), and boundary \(\Theta^+_t\).
This retention property survives the finite field extensions used below.
After normalization and restriction to a dense open, a finite extension of
\(\C(T)\) gives an irreducible finite surjective cover \(T'\to T\).
Every proper closed subset of \(T'\) has proper closed image, since finite
maps preserve dimension and both bases have the same dimension. Therefore
later dense open restrictions of \(T'\) exclude lifts only over a proper
closed subset of \(T\). Choose lifts of the remaining selected points.
The same reasoning applies after any finite composite of these extensions.
\begin{lemma}[The geometric generic marked model]
\label{b:generic-marked-fano}
After the finite covers and dense open restrictions just described, the
geometric generic marked fibre \(X^0_{\bar\eta}\) is
\(\Q\)-factorial and of Fano type. Its marked divisor \(P'_{\bar\eta}\)
is big, and its marked boundary \(\Theta^+_{\bar\eta}\) is klt with
\(N(K+\Theta^+_{\bar\eta})\sim0\).
\end{lemma}
\begin{proof}
We first transport the integral divisor-class relations to the generic
fibre. We then transport the crepant snc calculation for
\(\Theta^+\) and the section map of a Cartier multiple of \(P'\);
these establish klt singularities and bigness, respectively.
Spread a projective smooth resolution of the geometric generic fibre,
after a finite extension, and restrict the base so that it resolves every
fibre under consideration. Track all exceptional prime divisors and all
strict transforms of the finitely many marked primes. This uses the
resolution and constructibility argument in
Lemma~\ref{b:class-lattices}: keep a birational isomorphism on dense opens,
spread the complementary components, and fix their fibre and image
dimensions. Shrink also so that the downstairs fibres are geometrically
normal and integral and the prime marking families have integral
geometric fibres. Write this simultaneous resolution as
\(b_t:W_t\to X^0_t\).
On every selected fibre, vanishing for a log Fano boundary
\cite[Theorem~3.2]{FujinoFundamental} and rationality of klt
singularities \cite[Theorem~0.1]{FujinoRational} give
\(H^1(\OO)=H^2(\OO)=0\) both downstairs and upstairs. Semicontinuity and
density of the selected points give these vanishings throughout an open
base. On its complex fibres, the exponential sequence and GAGA
\cite[Section~12, Theorems~1--3; Section~20, Proposition~18]{SerreGAGA}
identify \(\Pic\) with integral \(H^2\). On a connected dense stratum, topological
constructibility gives locally constant integral group data
\[
H^2(X^0_t,\Z)\ \xrightarrow{\ b_t^*\ }\ H^2(W_t,\Z)
\ \longrightarrow\ H^2(W_t,\Z)/
\langle[\text{exceptional primes}]\rangle.
\]
The last group is \(\Cl(X^0_t)\), and the image of the first group is
\(\Pic(X^0_t)\). The exceptional and marked upstairs divisor classes,
and the class of the relative canonical bundle of the smooth resolution,
are locally constant as well. These are integral diagrams, so their
cokernels, including torsion, are controlled; trivial monodromy is not
required.
On the selected fibres, the image of \(\Pic\) has finite index in
\(\Cl\), and \(N[K+\Theta^+]=0\) in \(\Cl\). The same assertions
therefore hold on this entire complex stratum. Here the canonical class
downstairs is the pushforward of the canonical class upstairs. They hold
also on the geometric generic fibre. For completeness, descend the finite
family, markings, resolution and open conditions to a countable
algebraically closed subfield of \(\C\), and choose a complex point
generic over that subfield. The geometric generic model is identified,
up to an isomorphism of algebraically closed fields preserving these
finite data, with this complex fibre: the two algebraically closed
extension fields have the same infinite transcendence degree over the
embedded defining function field. Thus the integral class conclusions
apply there. Finite index of \(\Pic\) in \(\Cl\) gives
\(\Q\)-factoriality, and the second conclusion gives the asserted
principal multiple of the log adjoint.
It remains to verify singularities and bigness; neither follows from the
integral class diagram alone. Take a log resolution of the geometric
generic marked pair, and a relative rational volume form and a rational
function witnessing \(N(K+\Theta^+)\sim0\). Spread all these data after
a further finite extension. On the smooth resolving family, retain the
divisor identities for the form and the witnessing function, the strict
transforms and exceptional components, and the snc pullback computation.
The coefficients and orders in these finite divisor formulas remain
unchanged on every fibre after shrinking. This can be arranged by spreading
each prime component of the formulas and retaining its geometric
integrality and its image dimension; restricting the formulas on the
smooth models then preserves the divisor orders. Pushing down preserves
the marked principal relation. The snc crepant boundary coefficients are now
constant, and one selected fibre is klt. They are therefore all less than
one on the geometric generic resolution, proving that
\(\Theta^+_{\bar\eta}\) is klt.
Choose a positive Cartier multiple \(tP'_{\bar\eta}\), and spread its
line bundle with a rational section whose divisor is exactly this marked
divisor. Retain the divisor identity by the same resolution argument.
After further shrinking, formation of its space of sections commutes with
base change. The dimension of the image of the associated rational map
is generically constant, by constructibility. On each selected fibre,
the system \(|tP'_t|\) contains products from the birational system
\(|P'_t|\), so its image has dimension \(\dim F'\). The generic system
has that same dimension. Hence \(P'_{\bar\eta}\) is big.
Finally \(\Theta^+_{\bar\eta}\geq e'P'_{\bar\eta}\) is big. To see
Fano type explicitly, write
\(\Theta^+_{\bar\eta}\sim_{\Q}A+E\) with \(A\) ample and \(E\geq0\).
For sufficiently small rational \(t>0\), the effective boundary
\((1-t)\Theta^+_{\bar\eta}+tE\) is klt, and its negative log adjoint
is rationally equivalent to \(tA\). This is a log Fano boundary.
\end{proof}
\subsection{Facet curves and one effective witness}
The geometric generic marked fibre now has the Fano type and rational
factoriality needed for its effective cone. We use this one cone in two
ways: finitely many sweeping curve families test every candidate on a
retained fibre, and one fixed rational solution supplies an effective
section that spreads to those fibres. These two implications suffice for
the degree-zero obstruction.
\begin{lemma}[Specializing a finite cone test]\label{b:marked-cone-test}
In \eqref{eq:D2}, replace the last condition by
\[
\sum_{P\ \mathrm{marked}}r_P[P_{\bar\eta}]
\in\overline{\operatorname{Eff}}(X^0_{\bar\eta}).
\]
This is a finite rational polyhedral test. After the allowed covers and
open restrictions, every actual candidate satisfies this test. Conversely,
each fixed rational solution has, after one clearing multiple fixed
for that solution, regular eigenfunction realizations on all retained
sufficiently late members of the sequence.
\end{lemma}
\begin{proof}
By Lemma~\ref{b:generic-marked-fano} and
\cite[Corollary~1.3.1]{BCHM}, \(X^0_{\bar\eta}\) is a Mori dream space.
Its pseudo-effective cone in numerical divisor space is rational
polyhedral. We first show that each facet is tested by a family of curves
sweeping a dense open of \(X^0_{\bar\eta}\).
Choose a rational divisor class \(D\) in the relative interior of that
facet. The class is pseudo-effective. Its divisor MMP ends in a minimal
model \(Y_D\) on which the transform \(D_Y\) is nef and semiample.
As \(D\) lies on the boundary of the pseudo-effective cone, \(D_Y\) is
not big. A sufficiently divisible multiple defines a contraction
\(\varphi:Y_D\to Z_D\) with positive-dimensional general fibres.
The birational program has no extraction. There is consequently an open
\(U\subset Y_D\), with complement of codimension at least two, isomorphic
to an open of \(X^0_{\bar\eta}\). On a common resolution the comparison
has the form
\[
p^*D=q^*D_Y+E_D,\qquad E_D\geq0,
\]
where the correction is exceptional over \(Y_D\). Enlarge the excluded
codimension-two subset to contain its image and all inverse-map
indeterminacy. Curves in \(U\) then lift to the original variety and
avoid this correction, including the correction from any contracted
fixed divisorial part of \(D\).
A general fibre of \(\varphi\) is geometrically integral. Each component
of \(Y_D\setminus U\) either does not dominate \(Z_D\), and hence misses
a general fibre, or has codimension at least two in that fibre. If the
relative dimension is one, the latter case is impossible: a bad component
has dimension at most \(\dim Y_D-2<\dim Z_D\). Thus the entire general
curve fibre lies in \(U\). In larger relative dimension \(t\), intersect
a general fibre with \(t-1\) sufficiently general very ample hyperplanes.
Bertini gives an integral curve, and the codimension estimate makes it
miss the bad locus entirely.
These curves vary in a projective flat family over an integral parameter
space after restriction to a dense open of the fibre-and-hyperplane
parameter space. Its evaluation map dominates \(Y_D\). Indeed the
incidence obtained by choosing a point and hyperplanes through it is
irreducible and dominates \(Y_D\); restriction to the dense open of flat
integral curves avoiding the bad locus preserves dominance. The lifted
curves therefore sweep a dense open of \(X^0_{\bar\eta}\). Their
intersections with divisor classes are constant in this family. The
corresponding linear functional is nonnegative on every effective class,
because a member can be chosen not contained in the divisor's support;
it is positive on an ample class. Its value on \(D\) is zero by the
pullback comparison and contraction. A nonzero supporting functional
vanishing at a relative interior point of a facet defines exactly that
facet inequality. This constructs the required test for each facet.
There are finitely many facets. Spread the finitely many curve families,
their parameter spaces and evaluation maps after a finite extension of
the base function field. Spread also fixed Cartier multiples of all
marked divisors. After shrinking, the curves remain projective and flat
with integral geometric fibres, their intersection vectors with these
markings remain fixed, and their evaluation maps dominate dense opens of
every selected fibre. To justify the last assertion, the generic
evaluation image contains a dense open. Its image is constructible, so
after spreading and shrinking one retains an open in that image whose
intersection with each fibre is dense; fibre-dimension constructibility
excludes the exceptional parameter values. Equivalently one can track
the evaluation incidence and its generic dominance before the shrink.
Let \(\sum r_PP\) be the class furnished by an actual candidate on any
retained fibre. It has an effective rational Cartier representative.
For each of our curve families, density provides a member not contained
in that representative's support. Its intersection with the effective
representative is nonnegative, and is the fixed linear combination of
the marked intersection vector with coefficients \(r_P\). Thus every
generic facet inequality holds. This argument makes no boundedness
assumption on the coefficient vector of the candidate.
For the converse, fix one rational solution of the generic test.
On the generic Mori dream space, a rational class in the effective cone
has an effective rational representative. Numerical and rational linear
classes agree for rational Cartier divisors here by
Lemma~\ref{b:class-lattices}. Hence for some positive integer \(q\) there
is a rational function \(f\) with
\[
\operatorname{div}(f)+q\sum r_PP_{\bar\eta}\geq0.
\]
Replace \((q,f)\) by \((kq,f^k)\) for a sufficiently divisible
positive integer \(k\), and retain the notation \((q,f)\). This
preserves the effective divisor inequality while making the marked
divisor Cartier and all degree and charge exponents integral, with
the degree divisible by \(n\). Spread the line bundle
\(\OO(q\sum r_PP_{\bar\eta})\), its rational trivialization, and the
nonzero regular section represented by \(f\), after a further finite
extension. The divisor identity can also be retained on the simultaneous
resolution. On a dense open of the parameter, the section is regular and
not identically zero on any fibre. Thus its rational-function
representatives have exactly the permitted pole bound on each selected
fibre; possible additional zeros are harmless. Regularity as a section
of this specified line bundle excludes unmarked poles.
The minimal-closure retention property ensures that these open
restrictions keep infinitely many selected indices, and eventually all
retained ones. Transfer the witness through the marked identifications
with the corresponding \(F'\). The converse argument following
\eqref{eq:D2}, including its automatic unmarked tests, gives the asserted
regular eigenfunctions. Only this one rational solution, its clearing
multiple and its effective section were spread. No simultaneous family
of the algebras \(R\) is needed, and no equality between the effective
cones of different fibres is asserted.
\end{proof}
\begin{proposition}[The slow-charge bound]\label{b:slow-charge-bound}
For sequences of actual candidates above satisfying
\( (m'+\sum|a_j|)s_{\exp}\to0\), the ratio
\(\sum|a_j|/m'\) cannot tend to infinity, with the convention already
specified at degree zero.
\end{proposition}
\begin{proof}
Suppose first that \(\dim F>0\). Use the stabilized rows of
\eqref{eq:D2} and the generic cone test of Lemma~\ref{b:marked-cone-test}.
They define a rational polyhedral cone in the variables \((m',a,(r_P))\).
Its projection \(\mathcal P\) to \((m',a)\) is again a rational
polyhedral cone and is therefore closed. For example, express the
original cone by finitely many rational generators and project those
generators. This proof of closedness imposes no boundedness condition
on the auxiliary coefficients \(r_P\).
Actual candidates lie in \(\mathcal P\). Divide an alleged obstruction
sequence by \(\sum|a_j|\). After a subsequence, their charge coordinates
converge to \(a_\infty\) of \(\ell^1\)-norm one, whereas their degree
coordinates tend to zero. Closedness gives
\((0,a_\infty)\in\mathcal P\). The rational face
\(\mathcal P\cap\{m'=0\}\) therefore contains a rational point with
nonzero charge: one can use its rational generators, or density of
rational points in its relative interior. A nonempty fibre of a rational
linear inequality system over a rational point has a rational point, so
choose rational auxiliary coordinates giving a fixed rational solution
of \eqref{eq:D2} with \(m'=0\) and \(a\ne0\).
By Lemma~\ref{b:marked-cone-test}, one fixed clearing multiple and one
spread effective witness produce actual nonzero regular eigenfunctions
on all sufficiently late retained rings \(R\), of degree zero and fixed
nonzero charge coordinates. If \(\dim F=0\), the same conclusion follows
directly: pure descent makes \(g\) constant, all restrictions to \(K_f\)
are trivial, and the stabilized trivial-projection inequalities in
\eqref{eq:D2} themselves give the rational polyhedral cone. There are no
markings or effectivity variables in this case.
The resulting eigenfunctions have positive \(\tilde v\)-cost, since a
nonzero charge is nonconstant and the grading is positive. These costs
are uniformly bounded: only cheap axes occur, their widths are bounded
above, the charge coordinates and the \(d_j\) are fixed or bounded, and
\(\tilde v\in Q_T\). Each eigenfunction lifts, in physical degree zero,
to a function of \(\mathcal A_0=k[Z]\) with the same positive value.
The lift therefore belongs to \(\mathfrak m_z\). Its bounded value
contradicts \eqref{eq:A7}, which asserts
\(\tilde v(\mathfrak m_z)\to\infty\) uniformly over all nonzero
functions in this ideal. This rules out the obstruction sequence.
\end{proof}
\subsection{The cost gap and the section count}
\begin{proof}[Proof of Proposition~\ref{b:bounded-axes-estimates}]
The residual widths \(1/s_i\) are already at least one. If some torus
width \(L_j\) tended to zero on a subsequence, choose integers
\(k\to\infty\) sufficiently slowly that \(kL_j\) remains bounded and
\(ks_{\exp}\to0\). Apply \eqref{eq:C5} to \(k\beta_j\), using a fixed
scalar norm bound at least one. It gives actual opposite weights of
bounded physical degree and charges \(\pm d_0k\beta_j\), with
\(d_0\) positive and bounded. These weights are supported on a cheap
axis, satisfy the slow condition, and have unbounded charge-to-degree
ratio. Raising to the earlier bounded common clearing power does not
change that conclusion. This contradicts
Proposition~\ref{b:slow-charge-bound}. A further-subsequence argument
therefore gives a uniform positive lower bound for every \(L_j\).
Choose \(S_0\to\infty\) so slowly that
\[
S_0s_{\exp}\longrightarrow0,\qquad
S_0/W_j\longrightarrow0\quad\text{for each width tending to infinity}.
\]
For a weight \(m\chi+\sum\alpha_j\beta_j\) of positive cost at most
\(S_0\), the weight comparisons preceding \eqref{eq:C7} give
\[
m\lesssim S_0,\qquad
\sum_j|\alpha_j|L_j\lesssim S_0.
\]
An expensive torus coordinate must vanish, since it is integral and its
width eventually exceeds every fixed multiple of \(S_0\). The lower
width bound then gives
\(m+\sum|\alpha_j|\lesssim S_0\), so these weights satisfy the slow
condition. Proposition~\ref{b:slow-charge-bound} now implies a uniform
\(\sum|\alpha_j|\lesssim m\) for all such weights: failure would give
an obstruction sequence on late members. The bounded clearing factors
and the bounded positive \(d_j\) preserve this equivalence.
Since the remaining widths are bounded above,
\[
b=m+\tilde v\Bigl(\sum_{j\in J}\alpha_j\beta_j\Bigr)
\leq m+\Bigl\|\sum_{j\in J}\alpha_j\beta_j\Bigr\|
\lesssim m.
\]
Thus every initial at fixed degree \(m\) through \(S_0\) already occurs
through \(\min\{S_0,C_4m\}\). Apply \eqref{eq:C8} with this cutoff.
The factors for expensive widths are uniformly bounded, while each cheap
factor is at most a fixed multiple of \(1+m\). There are exactly
\(|J|+p_f=r\) cheap axes. This proves both estimates in \eqref{eq:D3},
including the constant initial at degree zero.
Finally the probe construction after \eqref{eq:C8} had box denominators
\(\max\{1,W_j\}\). The positive lower width bound makes these uniformly
comparable to \(W_j\). Hence those lower bounds now match all widths.
The residual probes have independent initials along the field tower, and
the positive torus probes add independent Laurent charges over \(K_1\).
Their combined initials are algebraically independent. The stated
\(r\) probes at the cheap axes have bounded degree and cost.
\end{proof}
Through the angular identification of Section~\ref{a:angular},
\eqref{eq:D3} controls the actual systems \(V_m\). At each degree
\(m\in n\Z_{\ge0}\), no function has cost in
\(C_4m<b(F)\le S_0\), and the quotient of \(V_m\) by its subspace
of cost greater than \(S_0\), including zero, has dimension at most
\(C_4(1+m)^r\). The \(r\) cheap probes have actual lifts in these
systems with bounded degree and cost and independent initials. These
are the finite-cutoff estimates used when varying the valuation on
\(k(X)\).
==== END Bounded-klt-complements-for-Fano-contractions//bounded-axes.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//cheap-collapse.tex ====
\section{Collapsing the cheap parameters}
\label{c:cheap-section}
This section has two outputs. First we move the cheap probe values to
zero while retaining a lower bound on every angular section and a bounded
form discrepancy. We then prove a two-sided bound for all angular
values at valuations Gaussian on the cheap probes, vanishing there,
and satisfying the scaled lower bound and the stated discrepancy bound.
The bridge is the cost gap from
Proposition~\ref{b:bounded-axes-estimates}: it forces certain traced
outputs of bounded degree and cost to have exactly zero value. The
two-sided estimate will supply the lower dimension bound in the next
section.
We continue with the actual field $K$, the fractional form $\theta$,
and the nested multiplicative angular systems $V_m$.
Degrees are nonnegative multiples of $n$, and the cost of a nonzero
$F\in V_m$, with this assigned degree, is
$b(F)=m+v(F)$. In particular, changing the assigned degree changes the
cost. We retain the uniform comparison $m\le Cb(F)$ for positive
costs. Constants denoted by $C$ may increase, but are uniform along
the chosen sequence. Let $y=(y_1,\ldots,y_r)$ be actual angular lifts
of the cheap probes in \eqref{eq:C8} and \eqref{eq:D3}. Their degrees
and costs are uniformly bounded; their initials at $v$ are algebraically
independent. Thus $v$ is Gaussian on $y$, and the vector $v(y)$ ranges
in a fixed bounded set. As in the preceding section, the notation
$P_0=\mathfrak p^{\ell_0}$ refers to a Frobenius power in a reduction
of characteristic $\mathfrak p$; it is unrelated to the auxiliary
cone dimension $p$.
\par
\begingroup
\dimen0=\pagegoal
\advance\dimen0 by -\pagetotal
\ifdim\dimen0<7\baselineskip
\newpage
\fi
\endgroup
\subsection{A translated-box count}
A section-dimension bound will control the number of distinct values of
powers of one trace output multiplied by cheap probe monomials. The
following count turns that control into a rational linear expression in
the probe values, with bounded denominator and coefficients. These
coefficient bounds will control the slope of the optimization objective.
\begin{lemma}\label{c:cheap-box-count}
Fix $r\ge1$ and $A>0$. There are constants $N_0,D_0,C_0$, depending
only on $r,A$, with the following property. Let $T\ge1$, let
$t_1,\ldots,t_r$ be linearly independent over $\Q$, and let $s\in\R$.
For an integer $N\ge N_0$, set $L=\lceil NT\rceil$. Suppose that
\[
\#\{as+\alpha\cdot t:0\le a<N,\quad
\alpha\in\{0,\ldots,L-1\}^r\}
\le A(NT)^r.
\]
Then $s=\gamma\cdot t$ for a unique $\gamma\in\Q^r$. The order of
$\gamma$ in $\Q^r/\Z^r$ is at most $D_0$, and
$|\gamma|\le C_0T$.
\end{lemma}
\begin{proof}
If $s$ is outside the rational span of the $t_j$, all $NL^r$ displayed
values are distinct. This contradicts the bound once $N>A$.
Otherwise write $s=\gamma\cdot t$, uniquely by independence of $t$.
The distinct values are precisely the distinct vectors
$a\gamma+\alpha$. If $D$ is the order of $\gamma$ modulo $\Z^r$,
these vectors occupy $\min(N,D)$ disjoint lattice cosets, each
containing a box of $L^r$ points. Consequently
$\min(N,D)L^r\le A(NT)^r\le AL^r$; taking $N>A$ bounds $D$ by a
constant $D_0$.
Use only $a=0,D,2D,\ldots,qD$, where
$q=\lfloor(N-1)/D\rfloor$. For a coordinate $j$, the projections of
the corresponding boxes are integer intervals of length $L$, whose
successive displacement is $D\gamma_j\in\Z$. Their union has
\[
L+q\min(L,D|\gamma_j|)
\]
points. Over each such point, one of the boxes supplies $L^{r-1}$
points in the remaining coordinates. Hence
$q\min(L,D|\gamma_j|)\le(A-1)L$. Choose $N_0$ so large, in terms
of $A,D_0$, that $q>A+1$ and $qD\ge N/2$. The minimum cannot equal
$L$, and therefore
\[
|\gamma_j|\le\frac{(A-1)L}{qD}\le C_0T.
\]
Increasing $C_0$ gives the asserted norm bound. If $A<1$, the
hypothesis is empty, so this case causes no exception.
\end{proof}
\subsection{Moving the cheap weights to zero}
\begin{proposition}\label{c:cheap-collapse}
There are fixed sufficiently large positive constants $B_0$ and
$\kappa$ such that every sufficiently late member of the sequence
admits a monomial valuation $w_c$, Gaussian on $y$, satisfying
\[
w_c(y)=0,\qquad A_\theta(w_c)\le\kappa,\qquad
w_c(F)+B_0m\ge\tfrac12 b(F)
\quad(0\ne F\in V_m).
\]
The choices are made in the order $B_0$, then $\kappa$, before
passing to a sufficiently late member.
\end{proposition}
\begin{proof}
If $r=0$, take $w_c=v$ and $B_0\ge1$; the assertion is immediate.
Assume henceforth that $r>0$. Use \eqref{eq:E3} with cap $q_*=1$, barrier $q_{\rm bar}=1/8$,
and budget scalar $B_*=B_0$. Write
$L(t)=\min\mathcal L$ under the Gaussian constraint $w(y)=t$.
At $t=v(y)$ the competitor $v$ has $A_\theta(v)=0$ and $q(v)=1$
when $B_0\ge1$. Since $A_\theta\ge0$ and $q\le1$, this gives
$L(v(y))=-\kappa$. We shall move $t$ along the segment from $v(y)$
to zero. Work in a fixed bounded open parameter region containing
all these segments, and consider the part where $L(t)<-\kappa/2$.
Every optimizer there has $q>1/2$. Thus it lies strictly above
the barrier, and the openness, continuity, and local polyhedral
properties of the constrained optimization apply.
Take a full-dimensional cell, a point $t$ with rationally independent
coordinates, and the tiebroken optimizer $w$ there. Fix its local
affine continuation by optimizers with the same active mask.
Apply Corollary~\ref{c:strict-budget} and
Proposition~\ref{c:trace-budget} on the
geometric generic fibre over the $w$-initials $g=\bar y$,
without further Taylor
shifts. Select a homogeneous summand with surviving full-initial-field
trace and lift its factors to the actual field in a reduction.
The continuation is fixed before choosing the tiny exponents and
rational approximations to $\kappa/b_i$. Choose them so that, after
division by $P_0$, their total errors in degree, value, and Lipschitz
variation on that continuation are bounded by a fixed small constant.
For the resulting nonzero traced output $F'\in V_{m'}$, put
\[
\tau=\sum_i\tau_i,\qquad
\rho=\sum_i\tau_i\frac{m_i}{b_i}\le C.
\]
Here the $\tau_i$ are the active multiplicities in \eqref{eq:E10};
$\tau\le1$, and $\tau<1$ is possible only when the cap is active.
The degree bound and the equality case of \eqref{eq:E8} give
\begin{equation}\label{eq:F1}
\begin{split}
m'&\le C+\kappa\rho,\\
w(F')&=(1-P_0^{-1})
\bigl(\kappa\tau q-\kappa B_0\rho-A_\theta(w)\bigr)+O(1).
\end{split}
\end{equation}
Indeed, every active input satisfies
$w(F_i)=qb_i-B_0m_i$. The Taylor flag factors from \eqref{eq:E9}
have exponents between zero and $P_0-1$ in the expanded summands.
Their normalized degrees and values, and their slopes because
$w_{\widetilde t}(y)=\widetilde t$, are bounded. All formulas used
here are among the finite chart and filtration data retained before
reduction. The constants in the following counting argument are
independent of the large fixed $B_0,\kappa$.
Set $T_0=C+1+\kappa\rho$, increasing the constant in
\eqref{eq:F1} if necessary. For a sufficiently large fixed integer
$N$, consider
\[
(F')^a y^\alpha,\qquad 0\le a<N,\quad
0\le\alpha_j<NT_0,\quad\alpha_j\in\Z.
\]
These products lie in a common nested degree $O(NT_0)$ and have
values at most $C'(\kappa+T_0)N$. The same upper bound, after a
fixed enlargement, holds near $t$ on the chosen continuation.
Their number of distinct values is at most $C''(NT_0)^r$.
To justify this last assertion uniformly, before selecting the
output transfer the filtration comparisons for every degree up to
$C'''N(1+\kappa)$. At each such degree $m$, the original
high-cost part $b>S_0$ has
\[
w>S_0/2-B_0m,
\]
whereas \eqref{eq:D3} bounds the dimension of its quotient by
$C_4(1+m)^r$. For fixed $N,B_0,\kappa$, the right side of the
value inequality exceeds all the required upper cutoffs on late
members. Retain this implication with slack by the finite-filtration
spreading argument, including the zero vector in each high-cost
subspace. Its strict margin persists locally on the continuation.
The quotient by values above the tested cutoff therefore has the
same dimension bound in the reduction. Distinct values below the
cutoff give independent classes in this quotient, proving the count.
Lemma~\ref{c:cheap-box-count} now yields
\[
w(F')=\gamma\cdot t,\qquad
\gamma\in\Q^r,\qquad
\operatorname{ord}_{\Q^r/\Z^r}(\gamma)\le D_0,
\qquad |\gamma|\le C'(1+\kappa\rho).
\]
The same linear expression holds on a neighborhood of $t$ on the
affine continuation. Indeed, after fixing $F'$, the same count
applies at nearby rationally independent parameters, with the same
bounds. There are only finitely many rational vectors of bounded
norm and denominator. At $t$ their evaluations are distinct.
Continuity of monomial evaluation therefore excludes all but the
central vector on a sufficiently small neighborhood; density of
rationally independent parameters extends its formula to that whole
neighborhood.
Because $t$ is bounded, this gives
$w(F')\ge-C'(1+\kappa\rho)$. In \eqref{eq:F1} use
$A_\theta(w)\ge0$ and $q,\tau\le1$. Since $1-P_0^{-1}\ge1/2$,
we obtain
\[
\kappa(B_0-C')\rho\le C''(1+\kappa).
\]
Thus $\rho\le C'''/B_0$ once $B_0$ and then $\kappa$ are
sufficiently large.
We next differentiate the trace inequality, using equality at the
central point rather than assuming equality throughout the
continuation. Denote its lifted input by $G$ and its valuations by
$w_{\widetilde t}$. The function
\[
D(\widetilde t)=w_{\widetilde t}(F')
-P_0^{-1}w_{\widetilde t}(G)
+(1-P_0^{-1})A_\theta(w_{\widetilde t})
\]
is nonnegative near $t$, by \eqref{eq:E8}, and $D(t)=0$ by the
surviving initial trace. Active entries retain
$w_{\widetilde t}(F_i)=b_iq(w_{\widetilde t})-B_0m_i$.
Consequently the variation of their normalized contribution is
$(1-P_0^{-1})\kappa\,dq$ up to the chosen small Lipschitz error.
If $\tau<1$, the cap is active and $dq=0$, so the same statement
holds. The flag contributions and the remaining errors have a
uniformly bounded Lipschitz constant.
The optimizer value $L$ is affine on this cell and
$w_{\widetilde t}(F')=\gamma\cdot\widetilde t$. Subtracting
$D(t)=0$ from the inequality at $t+h$ therefore gives
\[
\gamma\cdot h+(1-P_0^{-1})\nabla L\cdot h\ge-C|h|.
\]
Using $-h$ gives the reverse bound. Together with the bounds on
$\gamma$ and $\rho$, these inequalities imply
\begin{equation}\label{eq:F2}
|\nabla L|\le C_5(1+\kappa/B_0).
\end{equation}
All degree and cutoff bounds used in this deduction are uniform.
The individual reductions, Frobenius powers, and approximations may
depend on the optimizer; no uniform choice of these is required.
Let $D_y$ bound the lengths of the segments from $v(y)$ to zero.
Choose first $B_0$ so that $D_yC_5/B_0<1/4$, and then $\kappa$
so that $D_yC_5<\kappa/4$, with strict room in both inequalities.
Integrating \eqref{eq:F2} keeps $L$ strictly below $-\kappa/2$
throughout any portion of the segment reached so far. The bound
on generic full-dimensional cells also applies along nongeneric
segments by continuity and local polyhedrality. At a first limiting
endpoint, compactness of the relevant sublevels supplies a feasible
optimizer with the same strict objective bound. Openness then extends the
continuation, so the whole segment is reached. At the endpoint $q\ge1/2$ and
$A_\theta(w_c)=L(0)+\kappa q\le\kappa$, which proves the
proposition.
\end{proof}
\subsection{Exact vanishing from a cost gap}
The following elementary limiting argument will turn a bounded
initial-value output into an output of exactly zero value at the
collapsed valuation. The conclusion is stronger than convergence
of those values to zero.
In the two-sided estimate below, a prescribed power of the tested
function will occur in every trace input. Once the trace output has
value zero, the lower trace inequality will bound that prescribed
factor's contribution from above.
\begin{lemma}\label{c:cheap-exact-zero}
For each sequence index $i$, let $V_{m,i}$ be nested multiplicative
$k_i$-linear subspaces of a field, containing $1\in V_{0,i}$,
and let $v_i,w_i$ be real
valuations trivial on $k_i$. Suppose there are $r$ probes $y_{j,i}$
of uniformly bounded degree, Gaussian for both valuations, such that
$|v_i(y_{j,i})|$ is uniformly bounded and $w_i(y_{j,i})=0$.
Suppose also that there are $M_i,S_i,R_i\to\infty$ and a fixed
$C_4$ with the following properties in every tested degree
$m\le M_i$. Costs $b_i(F)=m+v_i(F)$ are nonnegative,
\[
\begin{gathered}
\{b_i:C_4m<b_i\le S_i\}=\varnothing,\\
\dim_{k_i}\bigl(V_{m,i}/\{F:b_i(F)>S_i\}\bigr)
\le C_4(1+m)^r,\\
b_i(F)>S_i\ \Longrightarrow\ w_i(F)>R_i.
\end{gathered}
\]
The high-cost subspaces include zero, whose value is $+\infty$.
Then any sequence of nonzero $F_i$ of bounded degree and bounded
$v_i(F_i)$ satisfies $w_i(F_i)=0$ eventually.
\end{lemma}
\begin{proof}
Suppose that the asserted vanishing fails. Restrict to a subsequence
on which $w_i(F_i)\ne0$, and choose a nonprincipal ultrafilter
$\mathcal U$ on it. Let $k_{\mathcal U}=\prod_{\mathcal U}k_i$.
The nested union
\[
A=\bigcup_m\prod_{\mathcal U}V_{m,i}
\]
is a $k_{\mathcal U}$-algebra, because the spaces are nested and
multiply with addition of degrees. Membership in this union means
that the degree is bounded independently of $i$; it does not permit
an unbounded sequence of degrees. Define
\[
I=\{a=(a_i)\in A:
v_i(a_i)\longrightarrow_{\mathcal U}+\infty\}.
\]
In a fixed degree $m$, nonnegative cost gives $v_i(a_i)\ge-m$.
This uniform lower bound, the triangle inequality, and additivity
on products show that $I$ is an ideal. It is proper because it
does not contain $1$. It is prime: if neither $a$ nor $b$ belongs
to $I$, each has a finite upper valuation bound on a
$\mathcal U$-large set; their product has such a bound on the
intersection of those sets. Thus $A/I$ is a domain.
Every $a\in I$ also satisfies
$w_i(a_i)\to_{\mathcal U}+\infty$. This implication uses the
gap, not just slow growth of the cutoffs. Indeed, represent $a$ in
one fixed degree $m$. Divergence gives $b_i(a_i)>C_4m$ on a
$\mathcal U$-large set. The gap excludes all costs between this
bound and $S_i$, hence $b_i(a_i)>S_i$ there. The last hypothesis
then gives $w_i(a_i)>R_i\to\infty$.
The image in $A/I$ of degree $m$ has dimension at most
$C_4(1+m)^r$ over $k_{\mathcal U}$. To see this, take more than
this many elements and representatives in $V_{m,i}$. The finite
quotient-dimension bound gives, at each sufficiently large $i$, a
nontrivial linear combination in the high-cost subspace. There
are finitely many coefficients; on a $\mathcal U$-large set one
fixed coefficient can be normalized to $1$. The resulting
coefficient sequences define constants in $k_{\mathcal U}$, and
the combination has $v_i\to_{\mathcal U}\infty$. It therefore
gives a nontrivial dependence modulo $I$.
Let $\widehat y_j$ denote the classes of the probes in $A/I$.
They are algebraically independent over $k_{\mathcal U}$.
For a nonzero polynomial with coefficients in $k_{\mathcal U}$,
choose representatives of its finitely many nonzero coefficients.
On a $\mathcal U$-large set they remain nonzero. Gaussianity at
$v_i$ evaluates that polynomial by the minimum of finitely many
fixed monomial weights in the $v_i(y_{j,i})$. Those weights are
uniformly bounded, so the polynomial does not belong to $I$.
It follows that $\Frac(A/I)$ is algebraic over
$k_{\mathcal U}(\widehat y_1,\ldots,\widehat y_r)$.
Otherwise some element of $A/I$ would be transcendental over that
field. This element and the $r$ probes lie in some fixed nested
degree. Their monomials with every exponent at most $N$ would be
linearly independent, giving $(N+1)^{r+1}$ independent elements in
a degree $O(N)$, contrary to the established $O(N^r)$ bound.
This also proves the assertion when $r=0$.
The bounded degree and bounded $v_i$-value of $F_i$ show that its
class $\widehat F$ in $A/I$ is nonzero. Its minimal polynomial
over the cheap rational field therefore has nonzero leading and
constant coefficients. Clearing denominators yields
\[
\sum_{a=0}^{d_0}h_a(\widehat y)\widehat F^{\,a}=0,
\qquad h_a\in k_{\mathcal U}[Y_1,\ldots,Y_r],
\qquad h_0h_{d_0}\ne0.
\]
Choose representatives of all its finitely many scalar coefficients
and lift this relation to $A$. Its residual belongs to $I$, so its
$w_i$-value tends to $+\infty$ along $\mathcal U$.
For every nonzero $h_a$, Gaussianity on $y_i$ over the whole
constant field $k_i$, together with $w_i(y_i)=0$, gives
$w_i(h_{a,i}(y_i))=0$ on a $\mathcal U$-large set. Intersect
these finitely many sets. If $w_i(F_i)>0$, the constant term is
the unique least term of the lifted relation, of value zero. If
$w_i(F_i)<0$, its highest-degree term is uniquely least, of
negative value. In either case the residual has value at most
zero, a contradiction. No positive lower bound on
$|w_i(F_i)|$ is used.
The relation was obtained after forming the ultraproduct, but this
does not reverse the order of the choices. Its supports, its
coefficients after denominator clearing, and all its products have
some fixed finite nested degree $m_0$. The hypotheses were retained
for every $m\le M_i$, with $M_i\to\infty$, so they already apply
to $m_0$ on a tail. The proof does not require spreading this newly
chosen relation. The contradiction on any contrary subsequence
proves ordinary eventual vanishing, as asserted.
\end{proof}
\subsection{Two-sided angular budgets}
\begin{proposition}\label{c:cheap-two-sided}
Fix the constant $B_0$ of Proposition~\ref{c:cheap-collapse} and a
positive constant $c$. For each sufficiently small fixed
$\delta>0$, consider any actual monomial valuation $w$, Gaussian
on $y$, such that $w(y)=0$, $A_\theta(w)\le1$, and
\begin{equation}\label{eq:F3}
w(F)+\delta B_0m\ge c\delta b(F)
\qquad(0\ne F\in V_m).
\end{equation}
For all sufficiently late sequence members, uniformly over all
these valuations and angular functions,
\begin{equation}\label{eq:F4}
|w(F)|\le C_6\delta b(F).
\end{equation}
The constant $C_6$ is independent of the small fixed $\delta$;
the index after which the assertion holds may depend on $\delta$.
\end{proposition}
\begin{proof}
The lower bound follows from \eqref{eq:F3} and $m\le Cb(F)$.
Degree-zero units have zero value at $w$ by centering, so they
also satisfy the assertion. To prove the upper bound uniformly,
consider an arbitrary sequence of tests $w,F$ with $b=b(F)>0$.
We may pass to subsequences throughout. A contradiction for every
sequence violating one uniform upper bound gives the stated
uniform conclusion.
Return to the leading construction at $v$, and let $\bar F$ be
the $v$-initial of $F$, of charge $\beta\in M_0$. The occurring
weight comparisons give $\|\beta\|\le Cb$, and positive costs
are bounded away from zero. By \eqref{eq:C5}, choose an actual
angular $G$ whose initial has charge $-d'\beta$, for a positive
integer $d'$, and whose degree $m_G$ and cost are at most $Cd'b$.
Thus
\[
u=\bar F^{\,d'}\bar G\in K_1,\qquad
\operatorname{div}_{S_1}(u)+MH_1\ge0,\qquad
M=d'm+m_G\le Cd'b,
\]
by purity and \eqref{eq:A9}. Here $M>0$: if $M=0$, both degrees
vanish and the opposite proportional charges have opposite
proportional costs, one strictly positive, contradicting
nonnegative degree-zero costs.
Work on a geometric generic fibre field of $K_v$ over the cheap
initial probes $\bar y$. For a prime-divisor valuation $Q$ there,
over the geometric generic constants, restrict first to $K_v$ and
then put $P=Q|_{K_1}$. The algebraic-extension and relative
Abhyankar argument of \eqref{eq:E4} shows that $P$ is trivial or
prime-divisorial up to scaling. In particular, restriction
preserves Abhyankar equality, and a nontrivial subgroup of the
discrete value group remains discrete. Moreover, $P$ is trivial
on $K_f$: the cheap residual probes contain a transcendence basis
of $K_f$, and the remaining extension is algebraic. We have
\begin{equation}\label{eq:F5}
\frac{P(u)}{d'}
\le Cb\bigl(H_1(P)+s_{\exp}A_1^+(P)\bigr),
\qquad
A_{\theta_{v,\mathrm{rel}}}(Q)\ge A_1^+(P).
\end{equation}
The first inequality is \eqref{eq:C4} applied to the effective
pure divisor just constructed. For the second, first apply the
relative-form comparison in \eqref{eq:E4}, then invariantize the
restriction to $K_v=K_1({\bf z}_0)$ by \eqref{eq:E1}. The form is
\[
\theta_v=\pm\frac{\omega_S\wedge d\log{\bf z}_0}
{f_h^{1/n}},
\]
so the Gauss extension of $P$ has discrepancy exactly $A_1^+(P)$.
Equality in the second inequality of \eqref{eq:F5} therefore
requires the restriction to $K_v$ itself to be Gaussian over $P$
in these torus variables.
Fix a large $\ell$ with $1/\ell\le\delta$, before taking the
sequence limit. We now construct a strict trace budget in which
the factors $\bar F,\bar G$ are mandatory. Give them respective
rational exponents
\[
u_0\in[\ell/b,2\ell/b],\qquad u_0/d'.
\]
Choose a sufficiently divisible $D\in n\Z_{>0}$ for which
$DH_1$ is free Cartier. A basis of this complete pure angular
system has minimum order $-DH_1(P)$ at every $P$, by freeness
and \eqref{eq:A9}. Lift it by charge-zero angular functions and
give its minimum a rational exponent in
$[K_0\ell/D,2K_0\ell/D]$, where $K_0$ is a sufficiently large
fixed constant. The mandatory contribution equals
$u_0P(u)/d'$ and is at most
$C\ell(H_1(P)+s_{\exp}A_1^+(P))$. The pure system subtracts
at least $K_0\ell H_1(P)$.
For $P\ne0$, choose $K_0>C$ and then take a sufficiently late
member so that $C\ell s_{\exp}<1/2$. Since $H_1(P),A_1^+(P)$
are nonnegative and their sum is positive, these contributions
have sum strictly less than $A_1^+(P)$, and hence less than
$A_{\theta_{v,\mathrm{rel}}}(Q)$.
For $P=0$, both contributions vanish. Failure of strictness would
force equality in \eqref{eq:F5}, so the restriction to $K_v$ would
be a Gauss valuation over the trivial valuation of $K_1$.
That restriction is nontrivial, by the algebraic-extension argument
in \eqref{eq:E4}; hence some torus basis weight is nonzero.
Add finite minimum systems consisting of $1$ and angular initials
of both signs of the basis charges, allowing multiples as supplied
by \eqref{eq:C5}. One of their minima is negative in each such
direction. Give each an arbitrarily tiny positive rational
exponent. Finally add systems $(1,\bar U)$ for angular initials
generating the relative field, again with arbitrarily tiny positive
rational exponents, as in \eqref{eq:E5}. These extra minima are
nonpositive everywhere, make the budget strict also when $P=0$,
and retain birationality through their ratios.
Clear all rational denominators. The resulting finite system $I$
and integer $k_I$ satisfy \eqref{eq:E6}. Every member of $I$
contains the mandatory factors, while products of the finite
minimum systems realize exactly the specified remaining minimum
orders. Because all terms have the same mandatory factor,
retaining the field-generating ratios is unaffected by that factor.
Choose the tiny exponents to make their total normalized degree
and absolute initial-value contributions bounded by a fixed
constant. These choices may depend on the individual test.
Before invoking trace existence, also choose the following finite
data in its reductions. Along the sequence take slow cutoffs
\[
M_i\longrightarrow\infty,\qquad
M_i\ll S_i'\le S_{0,i},\qquad S_i'\longrightarrow\infty,
\qquad R_i'\longrightarrow\infty.
\]
For every degree $m\le M_i$, retain nonnegative costs, the exclusion of all costs in
$C_4m<b\le S_i'$, and the quotient-dimension bound
$C_4(1+m)^r$ from \eqref{eq:D3}. Retain also
\[
b>S_i'\quad\Longrightarrow\quad w>R_i'.
\]
This is possible by \eqref{eq:F3}: its lower bound is
$c\delta S_i'-\delta B_0M_i$ on these high parts, and we choose
$R_i'$ below this bound with slack. Here $\delta$ is fixed and
$S_i'$ dominates $M_i$. The finite-filtration comparisons from
the preceding section retain these statements for whole spaces
over the enlarged reduced constant field, using adapted bases and
high module tails. There are only finitely many degrees and
cutoffs at each index. Slowing the diagonal choices as necessary
allows all of them to be retained before choosing $P_0$ or a
traced output.
We also retain Gaussianity on $y$ at both valuations over the
entire new constant field, with bounded $v(y)$ and $w(y)=0$.
This does not follow merely from preserving finitely many polynomial
tests. On each of the two monomial charts, write the probe initials
as rational functions of the stratum coordinates and independent
normal variables. In characteristic zero these initials are
algebraically independent. Retain their finite expressions and a
nonzero differential minor witnessing this independence, together
with smooth chart data and separating residue coordinates.
A sufficiently general large-characteristic reduction preserves
that minor and those separating conditions. Over the perfect
extended constants the reduced initials are therefore still
algebraically independent. This proves the Gaussian rule for
every polynomial over those constants, including coefficients
chosen later and varying with $i$. The actual charts at $v,w$,
the form identities, the prescribed input functions, and their
needed values are spread simultaneously with these data.
Now apply \eqref{eq:E7} and \eqref{eq:E9} at $v$. A surviving
full-initial-field trace, followed by homogeneous expansion and
selection of a surviving term, lifts to an actual nonzero traced
output $F'$ in the reduction. All its input degrees and absolute
$v$-values, after division by $P_0$, are at most $C\ell$.
Indeed, $m,|v(F)|\le Cb$ and
$m_G,|v(G)|\le Cd'b$; multiplication by $u_0$ and $u_0/d'$
respectively cancels these costs. The pure system has degree $D$
and initial value zero, while its exponent times $D$ is at most
$2K_0\ell$. The tiny systems were chosen with bounded normalized
contributions, and the Taylor flags contribute bounded amounts.
The degree bound for actual trace and the equality case of
\eqref{eq:E8}, using $A_\theta(v)=0$, therefore give
\[
m'\le C\ell+n,
\qquad |v(F')|\le C\ell.
\]
These are bounds before knowing which surviving term is selected;
no bound on the Frobenius iterate is needed.
Apply Lemma~\ref{c:cheap-exact-zero} to these reduced systems,
with $S_i=S_i'$ and $R_i=R_i'$. Its hypotheses were retained
before trace existence, and $\ell$ is fixed, so the output degree
and initial value are uniformly bounded. It follows that
\[
w(F')=0
\]
eventually. In particular, this is an exact statement even if a
hypothetical sequence of nonzero output values were tending to zero.
The diagonal degree tests and full Gaussianity are precisely what
allow the lemma to handle the polynomial relation chosen later.
Finally apply only the lower inequality of \eqref{eq:E8} at $w$
to the product traced to this $F'$. The mandatory input power of
$F$ is exactly $F^{(P_0-1)u_0}$, by denominator clearing and the
chosen characteristic congruence. Every remaining factor has
value at least $-\delta B_0$ times its degree by \eqref{eq:F3}
and the retained comparisons; constants have value zero. Their
total degree divided by $P_0$ is at most $C\ell$. The chart
calculation also retains the value of $F$ and
$A_\theta(w)\le1$. Hence
\[
0=w(F')\ge(1-P_0^{-1})\bigl(u_0w(F)-1\bigr)
-C\delta B_0\ell.
\]
Since $1-P_0^{-1}\ge1/2$, $u_0\ge\ell/b$, and
$1/\ell\le\delta$, this yields
\[
w(F)\le b\bigl(\ell^{-1}+2C\delta B_0\bigr)
\le C_6\delta b.
\]
The constants are independent of small fixed $\delta$ and of the
chosen sequence of tests. This proves the upper bound and
completes \eqref{eq:F4}.
\end{proof}
==== END Bounded-klt-complements-for-Fano-contractions//cheap-collapse.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//compatible-filtrations.tex ====
\section{Leading forms and compatible further filtrations}
\label{a:filtrations}
The exact minimizer gives one finitely generated graded ring. To improve
that ring by a further minimization, we must preserve its canonical form,
the lc equations, and the values of specified functions on the original
variety. This section proves these compatibility statements. Its final
flattening lemma realizes two successive filtrations by a single real
valuation on the original function field.
\subsection{Full initial fields and leading forms}
All associated graded fields in this section include every homogeneous
degree. Thus, for a valued function field \((K,b)\), we write
\[
K_b=\Frac(\gr_b K).
\]
This is not the degree-zero residue field. Its transcendence degree is
\(\operatorname{ratrank}(b)+\trdeg_k\kappa(b)\): choose functions of
independent values and lifts of a residue transcendence basis; after
these choices any further homogeneous element is algebraic, by taking
monomial ratios. In particular \(\trdeg_k K_b=\trdeg_k K\) for an
Abhyankar valuation. In the rationally independent smooth monomial chart
of Lemma~\ref{a:abhyankar}, it is the rational normal-variable field over
the stratum field. Every homogeneous fraction is a stratum coefficient
times a Laurent monomial, and the normal variables are algebraically
independent because their weights are rationally independent. A Gaussian
extension by weighted variables adjoins their independent initial
variables to the previous full initial field.
These descriptions also justify the continuities used below. On a
positive monomial cone, the value of a fixed rational function is locally
a difference of minima on finite supports. Form discrepancy is linear
on the corresponding log smooth chart. Both are therefore continuous.
For normal test configurations with normal central fibre, preservation
of a canonical generator is the top-degree case of
\cite[Definition~3.3 and Proposition~3.5(2)]{XuZhuangForms}.
Here we identify the leading rational top form on the full initial field
and retain prescribed rational-function tests.
\begin{lemma}[Leading differential forms]\label{a:leading-form}
Let \(b\) be a real Abhyankar valuation of a characteristic-zero function
field \(K/k\), and choose \(s_1,\ldots,s_q\in K^*\) whose initials form
a transcendence basis of \(K_b/k\), where \(q=\trdeg_k K\).
If \(\eta=g\bigwedge_{i=1}^q d\log s_i\), then
\begin{equation}
A_\eta(b)=b(g),\qquad
\eta_b=\bar g\bigwedge_{i=1}^q d\log\bar s_i.
\label{eq:A6}
\end{equation}
The second formula is independent of the probes \(s_i\), is equivariant
for actions preserving \(b\), and applies to fractional forms by taking
tensor powers.
\end{lemma}
\begin{proof}
Use the rationally independent normal monomial chart, including the
divisors of the finitely many probes in the monomialization. A basis of
logarithmic normal differentials together with logarithmic differentials
of a separating unit tangential basis has wedge of form discrepancy
zero. Each probe is a unit times a Laurent normal monomial. In this
basis its logarithmic differential has coefficients of nonnegative
value. Their degree-zero parts are exactly the coefficients of its
initial logarithmic differential in the initial variables: a logarithmic
normal derivative of a unit reduces to zero, and a tangential derivative
differentiates its restriction. Lifted etale coordinate derivations give
these assertions in the local ring.
The determinant relating the probe wedge to this fixed wedge has
nonzero reduction, since the initial probes are algebraically
independent and the field has characteristic zero. Its value is
therefore zero. This proves the discrepancy formula. Comparing two
sets of probes by the same determinant computation proves the
independence of the leading form. The construction commutes with field
automorphisms preserving the valuation. Applying it to an integral
tensor power and then dividing discrepancies by that power gives the
fractional-form assertion.
\end{proof}
\subsection{Finite presentations and simultaneous specialization}
We next realize a finitely generated associated graded ring by a weight
family, retaining the rational functions that specify its canonical form
and boundary. Finite generating sets of initials and the corresponding
weight presentations are developed in
\cite[Theorem~2.17, Lemmas~3.2 and~3.4, and Section~3.3]{KavehManon}.
The proof below explains how log Izumi and maximal-ideal powers end the
subduction with an exact polynomial representative in our local setting.
\begin{lemma}[Finite adapted presentations]\label{a:adapted-presentation}
Let \(x\in U\) be an affine klt germ and let \(b\) be a quasi-monomial
valuation centred at \(x\). Suppose \(R_b=\gr_b k[U]\) is finitely
generated. One can choose affine algebra generators \(x_i\), all
vanishing at \(x\), whose initials generate \(R_b\). If
\(J\subset k[X_1,\ldots,X_N]\) is their relation ideal, then
\[
R_b=k[X_1,\ldots,X_N]/\operatorname{in}_{b(x_i)}J,
\]
where \(\operatorname{in}\) always means the \emph{minimum}-weight
initial ideal. Every regular function has a polynomial representative
attaining its valuation and prescribed initial; the analogous statement
holds for rational functions using fractions.
\end{lemma}
\begin{proof}
Add lifts of finitely many graded generators to any affine generating
set, subtracting constants from the latter. All generator weights are
positive. The minimum of a polynomial relation vanishes on their
initials, giving one inclusion. Conversely, a homogeneous relation
between these initials lifts to a function of strictly higher value.
Subtract a polynomial matching that new initial, and continue with
strictly higher-weight polynomials. Only finitely many monomial weights in these generators
occur below any fixed bound, so the residual value can be made
arbitrarily large. Log Izumi then puts the residual in an arbitrarily
high power of the maximal ideal of the local ring. The same holds in
the affine ring: the power of the closed-point maximal ideal is primary.
That ideal is generated by the \(x_i\), so a polynomial expression in
its sufficiently high power has all monomials of strictly higher weight
than the starting relation. This ends subduction with an actual
polynomial relation of the desired initial. The identical argument,
starting from a function instead of a relation, gives adapted
representatives. Local unit denominators have constant initial, so the
affine and local graded rings agree; taking fractions gives the final
assertion.
\end{proof}
\begin{lemma}[Canonical specialization]\label{a:canonical-specialization}
In Lemma~\ref{a:adapted-presentation}, suppose further that \(R_b\) is
normal and that \(U\) has a canonical generator \(\Omega_U\).
Then \((\Omega_U)_b\) generates the canonical divisor of
\(\Spec R_b\): its divisor is zero. There is a one-parameter weight
family realizing this specialization and preserving initial
identifications on any prescribed finite set of rational functions.
\end{lemma}
\begin{proof}
Choose finite relations whose minima generate the initial ideal in
Lemma~\ref{a:adapted-presentation}. Add adapted representatives of the
finite tests, including full probes and the coefficient \(g\) in
\eqref{eq:A6}. Their ties and strict weight comparisons form a finite
system of rational linear equalities and strict inequalities. Approximate
the generator weights, up to common scaling, by positive integers
\(m_i\) preserving that system. Consider the domain
\[
\mathcal R=k[s,s^{-m_i}x_i]
\subset k[U][s,s^{-1}].
\]
It is flat over \(k[s]\), since it is torsion-free. Its fibre at zero
has the minimum-weight initial presentation for these integer weights.
Indeed substitution and homogenization give one inclusion; conversely,
separating the powers of \(s\) in a homogeneous expression lying in
\((s)\) expresses its upstairs value by terms of strictly higher weight.
The chosen initial relations show that this fibre's ideal contains the
prime presenting \(R_b\). The zero-coordinate section makes the fibre
nonempty. Every minimal prime of the principal ideal \((s)\) has height
one in the finite-type domain \(\mathcal R\), so the fibre has dimension
\(\dim U\). The old initial prime has that same dimension, since its
fraction field is \(K_b\). A strict enlargement would have smaller
dimension. Thus the initial ideals coincide and the special fibre is
exactly \(\Spec R_b\), with the required test initials.
All fibres are normal: the nonzero ones are copies of \(U\), and the
zero fibre is normal by hypothesis. Flatness over the smooth curve
therefore gives normality of the total space. The form
\(\Omega_U\wedge ds\) has zero divisor away from the special fibre.
That fibre is irreducible, reduced and Cartier, so multiplication by a
power of \(s\) gives a rational form with zero divisor on the total
space, hence a canonical generator on its smooth locus.
We identify its relative restriction explicitly. At the generic point
of the special fibre, write the tested probes as
\(s_i=s^{e_i}S_i\) and \(g=s^{e_g}G\), with residues
\(\bar s_i\) and \(\bar g\). Because a factor involving \(ds\)
disappears after wedging with \(ds\),
\[
\Omega_U\wedge ds
=s^{e_g}G\bigwedge_i d\log S_i\wedge ds.
\]
The initial logarithmic probe wedge is nonzero. The normalizing power
is consequently exactly \(-e_g\), and the relative restricted form is
\(\bar g\bigwedge_i d\log\bar s_i=(\Omega_U)_b\), up to sign.
At every codimension-one point of the normal special fibre that fibre
is smooth; flatness and geometric smoothness there make the family
smooth over the base. Its relative generator restricts to a generator
there. Thus \((\Omega_U)_b\) has zero divisor on \(\Spec R_b\), as
claimed. This argument uses only codimension-one restriction, and makes
no canonical-sheaf base-change assumption at singular points.
\end{proof}
\begin{lemma}[Simultaneous boundary specialization]
\label{a:boundary-specialization}
In the preceding situation, a rational effective boundary expressed as
a sum of divisors of regular equations specializes to the same sum of
the divisors of their initial equations. This identification agrees with
specializing its prime components, with all multiplicities.
\end{lemma}
\begin{proof}
For a principal ideal, valuation multiplicativity makes its associated
graded ideal principal, generated by the equation's initial. We verify
that component ideals can be preserved at the same time. For each of
the finitely many component ideals, use its polynomial preimage in the
adapted presentation. Its minimum initial ideal maps onto the associated
graded ideal: the adapted representatives in
Lemma~\ref{a:adapted-presentation} prove this also for each element of
the ideal.
Homogenize all these polynomial ideals with one extra variable of
weight zero. Each homogeneous minimum ideal has finitely many generators
that are minima of homogeneous elements of the original homogenized
ideal. Preserve these generators, and the ring's initial ideal and the
equation tests, in one positive integral weight approximation. Inclusion
of the old minima in the new minima is then automatic. There can be no
extra elements, since both homogeneous minimum ideals have the original
Hilbert function, obtained by filtering each finite-dimensional
total-degree piece. Dehomogenization preserves the equality: distinct
terms of one homogeneous polynomial cannot coalesce after setting the
extra variable to one.
The component closures in the resulting weight family have these
specialized ideals, as follows by using the analogous family algebras of
the quotient rings. Rescale each boundary equation by its parameter
power. Its divisor has no vertical component and restricts to the
divisor of the nonzero initial equation. Intersection of codimension-one
cycles with the special-fibre parameter is additive. This can also be
checked at the regular codimension-one points of the normal special
fibre, where the total family is smooth over the base. Decomposing the
horizontal equation divisors into their prime components therefore gives
precisely the same restricted boundary cycles and multiplicities.
\end{proof}
\subsection{Torus fields and flattening on the original field}
We have now identified the specialized canonical form and boundary.
The remaining task is to combine two graded constructions while retaining
that information. Torus invariance separates the first grading from the
second and makes the required combination possible.
The value torus has character lattice generated by the occurring
valuation degrees. Its grading valuation is the real cocharacter that
evaluates these degrees, injectively on this lattice. When other
invariant torus actions descend, we also use their joint effective torus.
For a faithful torus grading on a field generated by homogeneous
functions, choose multiplicative rational variables \(z^\mu\) for a
basis of its character lattice \(M\). Such variables exist because the
occurring characters generate \(M\) as a group. If \(K^T\) is the
invariant field, then
\[
K=K^T(z^{\mu_1},\ldots,z^{\mu_r}).
\]
Distinct characters give algebraic independence, and division of every
homogeneous generator by its corresponding torus variable produces an
invariant generator. In particular invariants are generated by ratios
of equal-weight homogeneous elements.
A \(T\)-invariant valuation is Gaussian in these variables. To see the
minimum rule, span the finitely many character terms of a sum by finitely
many torus translates and invert their character matrix. Invariance
shows that each term has value at least that of the sum; the valuation
inequality gives the reverse comparison with the minimum. Formula
\eqref{eq:A6} thus applies using the torus variables and full independent
initials of the restricted valuation. In particular a pure cocharacter
valuation evaluates a homogeneous volume form's discrepancy by its
character.
The next construction refines the successive-degeneration argument of
\cite[proof of Theorem~4.16 and Remark~4.17]{LiXuStable}.
We retain a prescribed finite set of rational-function initials and the
leading top form, and record uniform comparison with the first valuation.
\begin{lemma}[Compatible real flattening]\label{a:flattening}
Let \(b\) be as in Lemma~\ref{a:adapted-presentation}, and let \(T\) be
the value torus of \(R=\gr_b k[U]\), so that \(b\) is injective on its
character lattice. Let \(u\) be a \(T\)-invariant quasi-monomial valuation
centred at the vertex of \(\Spec R\), with \(R'=\gr_u R\) finitely
generated. Any stipulated old grading on \(U\) is assumed to preserve
\(b\) and to act on \(R\) through \(T\).
For arbitrarily small positive real \(\xi\), there is an Abhyankar
valuation \(b_\xi\) on the \emph{actual} field \(k(U)\) with associated
graded ring \(R'\), graded by first weights plus \(\xi\) times second
weights. Its value torus is the joint torus generated by \(T\) and the
value torus of \(u\) on \(R'\). It preserves the stipulated grading.
For any finite set of rational functions, one can preserve both their
double initials and the formula
\[
b_\xi(F)=b(F)+\xi u(\operatorname{in}_b F).
\]
One can also arrange, on every nonzero regular function,
\[
\tfrac12 b(F)\leq b_\xi(F)\leq2b(F).
\]
If \(U,R\) have the canonical generators in
Lemma~\ref{a:canonical-specialization} and both \(b,u\) have their
respective form discrepancies equal to one, the finite tests can be
chosen so that
\[
A_{\Omega_U}(b_\xi)=1+\xi,\qquad
(\Omega_U)_{b_\xi}=((\Omega_U)_b)_u.
\]
\end{lemma}
\begin{proof}
First use the lexicographically ordered valuation
\[
V(F)=(b(F),u(\operatorname{in}_b F)).
\]
On each first-weight slice its second graded calculation is exactly the
one for \(u\). Invariance makes the \(u\)-filtration split over first
degrees. Hence its associated graded on \(k[U]\) is precisely \(R'\),
with the joint grading. Choose lifted joint homogeneous generators,
including the additional generators needed for \(R\) and \(k[U]\).
Take them homogeneous for the stipulated old grading and vanishing at
the point. In the lexicographic subduction procedure there are only
finitely many possible values with bounded first coordinate: the first
graded pieces are finite-dimensional and only finitely many occur up to
that bound. Subduction therefore works as in
Lemma~\ref{a:adapted-presentation}, using log Izumi for the first
coordinate. In particular the lexicographic minimum ideal is the prime
presenting \(R'\), and adapted polynomial and fractional representatives
exist.
Choose \(\xi>0\) small enough to preserve the minima of finitely many
relations generating that initial prime, and avoid all rational ties on
the joint character lattice. The flattened initial ideal contains this
prime. Suppose it contained an extra initial relation. Approximate the
positive flattened generator weights by positive integral weights,
preserving that extra minimum together with the previous finite minima.
The weight-family argument of Lemma~\ref{a:canonical-specialization}
then gives a nonempty special fibre of dimension \(\dim U\), whose
ideal strictly contains the prime of \(R'\), also of dimension
\(\dim U\). This is impossible. Thus the flattened minimum ideal is
exactly the desired prime.
For completeness, the resulting real quotient filtration does define a
valuation on the original ring. Its weight on a function is the maximum
of the minimum weights of all polynomial representatives. Positivity
of the finitely many generator weights makes the weights locally finite.
If the minima of representatives could be arbitrarily high, every
monomial in those representatives would have arbitrarily large ordinary
total degree, putting the function in every maximal-ideal power, hence
making it zero. The maximum is thus
attained for every nonzero function. At an attained maximum, polynomial
minima give the graded quotient by the minimum relation ideal: an
improvement of a representative differs from it by a relation with that
minimum. The quotient being a domain proves multiplicativity, and the
triangle inequality is immediate. Extending by fractions yields
\(b_\xi\). Its full graded field has transcendence degree \(\dim U\),
so it is Abhyankar. Avoidance of rational ties identifies its value
torus with the full joint torus. The homogeneous presentation preserves
the old grading.
To preserve any finite test list, choose its lex-adapted polynomial
representatives and denominators and retain their leading comparisons.
For uniform comparison on \emph{all} regular functions, take \(\xi\)
smaller until every generator's flattened weight lies between half and
twice its first weight. A \(b\)-adapted representative gives
\(b_\xi(F)\geq b(F)/2\). Conversely, a \(b_\xi\)-adapted
representative gives \(b(F)\geq b_\xi(F)/2\). This uses the attained
maximum description separately for the two valuations, and so includes
functions whose initially chosen representatives have cancellation.
Finally choose first-stage probes as follows. Use rational torus
variables and functions of the invariant field whose restricted
\(u\)-initials form a full transcendence basis there. Gaussianity and
Abhyankar equality show that these have full independent first and
double initials: the second-stage torus initials adjoin independent
variables to the full restricted initial field. The invariant probes
are ratios of equal-first-weight homogeneous functions, so they lift
to \(k(U)\), as do the torus variables. Include these probes and the
coefficient of \(\Omega_U\) in their logarithmic basis among the finite
tests. Applying \eqref{eq:A6} first to \(b\), then to \(u\), and then
to the flattening gives coefficient values \(1\), \(1\), and
\(1+\xi\), respectively. The same formula identifies the flattened
leading form with the iterated leading form. This proves the last two
assertions.
\end{proof}
\subsection{A cone proof of the comparison for Cartier-equation boundaries}
\label{a:comparison-cone-proof}
The preceding constructions give a second proof of
Lemma~\ref{a:minimizer-comparison} in the setting actually used later:
\(U\) has a canonical generator, and the klt boundary is a nonnegative
rational combination of regular equations. This proof explains why tests
with nonclosed centres on the stable cone cause no loss of uniformity.
It uses the torus-field description above and the algebraic degeneration
and canonical-form assertions of
Lemmas~\ref{a:canonical-specialization} and
\ref{a:boundary-specialization}.
Let \(F\) be regular affine with \(b(F)>0\); local denominators that are
units have value zero. Degenerate to the stable cone, retaining the
initial equations of the boundary and of \(F\).
The grading \(b\) is still minimizing with the same pair discrepancy,
and the canonical form is the leading form.
Normalize \(A(b)=1\). For an invariant prime-divisor valuation \(E\)
centred on the cone, not necessarily at the vertex, normalize \(A(E)=1\).
On homogeneous elements use \(w_s=b+sE\), and extend by the minimum rule.
Gaussianity in rational torus variables shows this is a quasi-monomial
valuation: its invariant-field restriction is multiplied by \(s\), and
its torus weights are translated by those of \(b\).
It is centred at the vertex since \(w_s\geq b\) on the ring.
The leading-form rule and the homogeneous boundary equations give
\(A(w_s)=1+s\).
Put \(L_*=E(\bar F)/b(\bar F)>0\) and \(s=1/L_*\).
For \(k_m=\lfloor m/(2b(\bar F))\rfloor\), multiply the homogeneous
pieces whose \(b\)-weights lie in
\[
[\,3m/4-k_m b(\bar F),\ m-k_m b(\bar F)\,)
\]
by \(\bar F^{k_m}\).
They inject into the \(b\)-jets of weight below \(m\).
Their \(b\)-weight is at least \(3m/4\), and their added \(sE\)-weight
is at least \(s k_m E(\bar F)\), so they vanish among the \(w_s\)-jets
for large \(m\). The lost dimension fraction tends to
\(c_p=2^{-p}-4^{-p}>0\). Therefore
\[
\vol(w_s)\leq(1-c_p)\vol(b),\qquad
1\leq (1+1/L_*)^p(1-c_p)
\]
by minimality. This bounds \(L_*\) by a constant depending only on \(p\).
Equivariant log resolution shows that invariant prime divisors suffice
to test the homogeneous pair together with \(\bar F\). Undoing the
normalizations, adding \(a\Div(\bar F)\) preserves lc near the vertex
whenever
\[
a<\frac{A(b)}{C_p b(F)}
\]
for a dimensional \(C_p\), which may be enlarged from the first proof.
For a strict inequality the cone pair is klt.
Transfer this test through the weight family of
Lemma~\ref{a:canonical-specialization}. Its total space is normal with
Cartier canonical divisor; the special fibre is reduced, normal and
Cartier. All other boundaries are nonnegative rational Cartier-equation
divisors without a vertical component. At codimension one of the
special fibre the family is smooth, and the different is exactly their
Cartier restriction. The ordinary inversion-of-adjunction theorem
\cite[Theorem on p.~1]{Kawakita} applies with coefficient one on the
special fibre. It makes the total pair lc near the vertex.
Away from parameter zero the family and pair are a product. The section
of the original closed point, with all chosen affine generators zero,
specializes to the vertex. Hence the original pair with \(a\Div(F)\)
is lc near that point. Letting \(a\) increase to the displayed bound
gives \eqref{eq:A3}. This argument uses stable degeneration only for
the ordinary klt minimizer, and makes no finite-generation assertion
about an arbitrary lc place.
==== END Bounded-klt-complements-for-Fano-contractions//compatible-filtrations.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//constrained-minimum.tex ====
\section{Exact normalized-volume minimization with an lc constraint}
\label{a:minimum}
The principal point of this section is that a constrained minimum is
already an ordinary klt minimum for a sufficiently small positive
perturbation. Convergence to the constraint would not suffice for the
finite-generation argument.
\subsection{The ordinary klt inputs}
For a valuation \(u\) centred at a closed point \(x\) of a
\(p\)-dimensional germ, put
\[
\mathfrak a_m(u)=\{F:u(F)\geq m\},\qquad
\vol(u)=\lim_{m\to\infty}
\frac{p!\,\dim_k(\OO_{U,x}/\mathfrak a_m(u))}{m^p}.
\]
The limit exists for these valuation-ideal families
\cite[Theorem~1.1 and Corollary~5.6]{CutkoskyVolumes}; the local rings
here are excellent and normal, so their completions are reduced.
For a klt pair with discrepancy \(A\), the normalized volume is
\(A(u)^p\vol(u)\).
We use the following established results in their ordinary klt setting.
The minimum is attained at a unique ray, its valuation is quasi-monomial,
and its associated graded algebra is finitely generated and defines a
klt K-semistable log Fano cone. The induced grading again minimizes:
see \cite[Main Theorem and Section~7.3]{BlumExistence},
\cite[Theorem~1.2 and Remark~3.8]{XuQuasimonomial},
\cite[Theorem~1.1]{XuZhuangUniqueness}, and
\cite[Theorems~1.1--1.2]{XuZhuangStable}, with the cone
characterization in \cite[Theorem~3.5]{LiXuStable}.
For the bare klt germ, the estimates
\[
\widehat{\vol}_U(u)\geq c_U\frac{A_U(u)}{u(\m_x)},\qquad
u(F)\leq C_U A_U(u)\ord_{\m_x}(F)
\]
are the properness and log Izumi estimates of
\cite[Theorems~1.1--1.2]{LiProperness}.
Their constants may depend on this individual germ.
If the boundary is a rational nonnegative combination of divisors of
regular equations, its specialized boundary is the corresponding
combination of divisors of their initial equations.
The precise simultaneous specialization, including possible component
multiplicities, is proved in Lemma~\ref{a:boundary-specialization}.
Thus all the uses below are of effective rational klt pairs.
\begin{lemma}[A uniform comparison at a minimizer]\label{a:minimizer-comparison}
Let \(b\) minimize normalized volume for a \(p\)-dimensional klt pair.
For every centred quasi-monomial valuation \(u\) and every nonzero
\(F\in\OO_{U,x}\),
\begin{equation}
u(F)\leq C_p\frac{A(u)}{A(b)}\,b(F),
\label{eq:A3}
\end{equation}
where \(C_p\) depends only on \(p\). Multiplying the discrepancy function
by a common positive constant does not change this inequality.
\end{lemma}
\begin{proof}
The relative expected-vanishing criterion
\cite[Lemma~3.3, Definition~3.4 and Theorem~3.10]{XuZhuangUniqueness}
gives \(A(u)\geq S(b;u)\). On
\(Q_m=\OO_{U,x}/\mathfrak a_m(b)\), let \(W_m(b;u)\) be the sum of the
jumping weights of the induced \(u\)-filtration, with integer rounding
if desired. Then
\[
\frac{S(b;u)}{A(b)}
=\lim_{m\to\infty}\frac{W_m(b;u)}{W_m(b;b)}.
\]
The induced filtration is the image filtration: its dimension at level
\(q\) is
\[
\dim_k\frac{\mathfrak a_q(u)}
{\mathfrak a_q(u)\cap\mathfrak a_m(b)}.
\]
Filtering this image by the \(b\)-grading gives the same dimension, so
no compatibility of bases is being assumed.
Suppose \(b(F)>0\), and put
\(k_m=\lfloor m/(2b(F))\rfloor\).
Multiplication by \(F^{k_m}\) embeds
\(\OO_{U,x}/\mathfrak a_{m-k_m b(F)}(b)\) into \(Q_m\).
Every vector in this image has induced \(u\)-level at least \(k_m u(F)\).
The image dimension is asymptotic to \(2^{-p}\dim_k Q_m\), whereas
\(W_m(b;b)\leq m\dim_k Q_m\).
Thus
\[
\frac{S(b;u)}{A(b)}
\geq 2^{-p-1}\frac{u(F)}{b(F)}.
\]
The integer-rounding errors vanish in this limit.
This proves \eqref{eq:A3}, for example with \(C_p=2^{p+1}\).
A unit has value zero at both valuations.
\end{proof}
A second proof for canonical-generator germs with Cartier-equation
boundaries appears in Section~\ref{a:comparison-cone-proof}, after the
specialization and torus-field constructions it uses.
\subsection{The constrained problem}
\begin{proposition}[Exact penalization]\label{a:exact-penalization}
Let \(x\in U\) be an affine klt germ of dimension \(p\), and let
\(I_0,J\) be nonzero ideals. Suppose \(c>0\) is rational,
\((U,I_0^c)\) is lc, and an lc quasi-monomial place is centred at \(x\).
Let \(b_0\geq1\) and \(c'\geq0\) be rational, and assume
\[
a(u)=b_0A_U(u)-c'u(J),\qquad A_U(u)\leq a(u)\leq b_0A_U(u).
\]
The minimum of \(a(u)^p\vol(u)\) among centred quasi-monomial lc places
of \(I_0^c\) is attained at a unique ray. This same ray minimizes
normalized volume for a family of ordinary effective rational klt
pairs for every sufficiently small positive rational parameter.
In particular its associated graded is finitely generated.
It is invariant, after an invariant normalization, under any torus
preserving \(x,I_0,J\).
All smallness thresholds and constants in the proof may depend on the
fixed germ and ideals.
\end{proposition}
\begin{proof}
\emph{From ideals to ordinary klt minimizers.}
Choose a common log resolution of \(U,I_0,J,\m_x\).
Replace the order of each ideal by the average orders of sufficiently
many general elements in a prescribed finite-dimensional generating
space. Write the resulting functions as
\[
C=A_U-B_0,\qquad a_D=b_0A_U-B_J,
\]
where \(B_0\) and \(B_J\) are rational nonnegative combinations of the
divisors of the chosen regular equations, including the coefficients
\(c\) and \(c'\). The resolving support and these general moving
hypersurfaces form a simple normal crossing divisor \(\Phi\):
choose successively transverse members on its finitely many strata.
Dividing each coefficient among enough members makes every new
moving coefficient sufficiently small.
The old component orders are unchanged. The simple normal crossing
discrepancy tests therefore give
\[
C\geq0,\qquad a_D\geq A_U/2,\qquad a_D\leq b_0A_U.
\]
They also give \(C=0\) exactly on the original ideal-lc places and
\(a_D=a\) there. Indeed the discrepancy expression for \(C\) has
nonnegative residual term and component terms, strictly positive on
every added moving component, including those used for \(J\).
An ideal-lc place has \(C=0\) by comparison with the fixed ideal orders;
conversely, \(C=0\) forces all these moving orders to vanish and recovers
the original ideal equality. Every chosen equation then has exactly its
ideal's order.
For positive rational \(t'\), sufficiently small, the function
\[
\frac{C+t'a_D}{1+t'b_0}
\]
is the discrepancy of the effective rational klt pair with boundary
\[
\Delta_{t'}=\frac{B_0+t'B_J}{1+t'b_0}.
\]
Let \(u=u_{t'}\) minimize its normalized volume, with \(u(\m_x)=1\).
Comparison with one fixed centred lc place and \(a_D\geq A_U/2\)
bounds \(A_U(u)^p\vol(u)\) independently of \(t'\).
Properness bounds \(A_U(u)\); log Izumi consequently bounds
\(\vol(u)\) away from zero. The same comparison now gives
\(C(u)=O(t')\).
\smallskip
\noindent\emph{Retraction to the lc strata.}
Let \(\Sigma\) consist of the coefficient-one components for \(C\) on
the fixed resolution. Use the monomial retraction of
\cite[Lemma~4.7 and Corollary~4.8]{JM}: retract \(u\) to the generic
stratum of \(\Sigma\)
through its centre, retaining the same positive normal weights, and
call the retraction \(w\). The component formulas give
\[
\begin{aligned}
d_{\log}:=A_{(Y,\Sigma)}(u)&\leq C_1C(u),&
C(w)&=0,\\
a_D(w)&\leq a_D(u),&
w(\m_x)&\geq1-C_1C(u).
\end{aligned}
\]
To see the first bound, express \(C\) as \(A_{(Y,\Phi)}\) plus the
component values multiplied by their discrepancies. These coefficients
are zero exactly on \(\Sigma\), and positive on the omitted components.
For \(d_{\log}\), the omitted coefficients are instead one.
The formula for \(a_D\) has a nonnegative residual term and positive
component coefficients. The maximal ideal is a fixed component sum.
These observations give all four comparisons. In particular, for small
\(t'\), \(w\) is nontrivial and centred at \(x\).
Retraction has removed the lc defect and has not increased \(a_D\).
To compare the two objectives, we must also control its possible increase
in volume. We prove the following quantitative claim for the minimizers
\(u=u_{t'}\) and their retractions \(w\) constructed above:
\begin{equation}
u(F)\leq(1+C_2C(u))w(F)
\qquad(0\ne F\in\OO_{U,x}).
\label{eq:A4}
\end{equation}
The constant \(C_2\) is independent of \(F\) and of sufficiently small
\(t'\) on this fixed resolved germ. We first isolate a polynomial's
least normal terms by a differential operator of controlled order, then
bound the value lost under differentiation. A local truncation will give
the claim for every regular \(F\).
\smallskip
\noindent\emph{Polynomial isolation.}
Cover the relevant strata by finitely many etale-coordinate charts
\((x_1,\ldots,x_s,y_1,\ldots,y_{p-s})\), with \(x_i=0\) the components
of \(\Sigma\) through the centre and \(w_i=u(x_i)>0\).
For a polynomial \(G\) of degree at most \(M\) in fixed affine generators
of \(k[U]\) vanishing at \(x\), we claim
\begin{equation}
u(G)-w(G)\leq C_3(1+M)d_{\log}.
\label{eq:A5}
\end{equation}
We will obtain a differential expression with \(u\)-value exactly
\(w(G)\), using \(O(1+M)\) tangential and logarithmic normal
derivatives. The order bound must remain independent of the smallest
positive normal weight; otherwise it would not control the minimizers
as their weights approach a face.
Expand \(G\) normally in the \(x_i\) at the generic stratum.
Etaleness identifies the coefficient field as the finite separable
residue-field lift over \(k(y)\). Its normal Taylor coefficients are
restrictions of regular lifted \(x\)-derivatives divided by factorials;
they are regular on the stratum. Lifted \(y\)-derivatives differentiate
these coefficients without changing normal exponents. At a closed
point this agrees with the expansion in \(x\) and translated tangential
coordinates.
Let \(H\) be the sum of the least-weight normal terms of \(G\).
We first bound every active exponent by \(O(1+M)\), independently of
the smallest positive weight. The extension
\(k(U)/k(x_1,\ldots,x_s,y_1,\ldots,y_{p-s})\) is finite.
Fixed multiplication matrices for the affine generators give a relation
\[
\sum_j a_j(x,y)G^j=0
\]
of fixed degree in \(G\), with polynomial coefficient degrees
\(O(1+M)\). Indeed entries in a degree-\(M\) polynomial of these
matrices have denominators bounded by powers of fixed denominators;
the characteristic polynomial and denominator clearing preserve a
linear degree bound.
For a positive normal weight vector \(\lambda\), write
\(f(\lambda)=\min_{\alpha\in\operatorname{Supp}_x(G)}
\langle\alpha,\lambda\rangle\).
At least two different powers \(j,j'\) tie at the least value in the
relation. Choose normal exponents \(\beta,\beta'\) in the finite
supports of \(a_j,a_{j'}\) that attain their respective minimum weights.
Here the supports retain only coefficients nonzero in the stratum field.
The tie gives
\[
f(\lambda)=
\frac{\langle\beta-\beta',\lambda\rangle}{j'-j}.
\]
Thus \(f\) takes its values among finitely many linear forms whose
coefficients are \(O(1+M)\): the coefficient degrees bound
\(\beta,\beta'\), and \(|j'-j|\geq1\).
The Taylor minimum is locally finite on the positive orthant, since
the coordinates of \(\lambda\) stay bounded below in a small
neighbourhood and all normal exponents are nonnegative.
Let \(\mathcal A(\lambda)\) be the finite set of active Taylor
exponents. For every coordinate \(i\) and all sufficiently small
\(h>0\), local finiteness gives
\[
\frac{f(\lambda)-f(\lambda-he_i)}{h}
=\max_{\alpha\in\mathcal A(\lambda)}\alpha_i.
\]
On this segment, \(f\) agrees with the restriction of one of the
finitely many linear forms above, so the quotient is \(O(1+M)\).
It bounds every active coordinate, including tied exponents.
The permissible \(h\) may shrink as the weights approach a face,
but the bound does not: no division by a normal weight has occurred.
Take the least parts of the relation at the given weights and divide
out the lowest power of \(H\). The resulting polynomial relation in
\(H\) has a nonzero constant term, an \(x\)-polynomial with coefficients
polynomial in \(y\) of degree \(O(1+M)\). At a general closed point of
the centre, not every coefficient of \(H\) can vanish to order exceeding
that bound. Otherwise the relation would force every coefficient of
its nonzero constant term to vanish to that order, impossible for a
nonzero polynomial of the stated degree in etale tangential coordinates.
All other coefficients in the relation are regular there, so their
possible vanishing cannot invalidate this implication.
A tangential derivative of order \(O(1+M)\) therefore makes one active
coefficient a unit. Let its normal exponent be \(\alpha\). For each
normal coordinate apply the product of
\(x_i\partial_{x_i}-j\) over the integers from zero to the active
coordinate bound other than \(\alpha_i\).
This isolates the chosen active monomial, with total differential order
\(O(1+M)\). Both sorts of derivatives preserve normal exponents;
lower-weight coefficients remain zero. To interpret the resulting
congruence in the local ring, work at the chosen closed point of the
centre and put
\[
I_{>}=\bigl(x^\beta:\textstyle\sum_i\beta_iw_i>w(G)\bigr).
\]
This monomial ideal is finitely generated for the individual positive
weights. In the completion, the output is congruent to \(x^\alpha\)
times a coefficient whose residue on the stratum is a unit. Lift that
residue coefficient from \(\OO_Y/(x_1,\ldots,x_s)\) to an ordinary
local unit \(h\). Since \(x^\alpha(x_1,\ldots,x_s)\subset I_{>}\),
the output minus \(x^\alpha h\) lies in the completed ideal
\(I_{>}\), hence in \(I_{>}\) by faithful flatness. Localize this
congruence at the generic centre of \(u\).
Every ideal generator has \(u\)-value strictly larger than \(w(G)\).
The isolated output consequently has \emph{exactly} \(u\)-value \(w(G)\).
The positive gap here may depend on the valuation; its size is never
used in an estimate.
\smallskip
\noindent\emph{The differential loss.}
Let
\(\lambda_0=\bigwedge_i d\log x_i\wedge\bigwedge_jdy_j\).
For a prime divisor \(E\) centred in the chart,
\(\ord_E(\lambda_0)=A_{(Y,\Sigma)}(E)-1\).
Replacing a factor by \(dG_*/G_*\) multiplies the wedge, up to sign,
by \(x_i\partial_{x_i}G_*/G_*\) or
\(\partial_{y_j}G_*/G_*\). The new wedge has order at least \(-1\):
at a uniformizer of \(E\), each logarithmic differential has only a
possible simple logarithmic pole, and a nonzero wedge can contain that
normal logarithmic factor at most once. Pullbacks of regular tangential
differentials have no poles. Therefore each indicated derivation lowers
value by at most \(A_{(Y,\Sigma)}(E)\), including over a proper
subcentre of the stratum. Approximate real monomial weights on a higher
adapted chart by rational divisorial weights. Continuity of the fixed
function values and discrepancy gives the same inequality for \(u\).
The operator \(x_i\partial_{x_i}-j\) obeys it too, because constants
have value zero and \(d_{\log}\geq0\). Composing the bounded number of
factors loses at most their total order times \(d_{\log}\).
The exact value of the isolated output proves \eqref{eq:A5}.
\smallskip
\noindent\emph{From polynomials to local functions.}
For a nonunit \(F\in\OO_{U,x}\), take
\(M=\lceil3w(F)\rceil+1\).
Modulo \(\m_x^M\), the local ring is spanned by polynomials of degree
less than \(M\) in the fixed affine generators. In particular a unit
denominator is inverted by its finite geometric series in this quotient.
Choose such a polynomial \(G\) representing \(F\).
For small \(t'\), \(w(\m_x)\geq1/2\), so the remainder has
\(u\)-value at least \(M\) and \(w\)-value at least \(M/2>w(F)\).
Thus \(w(G)=w(F)\). Also \(w(F)\geq1/2\), so \(1+M\leq C'w(F)\).
Equation~\eqref{eq:A5} and \(d_{\log}\leq C_1C(u)\) now show that
\[
u(G)\leq(1+C_2C(u))w(F)<M
\]
for all sufficiently small \(t'\). Hence \(u(F)=u(G)\), proving
\eqref{eq:A4}. Units have value zero at both valuations.
\smallskip
\noindent\emph{Exactness of the minimum.}
We now compare the objectives at \(u\) and \(w\).
Put \(K=1+C_2C(u)\). Equation~\eqref{eq:A4} gives
\(\mathfrak a_m(u)\subseteq\mathfrak a_{m/K}(w)\), and therefore
\(\vol(u)\geq K^{-p}\vol(w)\).
On the fixed germ, properness has bounded
\(a_D(w)\leq b_0A_U(u)\leq M_0\).
Choose \(t'\) so small that \(C_2t'M_0<1\).
If \(C(u)>0\), then
\[
(C+t'a_D)(u)-K\,t'a_D(w)
\geq C(u)\bigl(1-C_2t'a_D(w)\bigr)>0.
\]
Together with the volume inequality this contradicts minimality of
\(u\), because \(C(w)=0\). Thus \(C(u)=0\) exactly.
On the constraint, \(a_D=a\), so every sufficiently small parameter
gives a constrained minimum. Conversely any other constrained minimum
would also minimize that ordinary klt objective, whose ray is unique.
The ray is consequently independent of the small parameter.
Stable degeneration gives finite generation. A torus preserving the
ideals and point preserves the constrained objective and its unique ray;
an invariant normalization, for instance by \(u(\m_x)\), makes the
valuation invariant. This proves the proposition.
\end{proof}
==== END Bounded-klt-complements-for-Fano-contractions//constrained-minimum.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//discrepancy-push.tex ====
\section{Valuative pushforward of discrepancy functions}
\label{b:discrepancy-push}
The residual system from Section~\ref{a:essential} will be studied through
successively smaller function fields. We need discrepancy data that pass
through these contractions with an exact rule for pullbacks and with
controlled denominators. The construction below uses the least normalized
discrepancy among divisors lying over a given base divisor. We first prove
attainment and compatibility, then construct its nef moduli presentation.
We continue over \(\C\), and retain the chosen rational volume forms.
A prime divisor is identified with its normalized integral valuation.
The coefficient of a rational b-divisor \(\mathbf D\) at a valuation
\(P\) is denoted by \(P(\mathbf D)\). All b-divisors are evaluated on
models on which the displayed Cartier data descend.
\begin{definition}\label{b:discrepancy-presentation}
A discrepancy presentation on a normal projective model \(Y\) is
\[
a=a_{K,Y}+\mathbf M-\overline D.
\]
Here \(a_{K,Y}\) is the form discrepancy, \(\mathbf M\) is b-nef and
b-\(\Q\)-Cartier, and \(D\) is a rational Cartier divisor on \(Y\).
The boundary trace \(B=\sum_P(1-a(P))P\), summed over strict primes
on \(Y\), satisfies
\[
K_Y+B+M_Y=D
\]
as divisors. The presentation is \emph{Calabi--Yau} if \(D=0\).
Sub-generalized lc and sub-generalized klt mean, respectively,
\(a(P)\ge0\) and \(a(P)>0\) for every relevant prime divisor on every
higher model. Effectivity means effectivity of the boundary trace
on the specified model, not on every higher model.
\end{definition}
The effective unscaled case is the generalized-pair framework of
Birkar--Zhang \cite[Definitions~1.4 and~4.1]{BirkarZhang}.
Positive rational scalings are also allowed. For
\(\mu a_K+\mathbf M-\overline D\), trace effectivity means that strict
prime discrepancies are at most \(\mu\). Bars denote Cartier closure
and may be suppressed when there is no ambiguity.
\begin{definition}\label{b:valuative-push}
For a projective contraction \(\pi:Y\to V\), put
\begin{equation}\label{eq:B1}
(\pi_\#a)(P)=
\min_{\substack{F|_{\C(V)}=mP\\m>0}}\frac{a(F)}m.
\end{equation}
We use this operation for data sub-generalized lc over the generic
point of \(V\), with total divisor pulled back from \(V\), and for
positive scalings of such data. A prime valuation upstairs is
\emph{horizontal} when its restriction to \(\C(V)\) is trivial.
\end{definition}
For an unscaled discrepancy presentation, \((\pi_\#a)(P)\) is one
minus the discriminant coefficient at \(P\) in the canonical bundle
formula \cite[Section~2]{FilipazziGeneralized}. Keeping it as a
function on all prime valuations makes composition and base pullbacks
explicit.
\begin{lemma}\label{b:push-threshold}
The minimum in \eqref{eq:B1} is finite and attained. Pullback terms
from the base translate it exactly, and successive pushes compose.
\end{lemma}
\begin{proof}
Place \(P\) on a smooth projective base model. Resolve the map upstairs,
the boundary and nef data, and the pullback of \(P\), so that the nef
part descends and the relevant divisors have simple normal crossings.
Over \(\eta_P\), the largest real number that can be subtracted times
the pullback of \(P\) while preserving sub-lc discrepancies is the
least component-discrepancy-to-multiplicity ratio among components
of positive multiplicity. This number may be negative. Horizontal
sub-lc takes care of components of zero multiplicity. The snc
discrepancy formula gives the same inequalities for every higher
valuation, proving attainment and \eqref{eq:B1}.
A nontrivial restriction of a prime-divisor valuation to a subfunction
field is a positive integral multiple of a prime-divisor valuation:
the relative Abhyankar inequalities force equality for the restriction,
and its nonzero value group is a subgroup of \(\Z\).
Apply this to the intermediate fields and choose attained minimizing
lifts successively to obtain composition.
Finally \(F(\pi^*D)=mP(D)\), proving translation.
\end{proof}
\subsection{Preparatory birational operations}
\begin{lemma}\label{b:ordinary-approximation}
Effective generalized klt data with big b-nef part admit an ordinary
effective klt representative in the same rational adjoint class, whose
discrepancies do not exceed the generalized ones. Generalized
\(\delta\)-lc data, \(0<\delta\le1\), admit such a representative which
is \(\delta/2\)-lc. A sufficiently small ample class on the low model
may be reserved before taking the representative. In particular,
effective relatively Calabi--Yau generalized klt data with big b-nef
part make the underlying variety of Fano type over the base.
\end{lemma}
\begin{proof}
On a projective descent write \(M\sim_\Q D_0+E_0\), with \(D_0\)
ample and \(E_0\ge0\). For small rational \(s>0\), use \(sE_0\) as
fixed part and the ample class \((1-s)M+sD_0\) as moving part.
A still smaller pullback ample reserve can be subtracted from this
moving part. Resolve fixed supports and take a sufficiently divisible
general free representative of the moving part, divided by its
multiplicity. Fixed coefficients change arbitrarily little and new
coefficients are arbitrarily small. The snc discrepancy formula
therefore retains klt, or \(\delta/2\)-lc, on every higher valuation.
Subtracting the effective Cartier b-divisor representing the nef part
only lowers the generalized discrepancies. For open bases use a
projective compactification and common higher projective descents,
testing these inequalities over the required open. The ample reserve
proves the Fano-type assertion.
\end{proof}
On a \(\Q\)-factorial model the negativity lemma
\cite[Lemma~4.16]{FujinoFundamental} gives
\[
\overline{M_Y}-\mathbf M\ge0.
\]
Removing effective boundary or forgetting b-nef data consequently
does not decrease discrepancies. Adjoint MMPs preserve lower
discrepancy bounds by pullback negativity on common resolutions.
We use the following established model results with their hypotheses.
An effective klt rational pair with big boundary and pseudo-effective
adjoint, projective over a quasi-projective base, has a log terminal
model; the model is nonextracting and \(\Q\)-factorial, and is good
because the boundary is big
\cite[Theorem 1.2 and Lemma 3.9.3]{BCHM}.
The nonextraction convention is explicit in
\cite[Definition~3.6.6]{BCHM}; all these locators refer to the same
arXiv version.
Relative bigness is tested on the generic fiber. A rational-function
witness can be extended globally after adding a sufficiently vanishing
effective Cartier divisor from the base to remove its finitely many
residual vertical poles.
A projective \(\Q\)-factorial variety of Fano type is a Mori dream
space \cite[Corollary 1.3.1]{BCHM}; in particular, its effective cone
is rational polyhedral and closed, rational classes in it are
rational-linearly effective, and their divisor MMPs end at nef
semiample models \cite[Definition 1.10 and Proposition 1.11]{HuKeel}.
This is a statement for each individual variety, with no uniform
chamber or generator assertion. Nonextracting \(\Q\)-factorial
outputs from a Fano-type model remain of Fano type: push an effective
klt rationally trivial adjoint with big boundary. Rational triviality
makes this crepant, and persistence of sections preserves bigness.
Perturbing the big boundary gives a log Fano boundary again.
\begin{lemma}\label{b:controlled-extraction}
Let \((Y,B)\) be an effective klt log Fano pair. Any finite collection
of exceptional prime divisors with discrepancies in \((0,1]\) can be
extracted on a projective \(\Q\)-factorial birational model having no
other exceptional prime divisors. This model is of Fano type.
An empty collection gives a small \(\Q\)-factorialization.
\end{lemma}
\begin{proof}
On a log resolution displaying the collection, retain their crepant
coefficients, which lie in \([0,1)\). Raise every other exceptional
coefficient to a strictly larger nonnegative rational number below
one. The resulting effective pair is klt, its boundary is big over
the birational base, and its adjoint is relatively the effective
exceptional divisor \(R\) supported exactly on the raised components.
Normality gives \(f_*\OO(mR)=\OO_Y\) for divisible \(m\).
The good log terminal model preserves these sections by negativity
and makes the transform of \(R\) relatively semiample. A free multiple
has only the relative generator \(1\), so its effective transform
vanishes. The pullback difference is zero at every requested divisor;
strict adjoint negativity for contracted divisors therefore prevents
their contraction. The MMP extracts no divisor. The original pair's
crepant trace on this extraction is effective and has nef big
anti-adjoint, so a small perturbation gives Fano type.
\end{proof}
\subsection{A uniform interval of primary minimization}
For a small perturbation of an lc discrepancy function, we need its
computing divisors to continue computing the unperturbed function. The
next lemma obtains one interval valid for every tested base prime from a
bounded index and generalized threshold ACC. This is the uniformity that
will later allow us to pass to a nef endpoint on a fixed model.
\begin{lemma}\label{b:primary-minimization}
Fix a positive integer \(r_{\mathrm{ind}}\). Let \(A,C\) have Calabi--Yau presentations on \(Y\), effective in
horizontal trace over \(V\). Assume that \(A\) is sub-generalized lc
over the generic point, that \(C\) is generalized klt there with big
b-nef part, and that \(r_{\mathrm{ind}}\mathbf M_A\) is integral b-Cartier.
There is \(t_0>0\), depending only on the dimension and \(r_{\mathrm{ind}}\), such
that for every rational \(0<t<t_0\), every prime \(T\) computing
\(\pi_\#((1-t)A+tC)(P)\) also computes \(\pi_\#A(P)\).
\end{lemma}
\begin{proof}
Fix a prime \(P\) and a lift \(T\) computing the mixture. We will
construct a model on which \(T\) is the only strict prime above \(\eta_P\), and
on which both individually shifted boundaries are effective. On that
model, generalized threshold ACC will force the shifted \(A\)-datum
to be lc.
\medskip\noindent
\emph{Shifting the two data at the computing prime.}
Work over an open neighborhood \(U\) of \(\eta_P\) on a smooth base
model and put
\[
T|_{\C(V)}=m_TP,\qquad
\alpha_A=A(T)/m_T,\qquad \alpha_C=C(T)/m_T.
\]
Subtract \(\alpha_A\pi^*P,\alpha_C\pi^*P\) from the respective
discrepancy functions. Their mixture is sub-generalized lc over \(U\)
after shrinking: its threshold at \(P\) is zero, and horizontal sub-lc
is already known. Both shifted totals are pulled back from \(U\).
Add \(\kappa\pi^*P\) to the shifted mixture discrepancy for a small
positive rational \(\kappa\). It becomes sub-generalized klt there,
with coefficient \(1-\kappa m_T\ge0\) at \(T\). On a high smooth
model displaying \(T\), resolve the boundary, exceptional, fiber and
nef data. Raise the coefficients of every other component over
\(\eta_P\), and every horizontally exceptional divisor over the
original generic fiber, to strictly larger nonnegative rationals
below one. Remove unwanted vertical supports away from \(\eta_P\)
by shrinking. Horizontal strict primes over the original generic
fiber already have effective trace. Thus we obtain effective
generalized klt data whose adjoint is relatively an effective divisor
\(R_{\mathrm{ex}}\), supported exactly on these raises and omitting
\(T\).
\medskip\noindent
\emph{A model whose only strict prime above \(\eta_P\) is \(T\).}
By Lemma~\ref{b:ordinary-approximation}, replace the big nef part by
ordinary effective data while keeping klt and big boundary.
Take its good \(\Q\)-factorial log terminal model over \(U\).
The reason every raised divisor disappears is that
\(R_{\mathrm{ex}}\) has no relative sections except those from
\(\OO_U\). We verify this before using semiampleness on the good
model.
For every divisible \(m\), a relative section of
\(\OO(mR_{\mathrm{ex}})\) has no nonexceptional horizontal poles
on the original normal proper generic fiber. It is therefore a
constant on that fiber, hence an element of \(\C(V)\).
A base pole at \(P\) is impossible because its pullback has a pole
along \(T\), absent from \(R_{\mathrm{ex}}\). At every other retained
base prime, a component missing from \(R_{\mathrm{ex}}\) likewise
prevents a pole. Such a component exists above each strict prime of
a normal contraction base, by pulling back a local equation near its
generic point; other vertical supports were removed by the shrink.
Consequently these relative section sheaves are precisely \(\OO_U\).
The ordinary approximation may change the adjoint by a rational
principal term and a base pullback. Clear denominators and identify
the corresponding section spaces by multiplication by that principal
function and local trivialization of the base term. Under this
identification the algebra just computed is unchanged. Negativity
preserves it on the good model. The effective transform of
\(R_{\mathrm{ex}}\) is relatively semiample; a divisible free multiple
is generated by the section \(1\), whose zero divisor is that multiple
of \(R_{\mathrm{ex}}\). Hence its support is empty.
The output must still have a divisor over \(\eta_P\), and the MMP
extracts none. Thus \(T\) survives and is the unique such divisor.
The pre-raise, \(\kappa\)-adjusted mixture is crepant on this output,
since its adjoint is pulled back from \(U\), and has effective trace.
Its big b-nef part gives relative Fano type by
Lemma~\ref{b:ordinary-approximation}. In particular the bare output
is klt. The surviving horizontal primes come from the original
generic fiber; both individual shifted boundaries are therefore
effective after shrinking. Their coefficient at \(T\) is one.
\medskip\noindent
\emph{The bounded-coefficient threshold test.}
We now return to the shifted mixture before the \(\kappa\)-adjustment;
it is lc on this same model. Remove \(t\) times the shifted
\(C\)-boundary and its corresponding nef datum. Effectivity and
negativity show that discrepancies only increase. Thus the
generalized threshold on the bare variety of the
shifted \(A\)-boundary \(B'_A\) together with \(\mathbf M_A\) lies in
\([1-t,1]\); the upper bound comes from \(T\).
The horizontal coefficients of \(B'_A\) lie in the finite
\(1/r_{\mathrm{ind}}\)-grid in \([0,1]\), because \(r_{\mathrm{ind}}A(F)\) is integral for Calabi--Yau
data with \(r_{\mathrm{ind}}\mathbf M_A\) Cartier. The only remaining nonzero vertical
coefficient is one. Use \((1/r_{\mathrm{ind}})(r_{\mathrm{ind}}\mathbf M_A)\) as a single nef
Cartier summand. These are exactly the effective test and nef
Cartier coefficient hypotheses of generalized threshold ACC
\cite[Theorem 1.5]{BirkarZhang}. The test scales the boundary and the
nef datum together; at parameter one it is precisely the shifted
\(A\)-presentation. More explicitly, the initial boundary and nef
datum in the threshold theorem are both zero; the effective test
boundary is \(B'_A\), and the nef test datum is
\((1/r_{\mathrm{ind}})(r_{\mathrm{ind}}\mathbf M_A)\).
The latter is required to be nef, with the stated Cartier coefficient;
its trace need not be effective. A Cartier divisor descending on the low model
does not affect the nef-data discrepancy
\(\mathbf M_A-\overline{M_{A,Y}}\), so any auxiliary effective
representative obtained by twisting on the affine base does not
alter the test.
If thresholds below one occurred for arbitrarily small \(t\), a
subsequence would increase strictly to one, contradicting ACC.
Thus the threshold is one for all sufficiently small \(t\), with a
bound depending only on dimension and \(r_{\mathrm{ind}}\). Closedness supplies the
inequalities at the endpoint. The shifted \(A\)-data are therefore
lc over \(\eta_P\), proving that \(T\) computes \(\pi_\#A(P)\).
\end{proof}
\begin{corollary}\label{b:principal-denominators}
In Lemma~\ref{b:primary-minimization}, with the fixed positive integer
\(r_{\mathrm{ind}}\) as there, assume in addition that
\(r_{\mathrm{ind}}\mathbf M_A\) is principal. Then every value \((\pi_\#A)(P)\)
has bounded denominator, in terms only of the dimension and \(r_{\mathrm{ind}}\).
\end{corollary}
\begin{proof}
Use the model constructed in the proof of
Lemma~\ref{b:primary-minimization}: its unique strict prime above \(\eta_P\) is
\(T\), and the shifted \(A\)-boundary is effective and lc. The
shifted \(A\)-data now have ordinary discrepancies, and
\[
K+B'_A=\alpha_A\pi^*P-M_{A,Y}.
\]
The pair is lc, its anti-adjoint is nef over \(U\), its coefficient
set is finite rational (hence contained in a fixed hyperstandard
set), and its underlying variety is of Fano type over \(U\).
Apply the bounded relative monotone lc complement theorem near a
closed point of \(P\cap U\)
\cite[Theorem 1.8]{BirkarComplements}.
The effective complement addition restricts to a rationally trivial
effective divisor on the proper generic fiber, so has no horizontal
component. It cannot add to \(T\), whose coefficient is already one.
Thus the boundary is unchanged near \(\eta_P\). Higher crepant
traces are determined by the same adjoint and introduce no new
independent addition.
For clarity, if the local complement identity is
\(n'(K+B'_A)=\Div(h)\) and
\(r_{\mathrm{ind}}M_{A,Y}=\Div(g)\), take
\(N=n'r_{\mathrm{ind}}\) and \(\phi=h^{r_{\mathrm{ind}}}g^{n'}\).
Thus a bounded integer \(N\) gives
\[
N\alpha_A\pi^*P=\Div(\phi)
\]
over a neighborhood still containing \(\eta_P\).
The principal adjustment from \(M_A\) is absorbed in \(\phi\).
This function has no horizontal poles. Generic-fiber properness and
the contraction property imply \(\phi\in\C(V)\), even though its
initial expression was obtained on an open neighborhood.
Evaluating at \(T\) gives
\[
N\alpha_A m_T=m_T\ord_P(\phi),
\qquad N\alpha_A=\ord_P(\phi)\in\Z.
\]
The fiber multiplicity cancels; neither a norm nor an extension-degree
factor is introduced.
\end{proof}
\subsection{Nef push presentations}
The preceding denominator argument applies to principal moduli data.
We now construct the moduli b-divisor for a general semiample or nef and
big presentation. The proof treats descent from generic coefficients,
tests on newly appearing primes, and nefness separately, before taking
the endpoint of a small perturbation. For the unscaled presentation, the
moduli conclusion is a case of the generalized canonical bundle formula
of Filipazzi \cite[Theorem~1.4]{FilipazziGeneralized}. We retain an
explicit proof in the chosen volume-form conventions, together with the
perturbation argument used here.
\begin{proposition}\label{b:push-presentation}
Let \(a=a_{K,Y}+\mathbf M\) be a Calabi--Yau presentation, sub-generalized
klt over the generic point of \(V\) and effective in horizontal trace.
If \(\mathbf M\) is b-semiample or b-nef and big, then
\[
\pi_\#a=a_{K,V}+\mathbf N
\]
for a b-nef b-\(\Q\)-Cartier divisor \(\mathbf N\).
Positive rational scaling and translation by base Cartier pullbacks
give the corresponding presentations with their scaled canonical
coefficient and prescribed total divisor.
\end{proposition}
\begin{proof}
We first assume b-semiampleness. Let \(k=\C\), and enlarge constants
to the algebraic closure \(k'\) of a purely transcendental extension
of finite transcendence degree. Use enough independent coefficients
to choose a member of a free Cartier system \(|m\mathbf M|\),
\(m>1\), generic over \(k\), on a high descent. Divide it by \(m\)
and denote its Cartier closure by \(\Delta\).
The data
\[
a^{k'}-\Delta
=a_{K,Y}^{k'}+\mathbf M^{k'}-\Delta
\]
are ordinary pair discrepancies, because the remaining moduli term
is rationally principal. Their ordinary boundary adds the trace of
\(\Delta\) to the generalized boundary. Bertini on a log resolution,
with moving coefficient \(1/m<1\), makes them sub-klt generically.
Their horizontal trace on the original low model remains effective.
The rank-one condition in the ordinary canonical bundle formula
holds as follows. On a resolution over the generic point, the
rounded discrepancy sheaf contains constants by sub-klt singularities.
Any additional poles allowed by it are exceptional over the original
normal generic fiber, because the low horizontal boundary is
effective. Such functions are regular on that proper low fiber and
therefore constant. The contraction condition gives rank one.
The rational principal adjustment makes the total adjoint pulled back
from zero. Ambro's ordinary klt-trivial-fibration theorem therefore
gives a b-nef b-\(\Q\)-Cartier divisor
\[
\mathbf N^\Delta
=(\pi^{k'})_\#(a^{k'}-\Delta)-a_{K,V}^{k'}.
\]
Here we use \cite[Theorem 0.2]{AmbroBoundary}; see also
\cite{AmbroModuli}.
Vertical negative boundary components are allowed in this theorem.
Alternatively, compensate them with a pulled-back base divisor,
which translates thresholds and total divisor compatibly.
For each prime \(P\) over \(k\), a resolution computing its threshold
is defined over \(k\). Extension does not change its finite snc
calculation, and the generic free divisor meets that calculation
transversely, without changing the threshold. Thus, writing
\(\mathbf N=\pi_\#a-a_{K,V}\) initially as a coefficient function,
\begin{equation}\label{b:old-prime-equality}
\mathbf N^\Delta(P_{k'})=\mathbf N(P).
\end{equation}
This holds for every old prime simultaneously in the requisite sense:
each transversality condition is a nonempty open condition defined
over \(k\), and the chosen coefficient tuple lies over the generic
point of its parameter space. No countable intersection of open
subsets of \(k\)-points is being asserted.
\medskip\noindent
\emph{Descent of the Cartier presentation.}
The restriction to old prime extensions of any rational b-Cartier
divisor over \(k'\) is rational b-Cartier over \(k\).
On a projective descent, taking differences and multiples reduces
this to a globally generated Cartier divisor \(J\). For finitely many
rational sections \(f_j\),
\[
Q(J)=-\min_j Q(f_j).
\]
Put these sections over a common denominator in
\(k(V)\otimes_k k'\), and express every numerator and the denominator
as a finite sum of coefficients in \(k(V)\) times constants linearly
independent over \(k\). At \(P_{k'}\), the value of such a sum is
exactly the least coefficient value. Indeed those constants remain
independent over \(k(P)\) in
\(\Frac(k(P)\otimes_k k')\); geometric integrality over algebraically
closed \(k\) makes this tensor product a domain.
Hence the restricted function is a difference of minima over finite
lists in \(k(V)\). Resolving the corresponding fractional ideals
makes it Cartier. Apply this to \(\mathbf N^\Delta\) and
\eqref{b:old-prime-equality} to obtain a b-\(\Q\)-Cartier presentation
of \(\mathbf N\) over \(k\).
\medskip\noindent
\emph{Testing new primes and nefness.}
The identity \(\pi_\#a=a_{K,V}+\mathbf N\) remains valid after
extension at every prime over \(k'\). To prove this, fix one such
prime on a smooth model dominating a descent of \(\mathbf N\).
Choose a smooth model upstairs, with descended \(\mathbf M\),
on which its threshold is computed by snc data including the
pullback of that prime. All models, morphisms, prime components,
rational forms and Cartier data in this single test are defined over
a finitely generated extension of \(k\). Spread them over an
integral parameter variety over \(k\). After shrinking, the models
remain projective birational resolutions fiberwise; the required
components remain geometrically integral; smoothness and relative
snc hold; and the same components dominate the tested prime.
Properness and constructibility permit these requirements.
The finitely many divisor identities for forms, functions and
pullbacks also persist with unchanged multiplicities: keep their
supports on the smooth models and remove the parameter locus where
a stated divisor identity or dominance fails.
Specialize to a sufficiently general closed \(k\)-point. Both the
proposed b-Cartier coefficient and the threshold are computed by
the same respective finite numerical lists as before specialization.
They agree there by the already proved identity over \(k\).
They therefore agree for the chosen prime over \(k'\).
This argument uses one finite spread per test prime, not a common
resolution of all primes.
Since \(\Delta\ge0\), threshold monotonicity gives the b-effective
difference
\[
\mathbf N^{k'}-\mathbf N^\Delta\ge0.
\]
Its value at every \(P_{k'}\) is zero by
\eqref{b:old-prime-equality}. Let \(C\) be a curve on a smooth
projective descent of \(\mathbf N\) over \(k\), and take a common
higher descent over \(k'\) where \(\mathbf N^\Delta\) is nef.
Choose an old prime valuation centered at the generic point of \(C\).
Its extended center \(Z\) on the common model dominates \(C_{k'}\).
After clearing denominators, if \(Z\) lay in the effective Cartier
difference, a local equation would have strictly positive valuation,
contrary to its zero value. Thus \(Z\) is not contained in that
support. General complete intersections in \(Z\) have a component
dominating \(C_{k'}\) and not contained in the support: their
restriction to the generic fiber of \(Z\to C_{k'}\) has positive
top ample intersection, and a general choice avoids containment in
a fixed proper closed subset. On this curve the effective difference
has nonnegative degree. The nef term and projection formula show
\(\mathbf N\cdot C\ge0\). If the base itself is a curve, use the
whole curve on its common smooth model; the avoidance issue is
automatic. A zero-dimensional base is immediate.
This proves b-nefness in the semiample case.
\medskip\noindent
\emph{The nef and big endpoint.}
On a projective descent write \(M\sim_\Q D_0+E_0\), with \(D_0\)
ample and \(E_0\ge0\). For sufficiently small rational \(s>0\),
\[
a_s=a-s\overline E_0
\]
retains horizontal klt and effectivity, and its moduli class
\(M-sE_0\sim_\Q(1-s)M+sD_0\) is ample, hence semiample.
Fix one such \(s_0>0\). Apply Lemma~\ref{b:primary-minimization}
to \(a,a_{s_0}\), choosing a positive integer \(r_{\mathrm{ind}}\) for
which \(r_{\mathrm{ind}}\mathbf M\) is integral b-Cartier. This index may
depend on the individual presentation; the interval below is uniform over
its test primes.
Since
\[
a_s=(1-s/s_0)a+(s/s_0)a_{s_0},
\]
the lemma gives one interval \(0<s<s_1\), independent of the
prime \(P\), on which every computing lift for
\(\pi_\#a_s(P)\) also computes \(\pi_\#a(P)\).
For a fixed \(P\), a computing lift at any one positive parameter
exists by Lemma~\ref{b:push-threshold}. Comparing it with every
other primary minimizer shows that it maximizes
\(E_0(F)/m\) among them. This maximum is attained and is independent
of \(s\). Thus the threshold is exactly
\[
(\pi_\#a_s)(P)
=(\pi_\#a)(P)-s e_P
\]
on this interval, including its stated value at \(s=0\).
The identity holds for every prime. If \(0<s'<s''<s_1\) are
rational, the endpoint moduli function is therefore
\[
\mathbf N_0=
\frac{s''\mathbf N_{s'}-s'\mathbf N_{s''}}{s''-s'}.
\]
The two terms on the right descend on a common model; hence so does
\(\mathbf N_0\). All other small positive-parameter terms descend
there by the same affine formula and are nef there, since nefness
on a higher descent descends by lifting curves and projection
formula. Taking the limit on this fixed model proves that
\(\mathbf N_0\) is nef. This proves the endpoint assertion.
Scaling and translation now follow from
Lemma~\ref{b:push-threshold}.
\end{proof}
==== END Bounded-klt-complements-for-Fano-contractions//discrepancy-push.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//essential-system.tex ====
\section{A deepest weakly essential residual system}
\label{a:essential}
We return to the sequence of anticanonical cones \(W=\Spec\mathcal A\)
from Section~\ref{a:cone}, retaining its fixed integer \(n>1\), degree
action \(D\), canonical generator \(\Omega\), and vertex \(o\).
Constants called uniform in this section are independent of the member
of that sequence. The small parameters realizing each exact minimum may
depend on the individual member.
\subsection{Initial minimizing data}
Apply Proposition~\ref{a:exact-penalization} on \(W\) with constraint
\(I_n^{1/n}\) and residual discrepancy \(a=A_W\). Proposition
\ref{a:initial-lc-place} supplies a centred quasi-monomial lc place.
Normalize the resulting \(D\)-invariant minimizer \(\tilde v\) by
\(A_W(\tilde v)=1\). In constructing its auxiliary klt pairs, take
the general core equations in degree \(n\); every such equation has
value \(n\) at \(\tilde v\). On both \(\tilde v\) and the lc place
\(u^0\) from \eqref{eq:A2}, the auxiliary discrepancy is the parameter
times bare discrepancy, up to their common scaling. Here rescale
\(u^0\) so that \(A_W(u^0)=1\), which preserves \eqref{eq:A2}. Applying
\eqref{eq:A3} with minimizer \(\tilde v\) and test valuation \(u^0\)
therefore gives, uniformly on the ideal \(\m_z\),
\begin{equation}
\tilde v(\m_z)\longrightarrow\infty.
\label{eq:A7}
\end{equation}
Indeed \(A_W(u^0)=A_W(\tilde v)=1\) and
\(u^0(\m_z)\to\infty\) by \eqref{eq:A2}.
Stable degeneration makes \(R=\gr_{\tilde v}\mathcal A\) finitely
generated and normal. Removing its effective specialized boundary shows
that \(\Spec R\) is klt. These homogeneous assertions, initially near
the vertex, hold globally: the positive grading makes every orbit
closure contain the vertex. Lemma~\ref{a:canonical-specialization}
gives the canonical generator \(\Omega_{\tilde v}\). Let \(T\) be the
joint effective torus of the value grading and \(D\), with character
lattice \(M_T\), and let \(\chi\) be the character of this form. Then
\[
D(\chi)=1=\tilde v(\chi).
\]
Every specialized core equation has pure weight \(n\chi\): its two
weights are degree \(n\) and value \(n\), and these determine the
joint character. Write \(I_R=(R_{n\chi})\).
The exact minimum fixes the normalized valuation and the chosen core
through all sufficiently small rational parameters on this member.
The induced auxiliary klt discrepancies, before their common canonical
scaling, are the specialized core discrepancy plus the parameter times
the bare discrepancy. Letting that parameter tend to zero shows that
the specialized core boundary is lc. The order of \(I_R\) is no larger
than the averaged order of the core equations; hence \(I_R^{1/n}\) is
lc as well. Moreover every quasi-monomial lc place \(u\) of this system
centred at the vertex satisfies
\begin{equation}
u(f)\leq C A_R(u)\tilde v(f)
\qquad(0\ne f\in R),
\label{eq:A8}
\end{equation}
with uniform \(C\). Apply \eqref{eq:A3} on the auxiliary klt stable
cone. At \(u\), the core equations have orders at least \(u(I_R)\),
so the auxiliary discrepancy is at most the parameter times \(A_R(u)\).
At the minimizing grading \(\tilde v\) it is exactly the parameter,
by \eqref{eq:A6}. The parameter cancels, giving the bound.
We may ensure that evaluation by \(\tilde v\) is injective on \(M_T\).
If necessary add a sufficiently small positive multiple \(\delta D\)
and renormalize by \(1+\delta\). On each original homogeneous degree
this translates values by \(\delta\) times that degree; extend by the
minimum rule. Generic \(\delta\) avoids ties of distinct joint weights
and retains the same joint graded ring and Abhyankar equality. To check
canonical normalization, choose homogeneous lifts of joint homogeneous
independent initial probes. In their logarithmic basis the coefficient
of \(\Omega\) has degree one and old value one, so its new value is
\(1+\delta\). Formula~\eqref{eq:A6} gives both the stated normalization
and the unchanged leading form. Homogeneous lifts exist by character
splitting in the \(D\)-invariant filtrations, also for fractions.
The actual ideal \(I_n\) still has normalized value \(n\).
All degrees on \(\mathcal A\) are nonnegative, and \(\m_z\) has degree
zero. Taking \(\delta\) individually small, the bounds
\eqref{eq:A7}--\eqref{eq:A8} persist with uniform factors. We henceforth
use the adjusted notation. To compare torus ranks, we will retain actual
valuations on \(W\), together with
\[
\begin{gathered}
A_W(\tilde v)=1,\quad \tilde v(I_n)=n,\quad
R=\gr_{\tilde v}\mathcal A\ \text{finitely generated and klt},\\
T\supset D\ \text{the value torus},\quad
D(\chi)=\tilde v(\chi)=1,\quad I_R^{1/n}\ \text{lc},
\end{gathered}
\]
and the uniform estimates \eqref{eq:A7}--\eqref{eq:A8}.
Among all sequences of such data and their subsequences, choose one
whose torus rank is constant and as large as possible. This is possible
because \(\rank T\leq\dim R=d+1\). All future rank comparisons concern
sequences retaining the uniform estimates, rather than isolated members.
For each member choose \(h\in\mathcal A_n\) general so that its divisor
divided by \(n\) is lc on \(W\) and the corresponding divisor is lc on
\(X\), and its initial \(\bar h\) is general nonzero in \(R_{n\chi}\).
Indeed this finite-dimensional weight space is exactly the leading
space at the minimal value \(n\) from degree \(n\). Include lifts of
its basis, together with system generators, in the finite-dimensional
space used for the general choice. On \(R\), the divisor
\(\Div(\bar h)/n\) is lc, and each of its discrepancy-zero primes is
also an lc place of \(I_R^{1/n}\). On a resolution, the moving part has
coefficient \(1/n<1\), so its general smooth components introduce no
additional discrepancy-zero primes. The fixed part is the ideal's part.
\subsection{The projective torus quotient}
\label{a:quotient}
The character \(\chi\) lies in the interior of the rational polyhedral
cone of occurring weights. Otherwise a nonzero rational dual
cocharacter would be nonnegative on all occurring weights and nonpositive
on \(\chi\). It defines a nontrivial centred valuation on \(R\), whose
bare discrepancy is its evaluation on \(\chi\), contradicting klt.
Also \(\chi\) is primitive, since \(D(\chi)=1\). Set
\[
R_\chi=\bigoplus_{m\geq0}R_{m\chi},\qquad
S=\Proj R_\chi.
\]
This ray ring is a subtorus invariant ring, hence finitely generated and
normal. Its field and the invariant field satisfy
\[
k(S)=k(R)^T,\qquad \Frac R_\chi=k(S)(z^\chi).
\]
Here and below \(z^\mu\) are multiplicative rational torus variables.
We give the monoid detail that ensures these identities in all degrees.
The finitely generated occurring monoid spans \(M_T\) as a group.
Its saturation in the weight cone is a finite module over it: subtract
integer multiples of its generators until the remainder lies in a
bounded set. For each of the finitely many module representatives, use
the group-span property to write its difference as a difference of
monoid elements. Adding a common monoid translate sends the entire
saturation into the monoid. Since \(\chi\) is interior, for every
fixed weight \(\mu\) the points \(m\chi-\mu\) lie in that translated
saturation for all sufficiently large \(m\). Thus a homogeneous
numerator and denominator of equal weight can be multiplied by a common
occurring element to have ray degrees. Likewise ray elements occur in
all sufficiently large integral degrees, so their degree group is
\(\Z\chi\). This proves the field identities.
Choose a rational volume form \(\omega_S\) so that
\[
\Omega_{\tilde v}
=z^\chi\omega_S\wedge d\log\mathbf z,
\]
where \(d\log\mathbf z\) is the wedge for a lattice basis of \(M_T\).
The quotient of the two sides is invariant, so it can be absorbed in
\(\omega_S\). There is an ample rational \(\Q\)-Cartier divisor \(L\)
on the normal projective variety \(S\) for which
\begin{equation}
fz^{m\chi}\in R
\quad\Longleftrightarrow\quad \Div_S f+mL\geq0
\qquad(m\geq0,\ f\in k(S)^*).
\label{eq:A9}
\end{equation}
Take a sufficiently divisible Veronese of the ray ring and write its
tautological polarization using the corresponding power of the rational
section \(z^\chi\). The equivalence holds in sufficiently high divisible
degrees by the Proj description. For any remaining degree, take a
sufficiently divisible power: membership of that power implies
membership of the element by normality of \(R_\chi\), and the divisor
inequality is equivalent to that for its power. If \(\dim S=0\), set
\(L=0\); all subsequent prime tests on \(S\) are then vacuous.
On \(S\) define
\[
f_h=\bar h/z^{n\chi},\qquad
H=L+\tfrac1n\Div_S f_h,\qquad
B=-K_S^{\omega_S}-L.
\]
In particular \(H\geq0\) and \(H\sim_{\Q}L\).
\begin{lemma}[Gauss lifts and the quotient boundary]\label{a:gauss-lifts}
For every prime-divisor valuation \(P\) over \(S\), there is a rational
Gaussian extension to \(k(R)\), centred on \(\Spec R\), whose restriction
is \(P\) and whose value on \(z^\chi\) is \(P(L)\). It is divisorial
up to scaling. The pair \((S,B)\) is klt log Fano, and \(B\) is effective
with coefficients in \(\{1-1/j:j\in\Z_{>0}\}\).
\end{lemma}
\begin{proof}
Impose nonnegativity on finitely many eigenfunction generators of \(R\).
For an extension of \(P\) these are linear inequalities in the rational
torus-variable weights, with the one weight on \(z^\chi\) fixed to
\(P(L)\). The criterion for a solution is that every nonnegative
combination eliminating the other weights has nonnegative right-hand
side. By rational polyhedrality it suffices to test rational
combinations. Clear denominators and multiply the corresponding
eigenfunctions. Their total weight is a multiple of \(\chi\); it is a
nonnegative multiple because the occurring cone is pointed. The required
inequality is exactly \eqref{eq:A9} pulled back to \(P\). The rational
linear system therefore has a rational solution. Its Gaussian valuation
has rational rank one and satisfies Abhyankar equality, so it is
divisorial up to scaling. Formula~\eqref{eq:A6} gives its discrepancy
\[
A_B(P)=a_{\omega_S}(P)+P(L)>0,
\]
where positivity follows from klt of \(R\). Thus \((S,B)\) is sub-klt;
we next verify effectivity and the coefficients.
For a strict prime \(P\) on \(S\), first lift \(P\) to a homogeneous
prime divisor \(P'\) of the affine ray cone. Use the punctured cone for
a degree-one generated Veronese, and then the finite map from the full
ray cone. The affine subtorus quotient \(\Spec R\to\Spec R_\chi\) is
surjective: projection to invariant summands splits the inclusion as an
\(R_\chi\)-module. Localize at the generic point of \(P'\) and pull
back its uniformizer. A height-one minimal prime of this nonzero
parameter ideal gives a prime divisor \(E\) on \(\Spec R\) dominating
\(P'\). Its contraction is the maximal ideal of that discrete valuation
ring. A connected torus preserves each of these finitely many components,
so \(E\) is invariant. Its restriction is \(jP\) for an integer
\(j>0\), by Lemma~\ref{a:abhyankar}.
Choose a section in a sufficiently divisible degree whose corresponding
section of \(L\) is nonvanishing at the generic point of \(P\).
It has order zero at the ray-cone divisor and at its dominating lift,
which gives \(E(z^\chi)=jP(L)\). As \(E\) is a strict prime of the
variety \(\Spec R\) with zero canonical divisor, \(A_R(E)=1\).
The leading-form formula now yields
\[
1=jA_B(P).
\]
The coefficient of \(B\) at \(P\) is therefore \(1-1/j\), proving
effectivity and the asserted standard coefficients. Together with all
prime tests already proved, this makes \((S,B)\) klt. Finally
\(-K_S-B=L\) is ample, so it is log Fano.
\end{proof}
Test the lc divisor \(\Div(\bar h)/n\) on the Gauss lifts of
Lemma~\ref{a:gauss-lifts}. This gives, for every prime over \(S\),
\begin{equation}
A^+(P):=A_B(P)-P(H)
=a_{\omega_S}(P)-\tfrac1n P(f_h)\geq0.
\label{eq:A10}
\end{equation}
The positive values of \(A^+\) on primes have gap \(1/n\): form orders
and function orders are integers. These are also generalized Calabi--Yau
data with effective boundary \(B+H\), principal nef b-divisor
\(-\frac1n\Div(f_h)\), and total adjoint divisor literally zero. The
principal divisor here is taken on every model. It is rational Cartier
and numerically trivial, so is b-nef. Indeed
\(K_S+B+H=\frac1n\Div_S f_h\).
For an invariant prime divisor \(E\) on \(\Spec R\), write
\(h_E=E(\bar h)/n\geq0\). When its restriction is nonzero,
Lemma~\ref{a:abhyankar} writes it as \(jP\), \(j\in\Z_{>0}\).
Then
\begin{equation}
E|_{k(S)}=jP\ne0
\quad\Longrightarrow\quad
1=jA^+(P)+h_E,\qquad jP(H)\leq h_E.
\label{eq:A11}
\end{equation}
For the equality, Gaussianity and \eqref{eq:A6} give
\[
1=j a_{\omega_S}(P)+E(z^\chi),\qquad
h_E=\tfrac jnP(f_h)+E(z^\chi).
\]
For the inequality, choose sufficiently divisible sections of \(L\)
avoiding the centre of \(P\). Their nonnegative values under \(E\),
together with \eqref{eq:A9}, give \(E(z^\chi)\geq jP(L)\).
If the restriction is trivial, the valuation is a pure toric
cocharacter and \(h_E=E(\chi)=A_R(E)=1\).
\subsection{Weak essentiality by maximality of the torus rank}
\begin{proposition}[Weak essentiality]\label{a:weak-essentiality}
On the chosen maximal-rank sequence, there are numbers \(\eta\to0\)
such that, for every prime divisor \(P\) over \(S\) with \(A^+(P)=0\),
\begin{equation}
\sup_{H'\geq0,\ H'\sim_{\Q}H} P(H')
\leq(1+\eta)P(H).
\label{eq:A12}
\end{equation}
\end{proposition}
\begin{proof}
Otherwise pass to a subsequence carrying lc primes \(P\) and effective
divisors \(H'\sim_{\Q}H\) with
\(P(H')>(1+c_*)P(H)\), for a fixed \(c_*>0\).
Here \(P(H)=A_B(P)>0\). The lift from
Lemma~\ref{a:gauss-lifts} has bare discrepancy \(P(H)\) and discrepancy
zero for \(\Div(\bar h)/n\). Normalize it to bare discrepancy one.
By our general choice of \(h\), it is also an lc place of \(I_R^{1/n}\).
Average this normalized lift and the grading valuation \(\tilde v\)
on eigenfunctions, extending by the Gaussian minimum rule. The resulting
valuation \(u_*\) restricts to \(P/(2P(H))\) on the invariant field.
It is quasi-monomial, has bare discrepancy one by \eqref{eq:A6}, and
is an lc place of \(I_R^{1/n}\). For the last assertion, the core
weight is pure: every core function has value at least \(n\) at the
lift and exactly \(n\) at the grading, with equality for \(\bar h\).
Also \(u_*\geq\tilde v/2\) on regular functions, so it is centred at
the vertex and
\[
\vol(u_*)\leq2^p\vol(\tilde v).
\]
Write \(H'=L+\frac1m\Div_S f'\) with \(m>0\) integral, clearing
rational denominators. By \eqref{eq:A9}, \(g=f'z^{m\chi}\in R_{m\chi}\).
Its value divided by \(m\) is \(P(H')/P(H)>1+c_*\) at the normalized
lift, and one at the grading. Hence
\[
u_*(g)/m>1+c_*/2.
\]
Choose rational \(1<c_0<1+c_*/2\) and a fixed rational \(\lambda>0\)
large enough that
\[
1+\lambda(c_0-1)>2C,
\]
where \(C\) is the old uniform constant in \eqref{eq:A8}.
On \(\Spec R\), minimize subject to the same lc constraint, now with
residual discrepancy
\[
a(u)=(1+\lambda c_0)A_R(u)
-\lambda\min\{u(g)/m,c_0u(I_R)/n\}.
\]
This has the ideal form of Proposition~\ref{a:exact-penalization}.
Explicitly take an integer \(q_0\) for which \(q_0/m\) and
\(q_0c_0/n\) are integral and put
\(J=(g^{q_0/m})+I_R^{q_0c_0/n}\); the subtracted term is
\((\lambda/q_0)u(J)\). Since the constraint ideal is lc,
\(u(I_R)/n\leq A_R(u)\), and consequently
\[
A_R(u)\leq a(u)\leq(1+\lambda c_0)A_R(u).
\]
The ideals are \(T\)-invariant, so the exact minimizing valuation is
\(T\)-invariant. Normalize it by \(A_R(u)=1\).
This \(u\) cannot be a cocharacter valuation in \(T\). For any centred
cocharacter competitor with bare discrepancy one, pure weights give
\(u(g)=m\) and \(u(I_R)=n\), hence
\(a=1+\lambda(c_0-1)\). Such competitors are lc places of the system;
\eqref{eq:A8} gives their volume at least
\(C^{-p}\vol(\tilde v)\). Their objective thus strictly exceeds
\(2^p\vol(\tilde v)\), while \(a(u_*)=1\) and the displayed volume
bound makes \(u_*\)'s objective at most that number. This contradicts
minimality if \(u\) were a cocharacter.
We have obtained a minimizer that is not a cocharacter of the old torus.
To contradict maximality, we must realize its joint graded ring on the
original \(W\) and preserve the system value \(n\), concentration
\eqref{eq:A7}, and the uniform comparison \eqref{eq:A8}.
Let \(R'=\gr_uR\). Stable degeneration for each sufficiently small
auxiliary klt parameter gives finite generation, normality and, after
removing boundaries, klt of \(R'\). Use the joint effective torus of
\(T\) and the new value torus. By
Lemma~\ref{a:canonical-specialization}, its canonical generator is
\((\Omega_{\tilde v})_u\); call its character \(\chi'\). It restricts
to \(\chi\) on \(T\), and \(u(\chi')=1\).
Choose the old core equations in \(R_{n\chi}\) for the exact-minimum
construction. Their next initials have old weight \(n\chi\) and
\(u\)-value \(n\), hence joint weight \(n\chi'\). The old and new
weights determine characters of the joint torus, so this is an equality
of characters, not only of their evaluations.
Let \(I_{R'}=(R'_{n\chi'})\). The core and normalized minimum remain
fixed as the auxiliary parameter tends to zero. Boundary specialization
and this pure weight therefore imply that \(I_{R'}^{1/n}\) is lc,
exactly as in the initial construction. The same auxiliary klt pairs give
a new uniform version of \eqref{eq:A8}, initially relative to grading
\(u\). Indeed at a centred lc place \(q\) of the new system, the core
term is at most zero; all perturbing boundary subtractions are
nonnegative, so the auxiliary discrepancy before common scaling is at
most
\[
t'(1+\lambda c_0)A_{R'}(q).
\]
At the minimizing grading \(u\) it equals \(t'a(u)\), which is at
least \(t'\). Formula~\eqref{eq:A3} cancels the parameter and yields
a sequence-uniform comparison with \(u\), since \(\lambda,c_0,p\) are
fixed.
On the old ring, \eqref{eq:A8} applies to the normalized lc place \(u\)
and gives \(u(f)\leq C\tilde v(f)\). On each new joint piece this says
that its new \(u\)-weight is at most \(C\) times its old
\(\tilde v\)-weight: use a homogeneous lift of that piece to \(R\).
We now flatten the two successive valuations on the actual original
\(W\), by Lemma~\ref{a:flattening}, and divide by \(1+\xi\).
This preserves \(D\), the iterated canonical generator, and discrepancy
one. The parameter \(\xi\) may be chosen individually for each member;
all bounds below use fixed factors.
Include a finite degree-\(n\) generating list for the actual ideal
\(I_n\) in the preserved tests. If an old value is greater than \(n\),
its normalized flattened value is still greater for small enough
\(\xi\). If the old value is \(n\), its first initial lies in
\(R_{n\chi}\), so the next value is at least \(n\). Include also a
degree-\(n\) lift attaining equality for this system minimum; such a
lift exists because \(R_{n\chi}\) is precisely that leading space.
The normalized flattened valuation therefore still has
\(\tilde v_{\mathrm{new}}(I_n)=n\).
The all-functions comparison in Lemma~\ref{a:flattening}, with
\(0<\xi<1\), bounds the new actual valuation below by
\(\tilde v/4\). Thus \eqref{eq:A7} persists. On each new joint weight,
its normalized value is
\[
\frac{\text{old weight}+\xi\,\text{new }u\text{-weight}}{1+\xi}
\geq\tfrac12\,\text{old weight}
\geq\tfrac1{2C}\,\text{new }u\text{-weight}.
\]
Both grading valuations use the minimum rule, so the same comparison
holds on all elements of \(R'\). The new comparison relative to \(u\)
therefore gives \eqref{eq:A8} relative to the normalized flattened
grading, with a uniform enlarged constant. All required sequence
properties survive.
The joint torus has strictly larger rank. Every old occurring character
still occurs after taking the invariant associated graded, so \(T\)
continues to act faithfully. If the \(u\)-grading belonged to this old
torus, each nonzero element in an old weight space would have its
\(u\)-value determined by that character and one fixed linear
functional. Invariance and the minimum rule would then make \(u\)
itself a cocharacter valuation on \(R\), which was excluded above.
Thus the rank increases. Passing to a constant-rank subsequence gives
an admissible sequence of larger rank, contrary to its maximal choice.
This proves \eqref{eq:A12}.
\end{proof}
Fix this sequence from now on. The dual section
\[
Q_T=\{\rho\in\Hom(M_T,\R):
\rho\geq0\text{ on occurring weights},\ \rho(\chi)=1\}
\]
is compact, because \(\chi\) is interior and the occurring cone is
full-dimensional. It contains \(D\), and its point \(\tilde v\) obeys
\[
\tilde v+c(\tilde v-Q_T)\subseteq Q_T
\]
for a uniform \(c>0\). To see this, apply \eqref{eq:A8} first to
interior cocharacters \(\rho\). They are centred at the vertex and are
lc places of the pure system, of bare discrepancy one. Therefore
\(\rho(\mu)\leq C\tilde v(\mu)\) for every occurring character
\(\mu\). Continuity extends this to all \(Q_T\); take, for example,
any \(c>0\) with \(c(C-1)\leq1\).
Use the integral splitting
\[
M_T=\Z\chi\oplus M_0,\qquad M_0=\ker D.
\]
At each invariant prime divisor \(E\) on \(\Spec R\), the values
\(E(z^\beta)\) are integral linear functions of \(\beta\in M_0\).
These primes, with the data in \eqref{eq:A11}, test regularity of
eigenfunctions by normality: any pole divisor of an eigenfunction is
invariant, and a normal affine ring is the intersection of its
height-one valuation rings.
\subsection{Angular forms and section costs}
\label{a:angular}
Distinguish the quotient function \(f_h\) on \(S\) from the actual
rational function
\[
f_h^W=h/t^n\in k(X)=k(Z).
\]
Set
\[
\theta=\frac{\omega}{(f_h^W)^{1/n}},\qquad
V_m=h^{-m/n}\mathcal A_m\subset k(X)
\quad(m\in n\Z_{\geq0}).
\]
The form is fractional, interpreted by its \(n\)-th tensor power.
The section systems multiply and are nested, since multiplication by
\(h\) identifies \(V_m\subseteq V_{m+n}\). For
\(v=\tilde v|_{k(X)}\), the Gaussian cone formula gives
\(A_\theta(v)=0\). On \(0\ne F\in V_m\), write its cost as
\[
b(F)=m+v(F),
\]
which is the value of its actual degree-\(m\) section under
\(\tilde v\). The label \(m\) is retained when an element belongs
to more than one system.
More generally, for a quasi-monomial valuation \(w\) of \(k(X)\), the
Gauss extension assigning \(h\) value \(nB_*\) has
\[
A_\Omega=A_\theta(w)+B_*,\qquad
\text{section value}=w(F)+mB_*\quad(F\in V_m).
\]
Indeed its value on \(t\) is \(B_*-w(f_h^W)/n\); substitution in the
cone formula gives the displayed identities.
The full initial field for \(v\) is
\[
k(X)_v=k(R)^D=k(S)(z^\beta:\beta\in M_0).
\]
This follows from Gaussianity of \(\tilde v\) in \(t\): its initial
field is the full initial field downstairs with one independent initial
variable of \(D\)-degree one. The leading \(V_m\)-spaces are the
leading degree-\(m\) pieces of \(R\) divided by \(\bar h^{m/n}\).
Apply \eqref{eq:A6} to the fractional identity
\[
\Omega/h^{1/n}=\theta\wedge d\log t.
\]
For a basis of \(M_0\), the result is
\[
\theta_v=\pm\frac{\omega_S\wedge
d\log\mathbf z_0}{f_h^{1/n}}.
\]
In fact \(d\log\bar t=d\log z^\chi\) modulo differentials from
\(k(R)^D\), since both variables have \(D\)-degree one; wedging with
the full downstairs form kills their difference.
Finally let \(w\) be any quasi-monomial valuation centred on \(X\),
including one lying over \(z\). The general divisor
\(\frac1n(\Div_X f_h^W-nK_X)\) is lc, so \(A_\theta(w)\geq0\), and
\begin{equation}
\begin{split}
A_X(w)&=A_\theta(w)
+\tfrac1n w(\Div_X f_h^W-nK_X),\\
\tfrac1n w(\Div_X f_h^W-nK_X)
&=\sup_{\substack{m\in n\Z_{>0}\\0\ne F\in V_m}}
\frac{-w(F)}m.
\end{split}
\label{eq:A13}
\end{equation}
Effectivity of every degree-\(m\) section divisor gives the upper bound
for the supremum. Conversely, sufficiently divisible multiples of
\(-K_X\) are relatively free. Over the affine base choose a section
avoiding the centre of \(w\); its effective section divisor has value
zero there and attains the asserted value. This proves the equality
and completes the angular data used below.
Equation~\eqref{eq:A13} specifies the final task. A lower bound close
to zero for all positive-degree ratios \(w(F)/m\), together with small
\(A_\theta(w)\), gives small \(A_X(w)\). To use \(\epsilon\)-log
canonicity, we also need a uniformly bounded positive integer \(e\) for
which \(ew\) is a positive integral multiple of a prime-divisor valuation.
We will obtain this control from independent initial coordinates with
integral values and the section counts. The tower and section estimates
in the next four sections supply the information needed to construct
such a valuation on the original field.
==== END Bounded-klt-complements-for-Fano-contractions//essential-system.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//initial-cone.tex ====
\section{The anticanonical cone and a distinguished lc place}
\label{a:cone}
We work with the contradiction sequence introduced after
\eqref{eq:A1}. Thus \(X\to Z\) is birational, \(Z\) is affine,
\(-K_X\) is relatively ample, \(z\in Z\) is closed, and
\(d=\dim X>0\). The integer \(n>1\) is fixed along the sequence and
\(n^{-1}|-nK_X|\) is an lc system. Put \(p=d+1\).
All valuations in this section are trivial on \(k=\C\).
\subsection{Valuations and forms}
We shall use the full initial field of a valuation, as well as its residue
field; these are different objects. We first record the monomial facts
needed to distinguish them. The smooth monomial description and its
finite-ideal refinements are standard in valuation theory; see
\cite[Proposition~3.7 and Lemma~3.6]{JM}. We include the argument in the
form needed for restrictions to subfields.
\begin{lemma}[Monomial charts and restriction]\label{a:abhyankar}
A real valuation of a function field of characteristic zero satisfying
Abhyankar equality is quasi-monomial on a smooth proper model.
The model can dominate a prescribed model and resolve finitely many
additional functions and divisors. Its restriction to a subfunction field
again satisfies Abhyankar equality. In particular, the restriction of a
prime-divisor valuation is either trivial or a positive integral multiple
of a prime-divisor valuation.
\end{lemma}
\begin{proof}
Choose functions with rationally independent values spanning the value
group over \(\Q\), and functions of value zero whose residues form a
transcendence basis of the residue field. Resolve the divisors of the
first functions and the maps to \(\PP^1\) defined by the second, together
with the prescribed data. The latter functions are regular and invertible
at the centre. Consequently the dimension of the centre is at least the
residue transcendence degree. At least as many transverse boundary
components as the rational rank pass through it: the independent values
of the first functions are integral combinations of their orders.
Abhyankar equality forces equality in both bounds. The centre is therefore
the generic point of their transverse intersection, and the normal
parameters have rationally independent values.
In the completed local ring at that generic point, each nonzero function
has a unique least-weight normal monomial with nonzero residue coefficient.
The monomial ideal of strictly higher weight is finitely generated.
The congruence to that monomial times a unit contracts to the local ring
by faithful flatness. This proves the monomial formula.
For a restriction, add the Abhyankar inequalities for the restricted
valuation and the valued field extension. Equality for the full field
forces equality for the restriction. A nonzero subgroup of the value
group \(\Z\) of a prime divisor is \(j\Z\), \(j>0\). The restricted
Abhyankar valuation of rational rank one is therefore \(j\) times a
prime-divisor valuation.
\end{proof}
For a rational top form \(\eta\), its \emph{form discrepancy} is
\(A_\eta(E)=1+\ord_E(\eta)\) at a prime divisor, extended by the
log smooth formula to quasi-monomial valuations. We also write \(a_\eta\).
For a fractional form whose \(r\)-th power is a rational \(r\)-canonical
form, use \(1+r^{-1}\ord_E(\eta^{\otimes r})\).
The top wedge of logarithmic normal differentials and regular tangential
differentials has form discrepancy zero at the monomial valuation;
multiplication by a rational function adds its value.
Pair discrepancy is form discrepancy minus the value of the
log canonical divisor expressed with that form. The trivial valuation
has discrepancy zero. Precise leading-form formulas are proved in
Lemma~\ref{a:leading-form}.
\subsection{The cone and its canonical generator}
Fix a rational volume form \(\omega\) on \(k(X)=k(Z)\) and use
\(K_X=\Div_X(\omega)\). For an indeterminate \(t\), set
\[
\mathcal A_m=
\{g t^m:\Div_X(g)-mK_X\geq0\}\cup\{0\},\qquad
\mathcal A=\bigoplus_{m\geq0}\mathcal A_m,\qquad
W=\Spec\mathcal A .
\]
Here \(\mathcal A_0=k[Z]\). Denote the degree cocharacter by \(D\), and
the point of the vertex section above \(z\) by \(o\).
\begin{lemma}[Canonical anticanonical cone]\label{a:canonical-cone}
The algebra \(\mathcal A\) is finitely generated and normal, with fraction
field \(k(X)(t)\). The variety \(W\) is klt, and
\[
\Omega=t\,\omega\wedge d\log t
\]
has divisor zero on \(W\). In particular it is a canonical generator,
of \(D\)-weight one.
\end{lemma}
\begin{proof}
Relative ampleness gives finite generation of a sufficiently divisible
Veronese and finite generation of each shifted-degree module over it.
Thus it gives finite generation in all degrees. The ring is the
intersection of \(k(X)[t]\) with the valuation rings imposing
\(\ord_P(g)-mP(K_X)\geq0\) at strict prime divisors \(P\) of \(X\).
Here and below, a strict prime of a specified model means a prime divisor
on that model itself; a prime over the model may lie on any higher
birational model. Thus a divisor extracted on a new model is strict on
that model. The displayed intersection is therefore normal. Over the birational generic point of \(Z\)
its field is \(k(X)(t)\).
Consider
\[
\widehat W=\Spec_X\bigoplus_{m\geq0}\OO_X(-mK_X)t^m .
\]
For a sufficiently divisible \(r\), the analogous relative spectrum in
degrees divisible by \(r\) is the total space of a line bundle. It is
proper over the Veronese cone: evaluation by generating sections embeds
it, relative to \(X\), into a product with affine space over \(Z\), and
\(X\to Z\) is proper. The original relative spectrum is finite over this
line-bundle model. Since \(W\) is separated over its Veronese cone, the
resulting map \(\widehat W\to W\) is proper. It is birational by the
fraction-field calculation. The same prime inequalities give normality
of \(\widehat W\).
Over the smooth locus of \(X\), if \(s\) is a local rational function with
\(\Div(s)=K_X\), the regular fibre coordinate is \(st\), and the displayed
form is a nowhere-vanishing top form. The complement of this locus in
\(\widehat W\) has codimension at least two, as is also seen by the finite
map to the line-bundle model.
For completeness, the local singularities of \(\widehat W\) are klt.
Trivialize \(rK_X\) locally and take the normalized cyclic cover adjoining
\(s\) with \(s^r\) equal to its rational trivializing function.
Because the coefficients of the Weil divisor \(K_X\) are integral,
this cover is quasi-etale. Finite crepant pullback preserves klt
singularities. On the index cover, form the line-bundle algebra
(the normalized reflexive pullback) with coordinate \(y=st\). The deck group acts on \(y\)
with the same character as on \(s\), keeping \(t\) invariant; the form
\(\Omega=(\omega/s)\wedge dy\) is invariant. Its cyclic invariants
recover the original algebra, by the codimension-one regularity inequalities.
The cover of total spaces is again quasi-etale. The finite-cover
discrepancy formula consequently proves that \(\widehat W\) is klt.
We have \(\Div_{\widehat W}(\Omega)=0\), and pushing forward gives
\(\Div_W(\Omega)=0\). Thus the proper birational map is crepant, and
\(W\) is klt as well.
\end{proof}
\begin{lemma}[Gaussian discrepancy]\label{a:cone-gauss}
For a prime divisor \(P\) over \(X\), extend \(P\) by the Gauss rule with
\(t\)-weight \(q-P(K_X)\). If \(q\geq0\), the extension \(\widetilde P\)
is centred on \(W\), and
\[
\widetilde P(gt^m)=mq+P(\Div_X(g)-mK_X),\qquad
A_W(\widetilde P)=q+A_X(P).
\]
If \(q\) is rational it is divisorial up to positive scaling.
Every \(D\)-invariant divisorial valuation centred on \(W\) has this form,
allowing a positive multiple of \(P\), or has trivial restriction to
\(k(X)\), in which case its discrepancy is \(q\).
The ideal \(I_n=(\mathcal A_n)\subset\mathcal A\) satisfies
\((W,I_n^{1/n})\) lc.
\end{lemma}
\begin{proof}
The value formula follows from the assigned weight of \(t\).
The divisor \(\Div_X(g)-mK_X\) is effective and rational Cartier,
so the values are nonnegative. A Gauss extension is Abhyankar; wedging
with \(d\log t\) preserves form discrepancy, while multiplication by
\(t\) adds \(q-P(K_X)\). Since
\(A_\omega(P)=A_X(P)+P(K_X)\), this proves the discrepancy formula.
The rational-weight assertion follows from Lemma~\ref{a:abhyankar}.
An invariant valuation evaluates a sum of different torus characters
by their minimum: finite torus translates separate the character
components. It is therefore Gaussian. Properness gives a centre on
\(X\) for its restriction, and Lemma~\ref{a:abhyankar} applies.
The same formulas hold for this restriction and for the trivial one.
The corresponding \(q\) is nonnegative: a sufficiently divisible
relatively free anticanonical section can be chosen with zero divisor
order at the centre, and its value on \(W\) is its degree times \(q\).
The lc assertion reduces to
\[
\frac1n P\bigl(|-nK_X|\bigr)\leq A_X(P),
\]
which is our choice of \(n\). Invariant divisors suffice: take an
equivariant log resolution of the invariant ideal; the connected torus
fixes its finitely many prime components.
We shall also use general members of these systems. On a log resolution,
the fixed part is the ideal's order and the residual system is free.
A general member with coefficient \(1/n<1\) is transverse to the
fixed support and preserves the lc inequalities. The same choice in a
finite-dimensional generating subspace can therefore be made lc on
both \(X\) and \(W\).
\end{proof}
\subsection{An lc place at the closed vertex}
We need a bounded hypersurface statement only for a fixed-index klt germ.
It is the argument used in \cite[proof of Proposition~5.1]{Chen}.
\begin{lemma}[A hypersurface on a fixed-index germ]\label{a:index-hypersurface}
Given \(p\geq2\) and an integer \(N>0\), there is
\(\tau(p,N)>0\) with the following property. If \((U,G)\) is klt near a
closed point \(x\), \(G\geq0\), \(\dim U=p\), and \(N(K_U+G)\) is
Cartier, then, after shrinking at \(x\), an effective Cartier divisor
\(H_x\) through \(x\) satisfies \((U,G+\tau(p,N)H_x)\) lc.
\end{lemma}
\begin{proof}
Take a plt blowup extracting only \(E\) over \(x\), as in
\cite[Lemma~2.27]{Chen}. For its strict boundary \(G_Y\),
\(-E\) and \(-(K_Y+G_Y+E)\) are relatively ample rational Cartier
divisors. Decreasing the coefficient of \(E\) slightly gives a klt
relative log Fano pair, so \(Y\) is of Fano type over \(U\).
The coefficients of \(G_Y\) belong to the finite \(1/N\)-grid below one.
The relative monotone lc complement theorem
\cite[Theorem~1.8]{BirkarComplements} gives a bounded-index lc complement
of \(G_Y+E\) near \(x\). Its coefficient at \(E\) remains one.
Push the complemented boundary down. The bounded-index principal
identity pushes down as well and makes the pullback crepant, so the
result is an lc boundary \(G^+\geq G\) with \(E\) an lc place.
For a bounded common multiple \(L\) of its complement index and \(N\),
the divisor \(H_x=L(G^+-G)\) is effective Cartier.
It passes through \(x\): its order at \(E\) is
\(L A_{(U,G)}(E)>0\). Taking \(\tau=1/L\), and then a common smaller
positive choice over the finitely many bounded indices, proves the claim.
\end{proof}
In the next proposition, \(N\) is a cutoff tending to infinity along
the contradiction sequence; it is independent of the final complement
index in Theorem~\ref{thm:main}.
\begin{proposition}[A centred lc place detecting the bad sequence]
\label{a:initial-lc-place}
After passing to a subsequence, there are divisorial valuations \(u^0\)
centred at \(o\), and integers \(N\to\infty\), such that
\begin{equation}
u^0(I_n)=nA_W(u^0),\qquad
u^0(\m_z)\geq N A_W(u^0).
\label{eq:A2}
\end{equation}
Here and below \(\m_z\) denotes also its extension to \(\mathcal A\).
\end{proposition}
\begin{proof}
Choose \(N\) divisible by \(n\), tending to infinity slowly enough that
the bad divisors in the contradiction sequence satisfy
\[
\frac{P(\m_z)}{A_X(P)}>\frac{N}{\tau(p,N)}.
\]
This is a diagonal choice: each fixed \(N\) is allowed before passing
sufficiently far along the bad sequence. For each individual algebra,
choose \(M>N\) sufficiently divisible that the degree-\(M\) elements
cut out the irrelevant locus set-theoretically. Set
\[
I=(\mathcal A_N,\m_z,\mathcal A_M).
\]
Its radical is
\(\m_o=\m_z\mathcal A+\mathcal A_{>0}\).
Since \(I_n^{N/n}\subset I\), Lemma~\ref{a:cone-gauss} implies
\((W,I^{1/N})\) lc.
Suppose it were klt near \(o\). A general element \(g\) of a finite
generating space of \(I\) would give
\(G=N^{-1}\Div(g)\) klt there. Indeed the moving part has coefficient
\(1/N<1\) on a log resolution. The canonical generator of \(W\) shows
that \(N(K_W+G)\) is Cartier. Lemma~\ref{a:index-hypersurface} supplies
an effective Cartier hypersurface \(H_o\) through \(o\) for which
\((W,G+\tau(p,N)H_o)\) is lc.
Choose rational \(q\) with
\[
\frac{A_X(P)}{\tau(p,N)}<q\leq\frac{P(\m_z)}N .
\]
The Gauss extension \(\widetilde P\) has positive value on every
positive-degree element and on \(\m_z\), hence is centred at \(o\).
It has value at least \(Nq\) on \(I\), whereas its bare discrepancy is
\(q+A_X(P)\). Consequently
\[
A_{(W,G)}(\widetilde P)\leq A_X(P).
\]
Moreover \(\widetilde P(\m_o)\geq q\): each positive-degree homogeneous
term has value at least its degree times \(q\), and each degree-zero
term vanishing at \(z\) has value at least \(Nq\).
A local equation of \(H_o\) therefore has value at least \(q\).
This contradicts lc, since
\[
A_{(W,G)}(\widetilde P)\leq A_X(P)
<\tau(p,N)q\leq\tau(p,N)\widetilde P(H_o).
\]
Thus \(I^{1/N}\) is lc but not klt at \(o\). An lc divisor on a log
resolution is centred there, because \(I\) is supported at \(o\) and
\(W\) is klt elsewhere. For this divisor,
\[
A_W(u^0)=\frac{u^0(I)}N
\leq\frac{u^0(I_n)}n\leq A_W(u^0).
\]
Both inequalities are equalities. Since \(\m_z\subset I\), the second
inequality in \eqref{eq:A2} follows too.
\end{proof}
==== END Bounded-klt-complements-for-Fano-contractions//initial-cone.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//introduction.tex ====
\section{Introduction}
\label{sec:introduction}
Let $f\colon X\to Z$ be a projective contraction of normal varieties,
meaning that $f$ is surjective and $f_*\OO_X=\OO_Z$. A complement near a
closed point $z\in Z$ is an effective rational boundary $B$ for which the
adjoint divisor $K_X+B$ becomes torsion on the inverse image of a
neighbourhood of $z$. More precisely, an index $N$ records that
$N(K_X+B)$ is integral Cartier and linearly trivial there. The complement
is klt if every log discrepancy of $(X,B)$ is positive. We ask whether one
index works for all contractions of a fixed dimension whose singularities
have a fixed positive lower discrepancy bound.
When $K_X$ is rational Cartier and $-K_X$ is ample over $Z$, we call $f$ a
\emph{Fano contraction}. Complements turn this positivity into an adjoint
divisor with controlled torsion. The distinction between lc and klt
complements is essential: an lc complement permits zero discrepancies,
whereas a klt complement must keep every discrepancy strictly positive.
We prove that a uniform positive bound for the singularities of $X$
suffices to achieve the latter conclusion with a bounded index.
\begin{theorem}\label{thm:main}
For every positive integer $d$ and positive rational number $\epsilon$ there
is a positive integer $N=N(d,\epsilon)$ with the following property. Let $k$
be an algebraically closed field of characteristic zero, and let
$f\colon X\to Z$ be a projective surjective morphism of normal integral
$k$-varieties such that $f_*\OO_X=\OO_Z$, $\dim X=d$, $K_X$ is
$\Q$-Cartier, $X$ is $\epsilon$-lc, and $-K_X$ is ample over $Z$.
For every closed point $z\in Z$ there are an effective $\Q$-divisor $B$ on
$X$ and a Zariski open neighbourhood $U$ of $z$ such that $(X,B)$ is klt over
$U$ and
\[
N(K_X+B)|_{f^{-1}(U)}\sim 0
\]
is an integral Cartier divisor.
\end{theorem}
The morphism in Theorem~\ref{thm:main} need not have relative Picard number
one, nor need its general fibre have positive dimension. The displayed
linear equivalence means triviality of the associated line bundle on
$f^{-1}(U)$, after the indicated shrinking; it is stronger than numerical
triviality. Both $B$ and $U$ may depend on $f$ and $z$, whereas $N$ does not.
A variety is of Fano type over its base if it admits an effective
$\Q$-boundary $\Delta$ for which $(X,\Delta)$ is klt and
$-(K_X+\Delta)$ is relatively ample. Equivalent versions using a nef and big
anti-log-canonical divisor may be converted to this one by a small boundary
perturbation.
The proof uses the equivalence between bounded klt complements and a local
threshold statement established by Chen \cite[Theorem~1.6]{Chen}. The
birational, boundary-free case of that threshold statement remains the main
issue. We formulate it in Proposition~\ref{prop:birational-threshold} and
prove it by contradiction. The role of bounded lc complements is different:
they supply an auxiliary linear system of a fixed degree $n$. Throughout the
proof, this $n$ is distinguished from the final klt complement index $N$.
\subsection{Earlier results and the finite-coefficient conjecture}
Shokurov introduced complements in his work on threefold log flips
\cite[Section~5]{ShokurovFlips} and formulated their boundedness
systematically in his surface theory \cite[Conjecture~1.3]{ShokurovSurfaces}.
Prokhorov--Shokurov developed the passage from global complements to local
ones and the higher-dimensional inductive program
\cite{ProkhorovShokurovFirst,ProkhorovShokurov}.
Shokurov's later treatment includes a bounded lc complement theorem
assuming the existence of a klt real complement
\cite[Theorem~3]{ShokurovComplements}. Here the klt condition concerns
the input real complement; the bounded-index output is lc.
An lc complement may have places of zero discrepancy. Reducing its
boundary coefficients need not preserve the linear triviality of the
adjoint divisor, so keeping the complement klt requires an additional
argument.
Birkar proved boundedness of lc complements for Fano type fibrations with
hyperstandard coefficients \cite[Theorems~1.7 and~1.8]{BirkarComplements},
and boundedness of klt complements for boundary-free Fano fibrations over curves
\cite[Corollary~1.4]{BirkarFibrations}. Chen related bounded klt complements
to uniform local hypersurface thresholds and established the conjecture for
bases of dimension at most two
\cite[Theorem~1.6 and Corollary~1.8]{Chen}.
Theorem~\ref{thm:main} also gives the following consequence in Chen's
complex quasi-projective setting.
\begin{corollary}\label{cor:finite-coefficients}
Fix a positive integer $d$, a real number $\epsilon>0$, and a finite set
$I\subset[0,1]\cap\Q$. There is a positive integer $M=M(d,\epsilon,I)$
with the following property. Let $(X,\Delta)$ be an $\epsilon$-lc pair of
dimension $d$ with coefficients in $I$, over $\C$. Let $f\colon X\to Z$
be a contraction of quasi-projective normal varieties, assume that $X$ is
of Fano type over $Z$, and assume that $-(K_X+\Delta)$ is nef over $Z$.
For every closed $z\in Z$, there is an effective $\Q$-divisor
$\Delta^+\ge\Delta$ such that, near $f^{-1}(z)$, the pair
$(X,\Delta^+)$ is klt and $M(K_X+\Delta^+)$ is Cartier and linearly
trivial after shrinking the base about $z$.
\end{corollary}
The proof, using Chen's equivalence and a crepant ample model, is given
in Section~\ref{sec:reduction}.
Corollary~\ref{cor:finite-coefficients} is the finite-rational-coefficient
form of Shokurov's bounded-klt-complement conjecture stated in
\cite[Conjecture~1.1]{Chen}. The same equivalence yields the local
hypersurface assertions in \cite[Conjectures~1.3 and~1.4]{Chen}.
The conclusion concerns klt complements; the stronger requirement of an
$\epsilon$-lc complement in \cite[Remark~1.2(1)]{Chen} is not asserted here.
\subsection{Methods and related work}
Our first tools come from normalized-volume minimization. Li established
properness estimates \cite{LiProperness}; Blum proved existence of a
minimizer \cite{BlumExistence}; Xu proved its quasi-monomiality
\cite{XuQuasimonomial}; and Xu--Zhuang proved uniqueness and finite
generation \cite{XuZhuangUniqueness,XuZhuangStable}. Li--Xu identifies
the minimizing Reeb valuation on the resulting K-semistable cone
\cite{LiXuStable}. We use this ordinary klt theory after proving an
additional exactness statement: one fixed small rational perturbation
has its actual global minimizer on the prescribed lc constraint.
This is Proposition~\ref{a:exact-penalization}; its constants are allowed
to depend on the individual germ and ideals.
The projective stage combines minimal models and Mori dream spaces
\cite{BCHM,HuKeel}, Ambro's canonical bundle formula
\cite{AmbroBoundary,AmbroModuli}, and Birkar's boundedness and
threshold theorems \cite{BirkarBAB,BirkarPolarised,BirkarFibrations}.
The needed uniformity concerns both divisor classes and effective
representatives. We retain those representatives in marked bounded
families before passing to generic fibres or taking limits.
The final stage uses Takagi's comparison between multiplier and test ideals
\cite{Takagi}, in the dense-open formulation recalled by
de Fernex--Docampo--Takagi--Tucker \cite[Theorem~11]{DFDTT}, together
with Schwede's Cartier-algebra formulation
\cite{SchwedeNonQG}. These results supply trace generation for a fixed
rational pair. The proof below additionally retains the finite output
spaces and their valuation bounds through two successive liftings.
Related local Cartier-threshold bounds under a positive normalized-volume
lower bound appear in \cite[Theorem~1.7]{HanLiuQiLocalVolumes}.
The companion article \cite[Theorem~1]{OpenAICartier} proves a uniform
Cartier-section threshold for rational-boundary Fano type contractions
by an independent character and pinning argument. Its endpoint is a
Cartier divisor through a prescribed base point with a uniformly lc
pullback. The proof here develops the complement index directly through
the birational threshold and the trace estimates above.
\subsection{Outline of the proof}
The reduction in Section~\ref{sec:reduction} asks for a uniform lower
bound on the log canonical threshold of the pullback of a closed base
point's maximal ideal in the birational, zero-boundary case. A sequence
violating this bound has divisors whose
orders on that ideal are arbitrarily large compared with their discrepancies.
We will construct a prime divisor of discrepancy less than $\epsilon$.
Small discrepancy at a real valuation is not by itself enough, since
rescaling the valuation also rescales its discrepancy. The proof must
therefore control both discrepancy and the eventual divisor normalization.
\smallskip\noindent
\emph{From concentration to a graded model.}
After shrinking each base to an affine neighbourhood of $z$ and replacing
the sequence by its crepant ample models,
Sections~\ref{a:cone}--\ref{a:essential} encode anticanonical sections in
an affine cone. Its bounded-degree lc system has a centred lc place
detecting the concentration of the base ideal. We minimize normalized
volume subject to this lc equality. Proposition~\ref{a:exact-penalization}
shows that the constrained minimizer is exactly the ordinary minimizer of
a small effective klt perturbation. This permits ordinary stable
degeneration to produce a finitely generated graded ring, while a
comparison at the minimizer preserves concentration of the base ideal.
Compatible further filtrations let us maximize the rank of the torus
whose characters record the occurring valuation degrees. Each successive
valuation remains realized on the original cone. The resulting projective quotient carries a
klt pair $(S,B)$ and an effective ample rational divisor
$H\sim_{\Q}-(K_S+B)$ for which $(S,B+H)$ is lc. At each lc place of
this latter pair, replacing $H$ by any effective rationally linearly
equivalent divisor increases its order by at most a factor $1+\eta$,
where $\eta\to0$ along the sequence. This is the weak
essentiality property of Proposition~\ref{a:weak-essentiality}.
We also keep actual sections on $X$. Choose a general degree-$n$
anticanonical section $h$, and let $B_h$ be one $n$-th of its effective
section divisor, so $(X,B_h)$ is lc. For $m\in n\Z_{\ge0}$, dividing
degree-$m$ anticanonical sections by $h^{m/n}$ gives a space of rational
functions $V_m\subset k(X)$. The normalized cone valuation restricts to
a valuation $v$ with $A_{X,B_h}(v)=0$. The cone valuation assigns to a
section the \emph{cost} $b(F)=m+v(F)$ of the corresponding
$F\in V_m$; the assigned
degree $m$ is retained in this notation. Costs are nonnegative,
$V_0=k[Z]$, and $v(\m_z)\to\infty$ along the sequence.
These systems and the exact
discrepancy identity below are constructed in
Subsection~\ref{a:angular}.
\smallskip\noindent
\emph{Upper counts and independent sections.}
Sections~\ref{b:discrepancy-push}--\ref{b:bounded-axes} use weak
essentiality to control these section spaces. A finite tower of projective
contractions removes one small scale of the residual field $k(S)$ at a time.
Horizontal discrepancy bounds give divisor-order estimates on each
relative field. Together with the torus directions, these fields determine
$d$ widths $W_j$: independent rational functions in direction $j$ can be
obtained at degree and cost $O(W_j)$.
The two needed estimates go in opposite directions. At any fixed physical
degree, the sum of initial spaces through cost $T$ has dimension at most
a uniform multiple of $\prod_j(1+T/W_j)$. Monomials in actual sections
with independent initials give lower box bounds at comparable degree and
cost. Here an
initial is the least-valued part of a function in the associated graded
field. The directions whose widths stay bounded need a further estimate:
through a slowly growing cutoff, cost is bounded by a uniform multiple
of physical degree, and
the dimension is $O((1+m)^r)$, where $r$ is their number. To prove this,
Section~\ref{b:bounded-axes} retains effective divisor representatives in
a marked projective family. An obstruction would produce a degree-zero
section of bounded positive cost, contradicting concentration of the
entire base maximal ideal.
\smallskip\noindent
\emph{A small discrepancy with controlled divisor normalization.}
The connection back to the singularities of $X$ is
\[
A_X(w)=A_{X,B_h}(w)+
\sup_{\substack{m\in n\Z_{>0}\\0\ne F\in V_m}}
\frac{-w(F)}m
\]
for quasi-monomial valuations centred on $X$; this is \eqref{eq:A13}.
Thus inequalities
$w(F)+B_*m\ge q\,b(F)$, with $B_*,q>0$, bound the second term by
$B_*$. In degree zero they give nonnegative values on $k[Z]$ and
positive values on $\m_z$, forcing the centre to lie over $z$.
Sections~\ref{c:optimization-section}--\ref{c:rank-section} seek such
valuations with small $A_{X,B_h}(w)$, while preserving independent
initial coordinates. The fractional-form discrepancy $A_\theta$ used
there is precisely $A_{X,B_h}$.
The optimization has a finite test and compact sublevels on a monomial
skeleton. A suitably chosen optimizer gives a strict divisorial budget on a
geometric generic fibre of the full initial field, which retains all
homogeneous degrees, rather than just residue classes. In suitable
positive-characteristic reductions, Cartier trace converts this budget
into sections of controlled degree and value.
Relative trace survival is first lifted to the full initial field and
then to the original field; the finite output spaces are fixed before
reduction to positive characteristic.
Comparing these trace outputs with the upper counts controls both changes
of coordinate values and growth of the number of independent initials.
First the bounded-width coordinates acquire value zero. A further trace
argument then gives two-sided bounds for section values. Through the
remaining width layers, independent initials are added and their values
are rounded to integers, preserving all earlier constraints. Once the
initials form a full transcendence basis, the section counts bound the
denominator of the entire value group. A bounded multiple of the resulting
valuation is an integral multiple of a prime divisor. The final choice of
tolerances and scaling makes that divisor's discrepancy less than
$\epsilon$, completing the contradiction.
\subsection{Conventions}
All varieties in characteristic zero are over an algebraically closed field.
After Section~\ref{sec:reduction} we work over $\C$ through
Section~\ref{b:bounded-axes}. In Section~\ref{c:optimization-section} we
descend the contradiction sequence and its specified data to a countable
algebraically closed field of characteristic zero; later constructions allow
the stated extensions of constants. From Section~\ref{c:trace-section},
positive-characteristic reductions are auxiliary trace calculations on
retained finite data: the chosen valuations, optimization, and strict
discrepancy assertions remain in characteristic zero. A contraction
is a projective surjective morphism with connected fibres in the sense
$f_*\OO_X=\OO_Z$.
We use log discrepancies. For a prime divisor $E$ on a smooth birational model
$\pi\colon Y\to X$,
\[
A_{X,B}(E)=1+\ord_E\bigl(K_Y-\pi^*(K_X+B)\bigr).
\]
Discrepancies, divisor orders, and ideal orders are extended homogeneously to
positive real multiples of valuations. The order of an ideal is the minimum
of the orders of local generators. The notation $A_X$ means $A_{X,0}$.
For a nonzero ideal $\mathfrak a$ on a klt variety,
\[
\lct(X;\mathfrak a)=
\inf_{E(\mathfrak a)>0}\frac{A_X(E)}{E(\mathfrak a)},
\]
where the infimum is over prime divisors over $X$. Rational Cartier divisors
are evaluated by pullback on a model on which the valuation has a centre.
In contradiction sequences we pass to subsequences and shrink the base about
the chosen point without repeating this convention. Constants called uniform
are independent of the sequence index after these operations; their allowed
geometric dependencies will be stated. The number of contractions, scale
layers, and rank increases is bounded. Small positive constants chosen before
a limit are kept separate from quantities tending to zero along the sequence.
The original dimension is $d$; the first section cone has dimension $p=d+1$.
Characteristic primes in the trace arguments are denoted by $\mathfrak p$.
==== END Bounded-klt-complements-for-Fano-contractions//introduction.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//main.tex ====
\documentclass[11pt]{amsart}
\input{preamble}
\title[Bounded klt complements]{Bounded klt complements for Fano contractions}
\author{OpenAI}
\date{September 25, 2026}
\subjclass[2020]{Primary 14E30; Secondary 14B05, 14J45}
\keywords{Fano contraction, klt complement, log canonical threshold, normalized volume, valuation, Cartier trace}
\begin{document}
\begin{abstract}
For fixed dimension $d$ and positive rational $\epsilon$, we prove that
every $\epsilon$-log-canonical Fano contraction over an algebraically
closed field of characteristic zero admits, near each closed base point,
a klt complement of index bounded only by $d$ and $\epsilon$.
Over $\mathbb C$, we also obtain monotone klt complements for Fano type
pairs with nef anti-log-canonical divisor and coefficients in a fixed
finite rational set. This proves the finite-rational-coefficient form of
Shokurov's bounded-klt-complement conjecture.
\end{abstract}
\maketitle
\enlargethispage{2pt}
\tableofcontents
\input{sections/introduction}
\input{sections/reduction}
\input{sections/initial-cone}
\input{sections/constrained-minimum}
\input{sections/compatible-filtrations}
\input{sections/essential-system}
\input{sections/discrepancy-push}
\input{sections/scale-tower}
\input{sections/width-estimates}
\input{sections/bounded-axes}
\input{sections/valuation-optimization}
\input{sections/trace-lifting}
\input{sections/cheap-collapse}
\input{sections/rank-growth}
\bibliographystyle{amsplain}
\bibliography{references/references}
\end{document}
==== END Bounded-klt-complements-for-Fano-contractions//main.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//preamble.tex ====
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pdftitle={Bounded klt complements for Fano contractions},
pdfauthor={OpenAI},
pdfsubject={Bounded klt complements and birational local thresholds},
pdfkeywords={Fano contraction, klt complement, log canonical threshold, normalized volume, Cartier trace}
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==== END Bounded-klt-complements-for-Fano-contractions//preamble.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//rank-growth.tex ====
\section{Expensive layers, rank growth, and the contradiction}
\label{c:rank-section}
The bounded-width coordinates now have value zero. We extend their
independent initials through the remaining width layers, while keeping
discrepancy small and making the new coordinate values integral. Once
there are $d$ independent initials, a section count bounds the denominator
of the entire value group. This produces the prime divisor that contradicts
$\epsilon$-log canonicity. The
\hyperref[c:choice-order]{completion proof at the end of this section}
chooses the tolerances and the final small scaling after all uniform size
bounds have been established. Proposition~\ref{c:divisorial-index} uses a
separate large-degree limit on one fixed member of the sequence.
After reordering and passing to a subsequence, partition the
unbounded widths \(W_j\) into successive layers. The widths in
one layer are comparable to a scale \(S\to\infty\), and each
scale divided by the next tends to zero. All cheap widths are
comparable to \(1\). If every width is cheap, the valuation
\(w_c\) constructed in the preceding section is Gaussian of
value zero on a full transcendence basis. It is therefore
trivial on its rational field and on the algebraic extension
\(K\), contradicting its positive barrier on \(\m_z\).
We may assume that an expensive layer exists.
For a small fixed scaling \(\delta>0\), use the optimization of
\eqref{eq:E3} with
\begin{equation}\label{eq:G1}
B_*=\delta B_0,\qquad q_*=\delta/4,\qquad
q_{\rm bar}=\delta/16,\qquad
\mathcal L_\delta(w)=A_\theta(w)+\delta^{-1}(q_*-q(w)).
\end{equation}
Only finitely many fixed scalings will be needed before taking
the sequence limit. We work with objective bounds strictly
below \(1/8\). At every such optimizer,
\[
q(w)>\delta/8,\qquad A_\theta(w)<1,
\]
because both summands in \eqref{eq:G1} are nonnegative.
At the cheap parameters \(t=0\), the starting optimum is at
most \(\delta\kappa\): use \(\delta w_c\), whose barrier is
at least \(\delta b/2\), so the cap is attained.
Consider a layer with scale \(S\), and let \(s\) be the number
of axes up to and including this layer. Inductively, the old
probes, from cheap and preceding layers, are actual angular
functions of degree and \(v\)-cost at most \(C W_j\), and their
Gaussian parameters are integral, satisfy
\(|t_j|\le C\delta W_j\), and are zero on the cheap axes.
Suppose the old minimum of \(\mathcal L_\delta\) is less than \(1/8\).
Constants below may depend on previous size bounds and fixed
width comparabilities, but not on the small fixed \(\delta\).
Choose an optimizer \(w\) with the tie rule from
Section~\ref{c:optimization-section}. The two-sided estimate
\eqref{eq:F4} applies to it.
For \(J\ge1\), let \(E_J\subset K_w\) be the span of the
homogeneous \(w\)-initials of elements of
\(V_{\lceil JS\rceil_n}\) with value at most \(K_2\delta JS\).
Here \(\lceil x\rceil_n\) denotes the least nonnegative multiple
of \(n\) at least \(x\), and \(K_2\) is a sufficiently large
fixed constant. Nesting of the systems makes these spaces
nested in \(J\).
\begin{lemma}[Layer dimension]\label{c:layer-dimension}
For every fixed \(J\ge1\), on sufficiently late members,
\begin{equation}\label{eq:G2}
\dim E_J\asymp
J^s\prod_{j\le s}\frac{S}{W_j}.
\end{equation}
The comparison constants are independent of \(J\) and
\(\delta\); the required lateness may depend on both.
\end{lemma}
\begin{proof}
At the tested degree, the barrier \(q>\delta/8\) sends costs
\(b>C'JS\), for a sufficiently large fixed \(C'\), to values
\(w>K_2\delta JS\). Equation~\eqref{eq:C8} bounds the quotient
through the former cost by a constant times the right side
of \eqref{eq:G2}, since fixed \(JS\) is negligible compared
with widths in subsequent layers. This proves the upper
bound. In particular the quotient by \(w>K_2\delta JS\),
with zero included in the high part, is finite-dimensional.
Its dimension equals \(\dim E_J\), by taking initials on its
successive value slices. Distinct occurring values already
give independent quotient classes.
For the lower bound, the original independent probe boxes
following \eqref{eq:C8} and \eqref{eq:D3} give products with
exponents
\[
0\le a_j\le\left\lfloor c'JS/W_j\right\rfloor
\qquad(j\le s),
\]
for a sufficiently small fixed \(c'>0\).
After nesting, they lie in the tested physical degree.
They are independent, and their \(v\)-Gaussianity bounds the
cost of every nonzero linear combination by \(C''JS\).
Equation~\eqref{eq:F4} bounds its absolute \(w\)-value by
\(C_6C''\delta JS\). Taking \(K_2\) above this constant
makes their span inject into the quotient. Their number
is bounded below by the required box product.
\end{proof}
\begin{proposition}[Rank increase in a layer]\label{c:rank-increase}
There is a bounded \(J\), independent of the small fixed
\(\delta\), such that the old probe initials extend to \(s\)
algebraically independent homogeneous initials in \(E_J\).
Their actual lifts can be chosen with degree and cost
\(O(S)\), and absolute \(w\)-value \(O(\delta S)\).
\end{proposition}
\begin{proof}
Choose a fixed \(J_0\ge1\) whose space contains the old
probe initials; their widths are \(o(S)\).
Suppose the transcendence degrees of the fields generated
by \(E_{J_0}\) and \(E_N\) are equal to \(a<s\), where \(N\)
is a sufficiently large fixed integer to be chosen in terms
of \(J_0\). The dimension of \(E_N\) grows as \(N^s\) times
the layer box factor in \eqref{eq:G2}. We will encode its elements
by Taylor orders in \(a\) variables and scale their order polytope
by \(1/N\). This loses only a factor \(N^a\). Trace will put
the resulting independent outputs into a degree and value budget
independent of \(N\), contradicting the upper section count when
\(a<s\).
Choose a transcendence basis \(g=(g_j)_{j\le a}\)
from the homogeneous generators of \(E_{J_0}\), extending
the old probes, with their actual lifts in that budget.
If \(a=0\), every element of \(E_N\) is algebraic over the
algebraically closed constant field, so \(\dim E_N\le1\).
This contradicts \eqref{eq:G2} for large \(N\), since
\(S/W_j\) is bounded below for \(j\le s\).
Assume henceforth \(a>0\).
The finite space \(E_N\) lies in a finite separable extension
of \(k(g)\).
Extend constants independently to \(k'=\overline{k(\xi)}\),
with \(\xi_1,\ldots,\xi_a\) independent, and use the generic
component at \(g=\xi\) from the Taylor construction
\eqref{eq:E9}. Put \(u'_j=g_j-\xi_j\).
Every nonzero pre-extension element of the finite algebraic
field is a unit at the generic point of this component.
Its Taylor expansion has coefficients in \(k'\).
Indeed take its separable polynomial over \(k(g)\).
At this generic specialization the coefficients are regular,
the chosen root remains simple, and its residue lies in
the algebraically closed field \(k'\). Simple-root lifting
gives the unique formal solution in \(k'[[u']]\).
This also describes all \(k'\)-linear combinations.
This operation keeps full Taylor series, not merely residue
values. Geometric integrality preserves linear independence
under the independent constant extension, and the local
completion is injective on the functions under consideration.
For example the independent functions \(1,g_j\) expand as
\(1,\xi_j+u'_j\), rather than being replaced by \(1,\xi_j\).
Thus the extended \(E_N\) retains its dimension and has
exactly \(\dim_k E_N\) distinct lexicographic Taylor orders.
Each leading-order slice has dimension one over \(k'\),
since its leading coefficient is a scalar; elimination then
gives the equality of counts. These orders lie in
\(\Z_{\ge0}^a\) and contain \(0\) and all standard basis
vectors, using \(1,u'_j\).
Let \(\mathcal P\) be their convex hull. It is full-dimensional
and
\[
\vol(\mathcal P)\ge c_d\dim E_N.
\]
To verify this bound even when many points lie on faces,
triangulate using all the given lattice points as vertices,
inserting them by successive subdivisions. Every such
vertex lies in a full simplex, so there are at least
\(\dim E_N/(a+1)\) full simplices. Each lattice simplex
has volume at least \(1/a!\).
Translation averaging gives \(z'\in[0,1]^a\) such that
\(z'+\mathcal P/N\) contains at least
\(\vol(\mathcal P)/N^a\) lattice points \(b'\).
They are nonnegative. For each occurring order \(\alpha_h\)
choose an element \(h'\) of the extended \(E_N\) with that
order and a nonzero scalar leading coefficient.
Retain their finite Taylor leading data and their expressions
in homogeneous initials with actual lifts, together with
those of \(g\), in the spread for
\eqref{eq:E7}--\eqref{eq:E9}.
We require shifts for whatever Frobenius power
\(P_0=\mathfrak p^{\ell_0}\), \(\ell_0\ge1\), that construction later supplies.
Write
\[
N(b'-z')=\sum_h\lambda_h\alpha_h,\qquad
\lambda_h\ge0,\qquad\sum_h\lambda_h=1.
\]
Use \(h'\) to exponent \(k_h=\lfloor P_0\lambda_h/N\rfloor\).
The remaining Taylor order is
\[
\begin{split}
P_0b'-\sum_h k_h\alpha_h
&=P_0z'+\sum_h
\left\{\frac{P_0\lambda_h}{N}\right\}\alpha_h.
\end{split}
\]
It is integral and componentwise nonnegative.
Its \(j\)-th coordinate is at most
\(P_0+\sum_h(\alpha_h)_j\), hence at most \(2P_0\)
once \(\mathfrak p\) exceeds the finitely many coordinate sums.
This one choice works for every \(\ell_0\ge1\).
Fill the remainder by powers of \(u'_j\).
The resulting shift has exact Taylor order \(P_0b'\),
a nonzero scalar leading coefficient, total \(E_N\)
multiplicity at most \(P_0/N\), and at most \(2P_0\)
filler factors per coordinate.
Use Corollary~\ref{c:strict-budget} and
Proposition~\ref{c:trace-budget}
over the probes \(g\), with \(\kappa=1/\delta\), at the
same old optimizer. Only the old probes are optimization
constraints; enlarging the list for the relative field
does not alter this optimization.
Apply every shift to the same surviving relative input
and the same trace power from \eqref{eq:E7}.
The full initial-field outputs have the distinct Taylor
orders \(b'\), so are independent over the enlarged
perfect constants. Their common relative leading
coefficient may be nonconstant in the residue field;
it is nonzero, which is all that distinct-order
independence needs. Semilinearity only takes the
\(P_0\)-th roots of scalar coefficients.
Expand the inputs in homogeneous angular initials and
discard summands with zero initial-field trace.
Each remaining summand lifts by equality in
\eqref{eq:E8} to an actual output initial.
Its physical degree and value satisfy
\[
m'\le C_7(1+J_0)S,\qquad
w(F')\le C_7(1+J_0)\delta S.
\]
These estimates are termwise. Each \(E_N\) generator has
degree at most \(\lceil NS\rceil_n\) and value at most
\(K_2\delta NS\), and there are at most \(P_0/N\)
occurrences. The filler factors and the flags of
\eqref{eq:E9} use the \(E_{J_0}\) budget with at most
\(3P_0\) occurrences per axis. Expand \(g_j-\xi_j\)
as a probe plus a scalar before estimating; both terms
satisfy that budget. Likewise use the fixed bounded
homogeneous expressions for each \(h'\), so no
cancellation is needed to improve an upper estimate.
The active terms of \eqref{eq:E10}, after division by \(P_0\),
cost only \(O(1/\delta)\) in degree and \(O(1)\) in value.
Indeed \(m_i/b_i\) is bounded and
\(w(F_i)=qb_i-\delta B_0m_i\).
The tiny factors and active-exponent approximation errors
can be made arbitrarily small. Since \(\delta\) is fixed
before \(S\to\infty\), these contributions are negligible
in the displayed budgets. Apply \eqref{eq:E8} with
equality, using \(A_\theta(w)\ge0\), and its degree bound.
Consequently the independent initial-field outputs belong
to the span of actual output initials in those two budgets.
Before choosing the characteristic, pad to one nested
degree dominating \(C_7(1+J_0)S\), retain the cost-dimension
bound through \(C'(1+J_0)S\) at \(v\), and retain the
inclusion of the larger-cost subspace into values strictly
above the required cutoff at \(w\).
The barrier \(q>\delta/8\) gives this inclusion with margin
for sufficiently large fixed \(C'\). Lemma~\ref{c:trace-finite-data}, with the budget
accounting of Proposition~\ref{c:trace-budget}, therefore
bounds that output span in reduction by
\[
C_8(1+J_0)^d\prod_{j\le s}S/W_j.
\]
It applies to all coefficient vectors of this predetermined
space, not to a list of outputs chosen after \(P_0\).
On the other hand, the number of distinct orders \(b'\)
is at least
\[
cN^{s-a}\prod_{j\le s}S/W_j
\]
by \eqref{eq:G2}. Choose \(N\) large in terms of \(J_0\)
and the preceding constants, independently of \(\delta\).
These bounds contradict each other.
Thus rank increases from \(E_{J_0}\) to \(E_{N(J_0)}\)
unless it has reached \(s\). Iterate, setting the next
\(J_0\) equal to that \(N\), at most \(s\) times, all at
the same optimizer. Choose the new probes among the
homogeneous generators, extending the old list.
Their degree is \(O(S)\); the barrier bounds their
cost by \(CS\) and their value below by
\(-\delta B_0 O(S)\), while the defining cutoff bounds
it above by \(C\delta S\). Their initials are independent,
so \(w\) is Gaussian on the enlarged list. This proves
all assertions.
\end{proof}
\begin{proposition}[Slopes after a rank increase]
\label{c:layer-slopes}
For the enlarged optimization with \(s\) probes, on
full-dimensional generic cells with
\(|t_j|\le C\delta W_j\) and minimum \(<1/8\),
\begin{equation}\label{eq:G3}
\left|\partial_{t_j}\min\mathcal L_\delta\right|
\le C_9\,S/W_j
\qquad(j\le s).
\end{equation}
The same assertion holds in slightly enlarged fixed
parameter boxes and at each of the later fixed scalings.
In particular the new-coordinate slopes are uniformly
bounded independently of the small fixed \(\delta\).
\end{proposition}
\begin{proof}
We repeat the slope mechanism from the proof of
Proposition~\ref{c:cheap-collapse}, using rectangular probe boxes
whose side lengths reflect the widths
\(W_j\). We retain its central trace equality and nearby lower
inequality. Here even cheap parameters may vary, so the lower section
barrier supplies the upper dimension count without using
\eqref{eq:F4}.
Choose a generic parameter \(t\)
whose coordinates are \(\Q\)-linearly independent, and
an optimizer with the tie rule. Use its affine
continuation with fixed stratum cone and active mask
from Proposition~\ref{c:optimization}.
Apply Corollary~\ref{c:strict-budget} and
Proposition~\ref{c:trace-budget} over
\(\overline y\), and lift a surviving initial trace
by \eqref{eq:E8}. Its actual output \(F'\), in the
reduction, satisfies
\[
m'\le C'S,\qquad
w(F')\le C'\delta S.
\]
The active contributions cost \(O(1/\delta)\) and
\(O(1)\) in the two budgets. The flag powers divided
by \(P_0\) cost \(O(S)\) and \(O(\delta S)\), by
the probe and parameter bounds. Choose tiny factors
and approximation errors after knowing this continuation,
including their Lipschitz errors. For fixed \(\delta\),
the claimed bounds follow on late members.
Fix a sufficiently large integer \(N\), and consider
\[
(F')^h y^\alpha,\qquad
0\le h<N,\qquad
0\le\alpha_j<L'_j:=\lceil NS/W_j\rceil.
\]
They lie in one nested degree \(O(NS)\) and have
upper value \(O(\delta NS)\). After enlarging the
bounds, these statements hold nearby on the same
continuation, by continuity after \(F'\) is chosen.
The quotient dimension through that value is at most
\(C''\prod_{j\le s}L'_j\).
Indeed, before applying \eqref{eq:E7}, retain
\eqref{eq:C8} in that predetermined degree and
the inclusion of costs \(b>C'''NS\) into larger
\(w\)-values, with strict margin. The barrier
\(q>\delta/8\) supplies this inclusion, and fixed
\(NS\) is smaller than all subsequent widths.
The constants do not depend on \(N,\delta\).
Finite-space transfer with margin gives the same
bound locally on the reduced continuation.
There are therefore at most \(C'''\prod_{j\le s}L'_j\)
distinct product values. At independent \(t\), this
forces a unique expression \(w(F')=\gamma\cdot t\)
with \(\gamma\in\Q^s\): otherwise all \(N\) layers
have different values, contradicting the count for
large \(N\). We record the quantitative consequence.
Let \(D'\) be the order of \(\gamma\) in
\(\Q^s/\Z^s\). Modulo \(\Z^s\), the first \(N\)
translated boxes have \(\min(N,D')\) distinct classes,
each with \(\prod_jL'_j\) points. Thus \(D'\) is
uniformly bounded. Restricting \(h\) to multiples
of \(D'\), the union of projections to coordinate
\(j\) contains
\[
L'_j+\lfloor (N-1)/D'\rfloor
\min(L'_j,D'|\gamma_j|)
\]
points, since these are translates of an integer
interval. Over each projected point lies a full
box with \(\prod_{l\ne j}L'_l\) points in the other
coordinates. Comparison with the total bound, for
sufficiently large uniformly chosen \(N\), gives
\[
|\gamma_j|\le C S/W_j.
\]
The same argument works at nearby independent
parameters on the continuation. There are only
finitely many possible rational vectors with the
stated denominator and coordinate bounds. Uniqueness
at the central independent parameter and continuity
therefore show that, in a neighborhood of that parameter,
\[
w_{\widetilde t}(F')=
\gamma\cdot\widetilde t
\]
with this same \(\gamma\).
Let \(G\) be the input term lifted for \(F'\).
Equation~\eqref{eq:E8} makes
\[
w_{\widetilde t}(F')-
\frac{w_{\widetilde t}(G)}{P_0}
+(1-1/P_0)A_\theta(w_{\widetilde t})
\]
nonnegative nearby and zero centrally. In the active
part of \(w_{\widetilde t}(G)/P_0\), each
\(w_{\widetilde t}(F_i)\) varies by \(b_i\,dq\).
Equation~\eqref{eq:E10} says that the resulting
variation is \((1-1/P_0)dq/\delta\), up to an
arbitrarily small Lipschitz slope error. If the cap
is active, \(dq=0\). Tiny systems give arbitrarily
small further slope errors, while the flag powers
in \eqref{eq:E9} have bounded slopes. All these
formulas use the chosen characteristic-zero continuation
and its chart valuations in reduction; finite input
data are retained and errors are chosen after that
continuation is known, as in the proof of \eqref{eq:F2}.
The displayed nonnegative function has a local minimum
at the central point. Using both signs of variation,
the formula for \(w_{\widetilde t}(F')\), and
\eqref{eq:G1}, bounds the objective slope by
\(C S/W_j\). Here bounded extra slopes are absorbed
because \(S/W_j\) is bounded below. This proves
\eqref{eq:G3} on the generic cells.
Openness, continuity, and local polyhedrality on the
strict sublevel set extend this bound to changes along
segments varying only the new coordinates, even when
the old coordinates are fixed at nongeneric values.
Equivalently, approximate such a segment by generic
ones, apply the bound piecewise on the finitely many
local cells, and pass to the limit using compact
sublevels. The assertions concerning enlarged boxes
and later fixed scalings follow from the same estimates.
\end{proof}
\begin{proposition}[Integral rounding]\label{c:integral-rounding}
For every absolute tolerance \(\sigma>0\), a rank
increase followed by a bounded integral scaling and
rounding of the new parameters changes a small objective
bound according to
\begin{equation}\label{eq:G4}
e_{\mathrm{new}}=K(\sigma,d)e_{\mathrm{old}}
+C_{10}\sigma.
\end{equation}
Old parameters remain integral and cheap parameters
remain zero. The constants for probe and parameter
sizes remain uniform at every later fixed scaling.
\end{proposition}
\begin{proof}
Simultaneous Dirichlet approximation gives a positive
integer \(u\le K(\sigma,d)\) for which every new
parameter of \(ut\) is within \(\sigma\) of an integer.
For example, subdivide the unit cube into boxes of
side at most \(\sigma\), apply the pigeonhole principle
to the fractional parts of finitely many multiples
of the new parameter vector, and subtract two multiples
in the same box. Multiply \(w,t,\delta\) by this
common \(u\). Integrality of the old parameters is
preserved. Both \(q\) and \(q_*\) scale by \(u\),
so the nonnegative penalty in \eqref{eq:G1} is unchanged;
discrepancy scales by \(u\). Thus the starting objective
for the newly scaled enlarged optimization is at most
\(u e_{\mathrm{old}}\).
Move only the new parameters to the chosen integers
along a line segment. Integrating \eqref{eq:G3} in
the new coordinates adds at most \(C_{10}\sigma\).
Choose tolerances so that the full resulting bound
is strictly less than \(1/8\). If continuation had
a first limiting endpoint, compact sublevels would
supply a feasible optimizer there with this same
strict bound, and local continuation would extend
the segment. Hence the whole rounding is possible.
At fixed scaling and tolerance, the rounding errors
are negligible relative to \(\delta W_j\) in the
new layer on sufficiently late members. The path
therefore stays within slightly enlarged parameter
boxes, and the stated size bounds persist under
subsequent scaling.
\end{proof}
\begin{proposition}[Bounded divisorial index]\label{c:divisorial-index}
At a final optimizer with \(d\) independent probe
initials and integral probe parameters, there is a
uniformly bounded positive integer \(e\) such that
\(ew\) is a positive integral multiple of a
prime-divisor valuation over \(X\).
\end{proposition}
\begin{proof}
First bound the image of the value group of \(w\)
in \(\R/\Z\). Choose any finite list of functions
whose values represent distinct classes. Multiplying
them all by a common denominator puts them in \(V_0\)
and translates all classes by the same amount, so
they remain distinct. Multiply each function by
all probe monomials in the boxes
\[
0\le\alpha_j\le\lfloor R/W_j\rfloor,
\qquad 1\le j\le d.
\]
Here \(R\to\infty\) on this one fixed member of the
sequence. The resulting initials are linearly
independent: between different functions their values
are in disjoint classes modulo \(\Z\), since all
probe values are integral; within one box Gaussianity
of the probes gives independence.
At a common nested degree \(O(R)\) all these products
have upper \(w\)-value \(O(\delta R)\). The constants
are uniform by the size bounds, once \(R\) is also
large enough for the finitely many chosen function
degrees and values. The barrier \(q>\delta/8\) and
\eqref{eq:C8} bound the dimension through this cutoff
by
\[
C_{11}\prod_{j=1}^d(1+R/W_j).
\]
Comparing with the number of independent products
and letting \(R\) tend to infinity bounds the length
of the finite list uniformly. Thus the image of the
entire value group modulo \(\Z\) is a finite subgroup
of uniformly bounded order. Its exponent is a uniformly
bounded positive integer \(e\), and \(ew\) is
integer-valued.
Equation~\eqref{eq:F3} makes \(w\) nonzero on the
degree-zero field. It is an Abhyankar valuation
with center on \(X\), as supplied throughout by
monomialization. A nonzero discrete rank-one
Abhyankar valuation is divisorial: on a monomial
chart, the rational weight ray is extracted by a
toric subdivision, and the corresponding primitive
ray defines a prime divisor. Since \(ew\) is
integer-valued, it is a positive integral multiple
of that prime-divisor valuation. This proves the
claim.
\end{proof}
\phantomsection\label{c:choice-order}
\begin{proof}[Completion of Proposition~\ref{prop:birational-threshold}]
We now specify the order of all choices. The rank,
probe-size, parameter-size, and slope constants in
\eqref{eq:G2}--\eqref{eq:G3} can be bounded successively
forward through the at most \(d\) layers. These
bounds are independent of the sufficiently small
fixed scalings and rounding tolerances: each step
uses only the preceding uniform size bounds, with
fixed enlargements of their constants. Fix these
constants first, including a final bound \(C_{12}\)
for the integer \(e\) in Proposition~\ref{c:divisorial-index}.
Prescribe objective tolerances backwards through
\eqref{eq:G4}. At each step choose \(\sigma\) first
so that \(C_{10}\sigma\) is smaller than the desired
next tolerance, then choose the previous objective
bound so small that its product with \(K(\sigma,d)\)
fits the remaining allowance. Make every intermediate
bound satisfy all strict-sublevel requirements, and
make the final bound less than \(\epsilon/(3C_{12})\).
Only after these finitely many choices choose the
initial \(\delta\) so small that \(\delta\kappa\)
is below the first starting tolerance. Require also
that after every bounded possible product of the
integer scalings, the resulting \(\delta\) remains
small and satisfies
\[
\delta B_0<\epsilon/(3C_{12}).
\]
Finally pass to sufficiently late members of the
sequence. All estimates involve fixed scalings;
only finitely many possible integer scaling products
have to be accommodated.
Cheap collapse, rank extension, and integral rounding
then give a final optimizer with
\[
A_\theta(w)<\epsilon/(3C_{12}),
\qquad
A_X(w)\le A_\theta(w)+\delta B_0.
\]
The second inequality is \eqref{eq:A13} and the
budget barrier. By Proposition~\ref{c:divisorial-index},
write \(ew=a\ord_E\), where \(E\) is a prime divisor
over \(X\), \(a\) is a positive integer, and
\(e\le C_{12}\). Consequently
\[
A_X(E)=\frac{e}{a}A_X(w)
\le e A_X(w)<\epsilon.
\]
This contradicts the assumed \(\epsilon\)-log canonicity
of \(X\). Hence the sequence violating
\eqref{eq:A1} cannot exist, proving
Proposition~\ref{prop:birational-threshold}.
\end{proof}
Theorem~\ref{thm:main} follows by
Lemma~\ref{lem:threshold-to-complement} and
Lemma~\ref{lem:field-descent}. The resulting complement
index \(N\) is distinct from the auxiliary degree
\(n\) fixed for the cone construction.
==== END Bounded-klt-complements-for-Fano-contractions//rank-growth.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//reduction.tex ====
\section{The birational threshold reduction}
\label{sec:reduction}
The central assertion is the following uniform estimate.
\begin{proposition}\label{prop:birational-threshold}
For every positive integer $d$ and every $\epsilon>0$ there is
$c(d,\epsilon)>0$ such that, over $\C$, the following holds. Let
$f\colon X\to Z$ be a birational contraction of $d$-dimensional normal
varieties, let $X$ be of Fano type over $Z$, and assume that $K_X$ is
$\Q$-Cartier, $-K_X$ is nef over $Z$, and $X$ is $\epsilon$-lc. For each
closed point $z\in Z$,
\begin{equation}\label{eq:A1}
\lct\bigl(X;\m_z\OO_X\bigr)\ge c(d,\epsilon).
\end{equation}
\end{proposition}
The subsequent sections prove Proposition~\ref{prop:birational-threshold}.
We first explain precisely why it suffices.
\begin{lemma}\label{lem:threshold-to-complement}
Proposition~\ref{prop:birational-threshold} in all dimensions implies
Theorem~\ref{thm:main} over $\C$.
\end{lemma}
\begin{proof}
Shrink $Z$ to an affine neighbourhood of $z$. Choose generators of $\m_z$
and take a log resolution of their pullback ideal and of $X$. The pullback
system has a fixed divisor and a base-point-free moving system. By
\eqref{eq:A1}, multiplying the fixed divisor by $c(d,\epsilon)$ preserves
log canonicity. For a general linear combination $h$ of the generators,
Bertini's theorem makes its moving divisor transverse to the resolution
strata. Consequently, with
$t=\min\{c(d,\epsilon),1/2\}$, the pair
$(X,t f^*\Div(h))$ is lc near $f^{-1}(z)$; the same conclusion follows
componentwise on the resolution. The divisor $\Div(h)$ passes through $z$.
This is the birational boundary-zero instance of the local hypersurface
assertion in \cite[Conjecture~1.3]{Chen}.
Apply the implication (4)$\Rightarrow$(1) in
\cite[Theorem~1.6]{Chen}, for every base-dimension bound needed and with
boundary coefficient set $\{0\}$. Its conclusion supplies a bounded klt
complement on an $\epsilon$-lc contraction of Fano type whose
anti-log-canonical divisor is relatively nef. A Fano contraction as in
Theorem~\ref{thm:main} satisfies these hypotheses, using the zero boundary.
Over our affine base its total space is quasi-projective, as required there.
Relative linear triviality becomes actual linear triviality after shrinking
about $z$ to trivialize the line bundle on the base.
The birational assertion in Chen's formulation allows a canonical Weil
divisor that is $\R$-Cartier. Such a divisor is already $\Q$-Cartier: on a
finite open cover, expressing its Cartier condition in terms of finitely many
integral divisors gives a finite system of rational linear equations. A real
solution of this consistent system can be replaced by a rational solution,
and a common denominator works on the finite cover. Finally a real positive
lower discrepancy bound may be decreased to a positive rational one. Thus
these harmless differences in formulation do not restrict the implication.
\end{proof}
\begin{lemma}\label{lem:field-descent}
If Theorem~\ref{thm:main} holds over $\C$ with a fixed index depending only
on $d$ and $\epsilon$, it holds over every algebraically closed field of
characteristic zero with a fixed multiple of that index.
\end{lemma}
\begin{proof}
Work over an affine neighbourhood of $z$. Descend the morphism, the point,
and the relevant canonical-divisor data to a countable algebraically closed
subfield $k_0$ of the given field $k$. All defining equations require only
finitely many coefficients, so such a $k_0$ exists and embeds in $\C$.
Normality, the contraction property, and relative ampleness descend and
ascend under these field extensions after enlarging the defining field when
necessary. A log resolution defined over $k_0$ checks the discrepancy bound:
its finitely many boundary coefficients, together with the smooth transverse
strata, compute the same lc condition after either extension. We may
therefore apply the complex assertion to this descended model.
Let $b>1$ be a fixed multiple of the complex complement index. Write $K_X$
using a rational top-degree form $\omega$ defined over $k_0$. The coherent
module of relative sections of the reflexive sheaf $\OO_X(-bK_X)$ has
finitely many generators over the affine base. Represent these generators
by rational functions $\phi_1,\ldots,\phi_s$ over $k_0$. Sections commute
with flat constant-field extension; equivalently one may compute the
reflexive powers on the smooth locus and use flat base change there.
We do not require $bK_X$ to be Cartier.
Take a log resolution $\pi\colon Y\to X$ over $k_0$ on which the fractional ideal
generated by the $\phi_i$ is principal, and its divisor together with
$\Div_Y(\omega)$ and the exceptional locus has simple normal crossings.
For every component $E$ of these divisors put
\[
a_E=\ord_E(\omega),\qquad
q_E=\min_i E(\phi_i).
\]
Over $\C$, the given klt complement is represented, after localizing on a
neighbourhood of $z$, by a section $\phi$ with
$\Div_X(\phi)=b(K_X+B)$. Its order at every $E$ whose proper image in the
base contains $z$ is at least $q_E$: the coefficients expressing it in the
chosen generators are regular on that neighbourhood. Klt of this complement
therefore gives the strict inequalities
\begin{equation}\label{eq:descent-fixed-part}
1+a_E>\frac{q_E}{b}
\end{equation}
for all such components. These are finite inequalities for the descended
resolution, so they remain true over $k$.
Remove from the base the proper images of the finitely many offending
components that do not contain $z$. The fractional ideal is now its fixed
divisor times a base-point-free system. A general $k$-linear combination
$\phi=\sum_i\lambda_i\phi_i$ has a moving divisor which is smooth and
transverse to the resolution strata, by Bertini in characteristic zero.
Its coefficient in
\[
\frac{1}{b}\Div_Y(\phi)-\Div_Y(\omega)
\]
is $1/b<1$; the fixed coefficients are strictly less than one by
\eqref{eq:descent-fixed-part}. This simple-normal-crossing subpair is klt.
Pushing down gives an effective boundary
\[
B=\frac{\Div_X(\phi)-bK_X}{b},
\qquad b(K_X+B)=\Div_X(\phi),
\]
which is klt on a neighbourhood of $f^{-1}(z)$. Properness justifies the
shrinking used above. Closing up this effective divisor in the original $X$
gives the divisor required by the theorem; its restriction to that
neighbourhood is unchanged. Thus the fixed index $b$ works over $k$.
\end{proof}
\subsection{The contradiction sequence}
\label{subsec:contradiction-sequence}
Suppose Proposition~\ref{prop:birational-threshold} fails for fixed
$d>0$ and $\epsilon>0$. Choose a sequence of its data with threshold tending
to zero. We will usually suppress the sequence index. Before constructing
the cone, we make two reductions that preserve this failure.
\begin{lemma}\label{lem:ample-model}
The sequence may be chosen with $-K_X$ ample over $Z$. This replacement
preserves all discrepancies and the threshold in \eqref{eq:A1}.
\end{lemma}
\begin{proof}
Choose a klt log Fano boundary $\Delta$ over $Z$ and a positive integer $r$
with $-rK_X$ Cartier. The divisor $D=-rK_X$ is relatively nef, and for a
positive integer $a$,
\[
aD-(K_X+\Delta)=-arK_X-(K_X+\Delta)
\]
is relatively ample. The relative klt base-point-free theorem
\cite[Theorem~1.3]{FujinoBPF} makes $-K_X$ semiample over $Z$. Let
$g\colon X\to X^{\mathrm a}$ be its relative ample model. Since $X\to Z$
is birational, so is $g$. Choose canonical divisors from the same rational top form. For a Cartier divisor $H$ on $X^{\mathrm a}$ ample over $Z$ and a sufficiently divisible $m$, write
$-mK_X=g^*H+\Div_X(\phi)$. Birationality identifies the function fields, so
pushing forward gives $-mK_{X^{\mathrm a}}=H+\Div_{X^{\mathrm a}}(\phi)$.
Pulling back this identity gives
\[
K_X=g^*K_{X^{\mathrm a}}.
\]
The morphism is crepant; the new variety is $\epsilon$-lc and has ample
anti-canonical divisor over $Z$. In particular it is of Fano type there.
Every divisorial valuation has the same discrepancy on these two models,
and its value on the ideal from $Z$ is the same. The valuative formula for
the threshold proves the assertion.
\end{proof}
We henceforth assume that $Z$ is affine and $-K_X$ is ample over $Z$.
By the valuative description of the failing threshold, there are prime
divisors $P$ over $X$ such that
\begin{equation}\label{eq:bad-prime-ratio}
\frac{P(\m_z)}{A_X(P)}\longrightarrow\infty.
\end{equation}
Here and below $P(\m_z)$ denotes the order of $\m_z\OO_X$.
Birkar's relative bounded lc complement theorem
\cite[Theorem~1.8]{BirkarComplements} applies to the pair $(X,0)$: the
coefficient set is fixed, the total space is of Fano type over the base,
and $-K_X$ is relatively nef. Our base has positive dimension $d$, so the
stated relative theorem applies. After shrinking about $z$, it supplies an
lc complement of an index bounded in terms of $d$. Taking a fixed multiple
of the finitely bounded indices and then a further multiple if necessary,
we obtain one integer $n>1$, independent of the sequence, for which
\begin{equation}\label{eq:auxiliary-lc-system}
\frac{1}{n}|-nK_X|\quad\text{is an lc linear system}.
\end{equation}
This means that for every divisorial valuation $E$ with centre in the
neighbourhood under consideration,
$\frac1n E(|-nK_X|)\le A_X(E)$, where the system order is the minimum
section order. One lc section already proves these inequalities for the
whole system. The integer $n$ will remain fixed in the cone construction;
it is not asserted to be a klt complement index.
\subsection{The finite-coefficient consequence}
The crepant model also gives the consequence stated in the introduction,
once Theorem~\ref{thm:main} has been proved.
\begin{proof}[Proof of Corollary~\ref{cor:finite-coefficients}]
We check item (2) of \cite[Theorem~1.6]{Chen} in every birational dimension.
For its boundary-zero, relatively nef data, use the crepant ample model
$g\colon X\to Y$ of Lemma~\ref{lem:ample-model}. Apply
Theorem~\ref{thm:main} to $Y\to Z$, decreasing $\epsilon$ to a fixed
positive rational number when needed. After shrinking about $z$, its
complement boundary $B_Y$ is $\Q$-Cartier, since both $K_Y$ and
$K_Y+B_Y$ are $\Q$-Cartier there. On this neighbourhood,
$B_X=g^*B_Y$ is effective and
\[
K_X+B_X=g^*(K_Y+B_Y).
\]
The pair upstairs is klt, and the same complement index makes its adjoint
divisor Cartier and trivial. Take the effective closure of $B_X$ outside
this neighbourhood if necessary. In particular the multiple used to construct
the ample model does not change the complement index. This proves Chen's
item (2) for every base-dimension bound. The implication (2)$\Rightarrow$(1)
of his theorem gives the assertion, with the stated dependence on $I$.
\end{proof}
In the rest of the paper we work with the contradiction sequence above,
with its fixed auxiliary lc degree $n$, and prove
Proposition~\ref{prop:birational-threshold}.
==== END Bounded-klt-complements-for-Fano-contractions//reduction.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//scale-tower.tex ====
\section{A tower on the residual field}\label{b:scale-tower}
We apply the discrepancy calculus to the residual data
$S,H,A^+$ of \eqref{eq:A9}--\eqref{eq:A12}. All varieties in this
section are over $\C$. The construction is performed along the
sequence fixed there. The symbols $\lesssim$ and $\asymp$ mean inequalities
with constants independent of its member. Constants may depend on the
original bounded dimension and index data and on constants already fixed
at an earlier level. There will be at most $\dim S+1$ levels; passing to
subsequences or discarding finitely many members at each level is therefore
harmless. A positive rational described as fixed is independent of the
member, whereas extraction models and sufficiently small perturbations
used only to construct them may depend on that member.
At a collapsing level, we isolate an lc divisor carrying the smallest
scale and construct a contraction for which it is the only horizontal
component of the positive support of $H$. Weak essentiality bounds the
contraction threshold from below; bounded fibre degrees and a birational
section system bound it from above. The same degree comparison then
excludes every other positive horizontal component.
\subsection{The data carried by one level}
The discrepancy functions and the divisor $H$ define the contraction step.
The invariant-prime inequalities retain the regularity tests for sections.
We also keep a model of the initial residual field: its principal moduli
will bound the index after pushing to each successive base, without
accumulating fibre multiplicities.
We maintain the following data on a projective normal variety $S$, whose
function field is denoted by $K$.
\begin{enumerate}
\item \emph{The lc presentation.} $H$ is an effective ample rational Cartier divisor. The discrepancy
function $A^+$ has an effective generalized lc CY presentation
$a_K+\mathbf M_0$, with boundary $B_0\ge H$ and a bounded b-Cartier
multiple of $\mathbf M_0$.
\item \emph{The positive discrepancy inherited from the preceding level.}
Except at the first level, there is an effective generalized
$\delta$-lc presentation $Q$, for a fixed $\delta>0$, with rational
b-Cartier b-nef part $\mathbf M_Q$, boundary $B_Q$, and total adjoint
$H/\lambda$, where
$\lambda>0$ is rational and tends to zero. Initially $Q$ is absent and
$\lambda=0$.
\item \emph{The scale.} Write
\[
\Lambda=H+\lambda Q,
\qquad s=\min\bigl(\{1\}\cup
\{\Lambda(P): A^+(P)=0\}\bigr).
\]
The primes in this definition are divisorial valuations anywhere over
$S$; the minimum is $1$ when there are none. It exists and is positive:
for each member the rational presentations give a discrete group of
values; away from the first level $\Lambda\ge\delta\lambda$ on these
primes, while initially $A^++H=A_B>0$. In particular $s\gtrsim\lambda$.
\item \emph{The section tests.} For every invariant prime $E$ of $\Spec R$ whose nontrivial
restriction to $K$ is $jP$, we have
\begin{equation}\label{eq:B2}
jA^+(P)\le1-h_E,\qquad jH(P)\le h_E,\qquad
j\Lambda(P)\le\lambda+C_0h_E.
\end{equation}
Every strict prime $P$ of $S$ with $H(P)=0$ has such a witness $E$ with
$h_E=0$ and bounded $j$. A positive strict coefficient $H(P)$ is at
least a fixed $c_0>0$ if $A^+(P)>0$, and at least $c_0s$ if $A^+(P)=0$.
\item \emph{Essentiality and the initial field.} Weak essentiality holds in the form
\begin{equation}\label{b:scale-essentiality}
P(H')\le P(H)+\eta\Lambda(P)
\quad\text{if }H'\ge0,\ H'\sim_{\Q}H,\ A^+(P)=0,
\qquad \eta\longrightarrow0.
\end{equation}
There is also a projective $\Q$-factorial Fano type model
$\mathcal Y$ of the initial field $K_1$, with a contraction to $S$,
such that $A^+$ is the valuative push of $A_1^+$ and $A_1^+$ has
effective trace on $\mathcal Y$.
\end{enumerate}
Here $A_1^+$ is the ordinary lc discrepancy with
$K_{S_1}+B_{0,1}=\frac1n\Div(f_h)$, equivalently with principal CY moduli of
denominator dividing $n$. Initially the asserted conditions follow
from \eqref{eq:A9}--\eqref{eq:A12}. For the strict coefficient bounds,
the witnesses there satisfy $jH(P)=h_E\in\frac1n\Z$; $j$ is bounded
when $A^+(P)>0$ by the discrepancy index gap. For a strict lc prime,
$H(P)\ge s$. Use a small $\Q$-factorialization for $\mathcal Y$;
the initial variety is of Fano type.
Pass to a subsequence on which either $s\to0$, called a collapsing
level, or $s$ stays bounded below. In the latter case this is the last
level and we replace $s$ by $1$ in subsequent formulas, retaining the
actual positive lower bound when choosing constants and decreasing
the strict coefficient-gap constants accordingly.
\subsection{A positive perturbation and a birational system}
Choose fixed small positive rationals $a,b$, with $b$ sufficiently small
relative to the preceding constants, and put
\begin{equation}\label{b:scale-perturbation}
\gamma=\frac{b\lambda}{s},\qquad
A'=(1-\gamma)A^++\gamma Q+aH,\qquad d'=\frac bs-a.
\end{equation}
At the first level the convention is $\gamma Q=0$ and $d'=-a$.
The total adjoint of $A'$ is $d'H$, and its b-nef part is
$\mathbf M'=(1-\gamma)\mathbf M_0+\gamma\mathbf M_Q$.
Since $\lambda/s$ is bounded, $\gamma$ is uniformly small. The trace is
effective because $B_0\ge H$ and both original boundaries are effective.
The datum is generalized klt: at later levels use $\gamma Q>0$, and at
the first use $A'=(1-a)A^++aA_B>0$.
\begin{lemma}\label{b:scale-bad-primes}
For a sufficiently small fixed $e>0$, every prime with $A'(P)<e$ is
lc for $A^+$. There are finitely many such primes for each member.
At a collapsing level they satisfy
\[
\frac s2\le H(P)\le\frac ea.
\]
At the last level the constants may be chosen so there are none.
One can extract exactly their exceptional members on a projective
$\Q$-factorial Fano type model $S'\to S$. At a collapsing level
one may additionally extract a prime $G_*$ with $A^+(G_*)=0$ and
$\Lambda(G_*)=s$.
On this model the trace of $A'$ is effective and $B_0\ge H$ still holds,
where $H$ now denotes its pullback.
\end{lemma}
\begin{proof}
The bounded b-index of $\mathbf M_0$ gives a uniform positive gap for
nonzero values of $A^+$. Choose $e$ below this gap multiplied by
$1-\gamma$. For a bad prime we therefore have $A^+(P)=0$, and
\[
aH(P)<e,\qquad \lambda Q(P)<\frac{es}{b}
\]
when $Q$ is present. Since $H(P)+\lambda Q(P)\ge s$, choosing
$e<b/2$ gives the asserted lower bound; initially it follows directly
from $H(P)\ge s$. At the last level, if the unmodified scale is at
least $s_0>0$, these same formulas give a uniform positive lower bound
for $A'$ on lc primes, in terms of $a,b,s_0$. Decrease $e$ further.
For completeness, the finiteness assertion uses more than the finiteness
of components on one resolution. On a smooth b-Cartier descent of any
generalized klt datum, let $\Sigma$ be the reduced snc support of its
boundary. Its discrepancy at $P$ is
\[
A_{(W,\Sigma)}(P)+\sum_D a_D\ord_P(D),\qquad a_D>0.
\]
The first term is a nonnegative integer. If the total is less than
$e<1$, this term is zero. We recall why this forces $P$ to be monomial
at a stratum of $\Sigma$. At the generic point of its centre choose
regular parameters $x_1,\ldots,x_\ell$, including the boundary normals,
and units $y_j$ lifting a separating transcendence basis of the centre.
The reduced-pair discrepancy is the discrepancy of
\[
d\log x_1\wedge\cdots\wedge d\log x_\ell
\wedge\bigwedge_jdy_j
\]
plus the orders of those $x_i$ which are not boundary normals.
On a smooth model carrying $P$, write $x_i=t^{w_i}u_i$, with $t$ a
uniformizer and $u_i$ units. The displayed form has at most a simple
pole. Equality zero thus forces every $x_i$ to be a boundary normal
and the logarithmic wedge to have nonzero residue. The centre is a
stratum, and the wedge of the initials
$\overline u_i\overline t^{w_i}$ and $\overline y_j$ is nonzero.
In characteristic zero these initials are algebraically independent
in the required normal directions, also over the centre's function
field, which is algebraic over $\C(\overline y_j)$. Expand an element
of the local ring to arbitrarily high finite order in the $x_i$, with
coefficients lifted from its residue field. Independence prevents
cancellation in its leading polynomial, while the remainder has
arbitrarily large value. Its value is therefore the minimum of its
monomial values. The stratum and the positive integral weights $w_i$
determine $P$. There are finitely many strata, and
$\sum a_Dw_D<1$ bounds all these weights. This proves finiteness.
In the collapsing case choose $G_*$ attaining the scale, and write
$H_*=H(G_*)$. Then
\[
H_*\le s,\qquad A'(G_*)\le b+as.
\]
To extract the indicated finite set, replace $\gamma$ temporarily by
a much smaller positive rational $\widetilde\gamma<a\lambda$,
leaving it zero at the first level. The resulting effective generalized
klt datum has ample anti-adjoint. Its discrepancy at every requested
prime is less than $1$: these primes are lc for $A^+$, their
$H$-values have the bounds just proved, and
$\widetilde\gamma$ can be chosen for this finite set in the given
member. Add a sufficiently small positive multiple of $H$ to both
moduli and total adjoint. This leaves discrepancies unchanged, makes
the moduli big, and preserves ample anti-adjoint. By
Lemma~\ref{b:ordinary-approximation}, replace the moduli by ordinary
effective klt data with no larger discrepancies. The resulting
ordinary pair is log Fano. Lemma~\ref{b:controlled-extraction}
extracts precisely the requested exceptional primes on a
$\Q$-factorial Fano type model.
For the original $A'$, their discrepancies are either less than $e$
or at most $b+as<1$, so its trace remains effective. At these primes
$B_0(P)=1$ and $H(P)\le\max(e/a,s)\le1$ after the indicated choices.
Consequently $B_0\ge H$ also survives the extraction.
\end{proof}
\begin{lemma}\label{b:scale-birationality}
At every level there is a positive integer $k\le C_1/s$ such that
\begin{equation}\label{eq:B3}
|kH|\text{ is birational on }S.
\end{equation}
For a rational divisor this denotes the space of rational functions
$f$ with $\Div(f)+kH\ge0$; no bounded Cartier index for $H$ is asserted.
\end{lemma}
\begin{proof}
There is nothing to prove in dimension zero. First extract just the
bad primes by Lemma~\ref{b:scale-bad-primes}, without an additional
pivot. Every positive strict coefficient of $H$ is now at least a
fixed positive multiple of $s$. Moreover the $A'$-boundary coefficient
at each such prime has a uniform positive lower bound. At a new prime
it is greater than $1-e$. At an old non-lc prime use
$B_0(P)\ge H(P)\ge c_0$ and the effectiveness of the other boundary;
at an old lc prime use $B_0(P)=1$ and $H(P)\le1$.
Let $G$ be the sum of the bad primes, which are now strict. With a
sufficiently large fixed $C_2$ and $k_0=\lceil C_2/s\rceil$, the integral
Weil divisor
\[
N=\lfloor k_0H\rfloor-G
\]
satisfies $N\ge k_0H/2$. Indeed the rounding error plus the coefficient
of $G$ is at most $2$ at each prime of the support, whereas each
positive coefficient of $k_0H$ is uniformly large.
Choose a fixed sufficiently small rational $\sigma>0$. Subtract
$\sigma(\{k_0H\}+G)$ from the $A'$-boundary and add
$(\sigma k_0-d')\overline H$ to its moduli; choose $C_2$ large after
$\sigma$ so this last coefficient is positive. The total adjoint is
$\sigma N$, the boundary is effective, and its discrepancy is
\[
A'+\sigma\overline{(\{k_0H\}+G)}.
\]
It is uniformly generalized positive-lc: nonbad valuations already
have discrepancy at least $e$, the added divisor is effective Cartier
up to a rational multiple on this $\Q$-factorial model, and every bad
prime has received an increment at least $\sigma$.
Run its adjoint MMP on this Fano type model. The Mori dream space
property and divisor MMP give a nef semiample transform of the big
$N$; see \cite[Corollary~1.3.1]{BCHM} and
\cite[Proposition~1.11]{HuKeel}. Denote that transform by $N'$ on $Z$.
It is an effective nef big integral Weil divisor. Negativity preserves
the uniform generalized discrepancy bound; forgetting the effective
boundary and b-nef part gives a uniform ordinary lc bound on $Z$.
The model remains of Fano type, so $-K_Z$ is big. In particular
$N'-K_Z$ is big. Theorem~1.1 of \cite{BirkarPolarised}, which permits
an integral Weil polarization, now makes $|qN'|$ birational for a
bounded positive integer $q$. Negativity on a common resolution
transfers every such section to $qN$ on the original model: the
pullback of the source divisor dominates that of its transform.
This applies to the rational Cartier divisors represented by Weil
sections as well. Finally $N\le k_0H$, so $|qk_0H|$ is birational and
$qk_0\le C_1/s$.
One can also obtain a bounded positive integer $u$ with $K_Z+uN'$ big.
This additional observation is useful when using the version of the
polarization result with both signs of $K_Z$. Let $u_0\ge0$ be the
pseudo-effective threshold of $K_Z$ with respect to $N'$. It is
rational by the Mori dream space property. If $u_0=0$, bigness follows
for every positive $u$. Otherwise take the adjoint MMP at $u_0$,
regarding $u_0\overline{N'}$ as b-nef big data. Negativity retains a
uniform generalized lc bound, and the nef semiample threshold adjoint
is nonbig, hence defines a contraction of positive relative dimension.
On its geometric generic fibre, ordinary approximation with a small
ample reserve gives an effective uniformly positive-lc log Fano pair.
Theorem~1.1 of \cite{BirkarBAB} bounds the fibre. For a bounded very
ample polarization, the degree of $-K$ is bounded above by adjunction
to a general section curve. The transform of $N'$ has positive integral
degree there: it is effective and integral, and its bigness, preserved
by sections, excludes purely vertical support. Relative adjoint
triviality gives $u_0\deg N'=\deg(-K)$, so $u_0$ is bounded.
Taking an integer $u>u_0$ proves the assertion.
\end{proof}
\subsection{Isolating the horizontal pivot}
Suppose the level is collapsing. Use
Lemma~\ref{b:scale-bad-primes} to choose an extraction $S'\to S$
containing the pivot $G_*$ as well as all bad primes, and put
$H_*=H(G_*)$. Set $b_*=1/4$, take $b\ll b_*$, and choose a fixed
rational $0<\nu\ll b_*$, sufficiently small also for the coefficient
bounds below. On $S'$ define
\[
J=b_*G_*+\nu\sum_{\substack{P\ {\rm bad}\\P\ne G_*}}P.
\]
The coefficient at the pivot will determine the contraction threshold.
The smaller coefficients at the remaining bad primes retain a uniform
positive discrepancy bound without leaving another horizontal component.
\begin{lemma}\label{b:scale-collapse}
At a collapsing level there are a projective $\Q$-factorial Fano type
model $Y$ of $K$, a contraction $\pi:Y\to V$ with $\dim V<\dim S$,
an effective ample rational Cartier divisor $H_V$ on $V$, and a rational
$x\asymp s^{-1}$ with the following properties. The pivot survives on
$Y$, and it is the only horizontal prime in the positive support of the
trace $H''$ of $H$. Moreover
\[
x=\frac{b_*}{H_*},\qquad H_*\asymp s,
\qquad xH''-J''=x\pi^*H_V,
\]
with $J$ as just defined. On common higher models,
\begin{equation}\label{eq:B4}
H\ge\pi^*H_V.
\end{equation}
The datum with b-nef part
$\mathbf M_C=\mathbf M'+(x-d')\overline H$ and boundary obtained by
subtracting $J$ from the $A'$-boundary has effective uniformly
generalized positive-lc trace on $Y$, with total adjoint $x\pi^*H_V$.
The geometric generic fibres of $\pi$ form a bounded family.
\end{lemma}
\begin{proof}
\emph{The threshold and its lower bound.}
On the chosen extraction set
\[
x=\min\{r:rH-J\text{ is pseudo-effective on }S'\}.
\]
Subtracting $J$ leaves the $A'$-boundary effective: its coefficient at
the pivot is at least $1-b-as$, and at the other indicated primes it
is greater than $1-e$. The discrepancy becomes $A'+\overline J$.
Every bad prime has received a fixed positive increment, so this datum
is uniformly generalized $\delta'$-lc for a fixed $\delta'>0$.
The threshold $x$ is positive and finite, because $J$ is nonzero
effective and $H$ is big. It is rational, and $xH-J$ is rationally
linearly equivalent to an effective divisor, by the Mori dream space
property. Adding $J$ to such a representative and dividing by $x$
gives an effective representative of the pullback of $H$ on $S'$.
It descends as $H+\Div(f)/q$ to an effective representative on $S$,
by pushing its coefficients down. Consequently weak essentiality at
the pivot gives
\begin{equation}\label{b:scale-pivot-lower}
\frac{b_*}{x}\le H_*+\eta s\le(1+\eta)s.
\end{equation}
It follows that $x\gtrsim s^{-1}$ and, after choosing $b$ small,
$x-d'\gtrsim x$. Thus $\mathbf M_C$ is b-nef big.
Adding $(x-d')\overline H$ to the moduli changes the total adjoint to
$xH-J$ without changing $A'+\overline J$.
Run the adjoint MMP to a nef semiample transform on $Y$. Its total
adjoint is the trace $xH''-J''$. This divisor is not big, since $x$ is
the threshold. Its semiample contraction $\pi:Y\to V$ therefore has
positive relative dimension. The transformed boundary is effective,
and negativity preserves the uniform generalized $\delta'$-lc bound.
All the birational models used here remain $\Q$-factorial and of Fano
type.
\smallskip\noindent
\emph{Bounded fibres and the upper bound for the threshold.}
We next bound the degree of $xH''$ on a geometric generic fibre. A proper
normal generic fibre of a contraction in characteristic zero is
geometrically integral and normal. Indeed its global functions are
$k(V)$, so $k(V)$ is algebraically closed in $k(Y)$; separability gives
the geometric assertions. Canonical forms on the fibre are obtained
by dividing out a base volume form. A resolution with snc data shows
that generic restriction and extension to an algebraic closure preserve
the required discrepancy bounds. Apply
Lemma~\ref{b:ordinary-approximation} to the b-nef big part, reserving a
tiny ample class. On the geometric generic fibre this gives an ordinary
effective $\delta'/2$-lc log Fano pair. Theorem~1.1 of
\cite{BirkarBAB} therefore bounds the underlying fibre. The theorem
applies over this algebraically closed characteristic-zero field; one
may also transport the data along an isomorphism of the algebraic
closure of $\C(V)$ with $\C$.
Fix a bounded very ample Cartier polarization on these fibres, and
write $\deg$ for intersection with its appropriate power. If $L$
is this polarization in dimension $r\ge1$, a general
complete-intersection curve avoids the singular locus and gives
\[
\deg(-K)\le (r-1)L^r+2.
\]
Thus this degree is bounded above. The trace of any b-nef part has
nonnegative degree: upstairs add an arbitrarily small ample rational
class, take an effective representative, push its trace down, and then
let that class tend to zero. On $Y$ the CY trace of $A^+$ still has
$B_0''\ge H''$ and is effective. Every component of $J''$ is lc for
$A^+$, so its $B_0''$-coefficient is $1$ and $J''\le B_0''$.
The CY equality gives
\begin{equation}\label{b:scale-fibre-degree}
x\deg H''=\deg J''\le\deg B_0''\le\deg(-K)\le C_3.
\end{equation}
Here traces are rational Cartier, since $Y$ is $\Q$-factorial.
The birational functions of Lemma~\ref{b:scale-birationality} remain
sections satisfying $\Div(f)+kH''\ge0$ after the non-extracting MMP.
If $sx$ were sufficiently large, \eqref{b:scale-fibre-degree} and
$k\le C_1/s$ would bound the degree of every possible polar divisor
on a geometric generic fibre by a number strictly less than $1$.
An integral nonzero effective divisor has positive integral degree,
so these polar divisors would vanish. Properness would then put all
the functions in $k(V)$, contradicting their generation of $k(Y)$
and the positive relative dimension. Thus $x\lesssim s^{-1}$.
Equation~\eqref{b:scale-pivot-lower} now also gives $H_*\gtrsim s$.
\smallskip\noindent
\emph{The pivot is the only positive horizontal component.}
Consider the coefficients of $H''-J''/x$. At every positive coefficient
other than the pivot, this divisor has coefficient at least
$H''(P)/2\gtrsim s$. Outside $J''$ this follows from the inductive
coefficient gaps and the extraction bounds. At a bad prime use
$H(P)\ge s/2$, $1/x\le(1+\eta)s/b_*$, and choose $\nu$ sufficiently
small compared with $b_*$. At the pivot its coefficient is at least
$-\eta s$, by \eqref{b:scale-pivot-lower}. If horizontal, that prime
has bounded degree, since its coefficient in $B_0''$ is $1$.
The total degree of $H''-J''/x$ on a generic fibre is zero.
As $\eta\to0$, a horizontal nonpivot component would contribute at
least a fixed multiple of $s$, and the pivot could contribute only
$-O(\eta s)$. This is impossible. There must nevertheless be a
horizontal component in the positive support of $H''$, since otherwise
the birational functions would again be fibrewise regular. Hence the
pivot survives and is the sole such horizontal component. The degree
identity forces its coefficient in $H''-J''/x$ to be zero, proving
$x=b_*/H_*$.
\smallskip\noindent
\emph{The exact divisor on the base.}
It follows that $D_Y=xH''-J''$ is effective and vertical. Semiampleness
gives $D_Y\sim_{\Q}\pi^*L$ for an ample rational Cartier divisor $L$
on $V$. The rational function implementing this equivalence has no
horizontal poles, and hence lies in $k(V)$ by normality and properness
of the generic fibre. Absorb its principal divisor into $L$ and divide
by $x$ to obtain an exact equality
$D_Y=x\pi^*H_V$. Every strict base prime has a strict lift, so
$D_Y\ge0$ implies $H_V\ge0$. Its rational linear class is ample.
Finally, on a common resolution, negativity gives the effective
rational Cartier b-divisor
\begin{equation}\label{b:scale-mmp-correction}
\mathbf E_{\mathrm{MMP}}
:=x\overline H-\overline J-x\overline{\pi^*H_V}\ge0.
\end{equation}
Since $J\ge0$, this proves \eqref{eq:B4}. The threshold datum after
the MMP has discrepancy
\[
(A'+\overline J)+\mathbf E_{\mathrm{MMP}}
=A'+x\overline H-x\overline{\pi^*H_V}.
\]
Thus its uniform positive lower bound holds on every prime over $Y$,
as required for the next step.
\end{proof}
\subsection{Passing the data to the base}
The contraction supplies $H_V$ and a uniformly positive discrepancy on
$Y$. We push this discrepancy to obtain $Q_{\rm next}$, and push $A^+$
separately to retain the lc places. Its bounded index will follow by
returning to the principal-moduli datum on the initial field.
\begin{proposition}\label{b:scale-inheritance}
At a collapsing level the construction of
Lemma~\ref{b:scale-collapse} supplies data at the next level satisfying
all the stated invariants. More precisely, with the CY discrepancy
\[
C=A'+xH=a_K+\mathbf M_C,
\]
put
\begin{equation}\label{b:scale-next-data}
\begin{aligned}
A^+_{\rm next}&=\pi_\#A^+,&
Q_{\rm next}&=\pi_\#(C-x\pi^*H_V),\\
\lambda_{\rm next}&=x^{-1},&
S_{\rm next}&=V,\qquad H_{\rm next}=H_V.
\end{aligned}
\end{equation}
Then $\Lambda_{\rm next}=x^{-1}\pi_\#C$,
$\lambda_{\rm next}\asymp s\to0$, and $s_{\rm next}\gtrsim s$.
The CY moduli of $A^+_{\rm next}$ have a bounded b-Cartier multiple,
and $Q_{\rm next}$ has a uniform positive discrepancy bound on every
prime over $V$.
\end{proposition}
\begin{proof}
\emph{Discrepancy presentations and their lower bounds.}
By \eqref{b:scale-mmp-correction}, the threshold datum on $Y$ has
discrepancy exactly $C-x\pi^*H_V$. Therefore $C$ has effective horizontal trace and is
uniformly generalized klt over the generic point of $V$. Its moduli
are b-nef big, so Proposition~\ref{b:push-presentation}, followed by
translation by the base divisor, gives a presentation for
$Q_{\rm next}$ with rational b-Cartier b-nef part and total adjoint
$xH_V$.
The CY datum $A^+$ has effective horizontal trace on $Y$. Apply
Lemma~\ref{b:primary-minimization} to $A^+,C$. For all sufficiently
small rational $t>0$, uniformly in the member and the base prime, a
lift computing the push of $(1-t)A^++tC$ also computes the push of
$A^+$. On primary minimizers its value is
\[
(1-t)A^+_{\rm next}(P)+t\frac{C(T)}{m_T}.
\]
A minimizer at any one such $t$ minimizes the second summand among
all primary minimizers. Thus the push is exactly affine on a common
interval including $t=0$. At positive $t$ the moduli are b-nef big
upstairs and the horizontal trace is klt, so
Proposition~\ref{b:push-presentation} supplies b-nef rational b-Cartier CY
moduli downstairs. Two distinct positive rational parameters give
a common descent for their affine endpoint. All smaller positive
parameters are nef on that descent; passing to the limit proves
b-nefness of the endpoint. This constructs the required CY presentation
for $A^+_{\rm next}$.
Nonnegativity of $A^+_{\rm next}$ follows from \eqref{eq:B1}.
For effectivity, take a strict lift $F$ of a strict prime $P$ of $V$,
with restriction $mP$. The effective trace upstairs bounds its
normalized discrepancy by $1/m\le1$, proving effectivity of the
pushed boundary. For $A^+$ the stronger inequality
$B_0''\ge H''\ge\pi^*H_V$ gives
\[
A^+_{\rm next}(P)\le\frac{A^+(F)}m
\le\frac1m-H_V(P)\le1-H_V(P).
\]
Its boundary therefore dominates $H_V$.
To obtain a bound for $Q_{\rm next}$ on all valuations, apply
Lemma~\ref{b:ordinary-approximation} to the effective uniformly
generalized $\delta'$-lc threshold datum on the whole of $Y$.
Use an ordinary effective $\delta'/2$-lc representative with the same
relative rational log CY class and no larger discrepancies.
The Fano type property implies that $-K_Y$ is big over $V$.
The discriminant bound of \cite[Theorem~1.1]{BirkarFibrations}
therefore gives a uniform positive bound for the valuative push of
this ordinary pair, on every model of $V$. Equation~\eqref{eq:B1}
then gives the same bound for $Q_{\rm next}$. Notice that this
application uses the uniform lc bound on the entire total pair,
not just its generic fibre.
\smallskip\noindent
\emph{A model of the initial field and the uniform index.}
To bound the new index, we retain a model of the initial field throughout. For each exceptional prime of $S'\to S$,
which is lc for $A^+$, attainment and composition in
Lemma~\ref{b:push-threshold} give a prime over $\mathcal Y$ that is
lc for $A_1^+$ and restricts to a positive multiple of the prescribed
prime. Extract these primes, if they are not already strict, without
any others. To justify this extraction, mix the effective ordinary
lc CY boundary of $A_1^+$ on $\mathcal Y$ with an effective klt log
Fano boundary, giving the latter a sufficiently small positive weight.
The mixture is log Fano and klt, and the requested discrepancies lie
strictly between $0$ and $1$. Lemma~\ref{b:controlled-extraction}
applies, producing a $\Q$-factorial Fano type model $\mathcal Y'$.
The trace of $A_1^+$ remains effective, since each new prime has
$A_1^+$-discrepancy zero. Every strict prime of $S'$ now has a strict
lift on $\mathcal Y'$; for the nonexceptional ones use the existing
morphism to $S$. The same is true for every strict prime of $Y$,
because the MMP extracts none.
There need not yet be a morphism $\mathcal Y'\to Y$, so we construct
one without losing these properties. Choose an ample Cartier divisor
$D_Y$ on $Y$, pull it back on a resolution of the rational map, and
let $D^*$ be its trace on $\mathcal Y'$. It is rational Cartier.
The rational-function sections satisfy, in every nonnegative degree,
\begin{equation}\label{b:scale-auxiliary-sections}
\{f\in K_1:\Div(f)+mD^*\ge0\}
=\{f\in k(Y):\Div(f)+mD_Y\ge0\}.
\end{equation}
Indeed a function on the left has no poles on the normal proper generic
fibre of $\mathcal Y'\to S$: the divisor $D^*$ has coefficient zero
at every prime dominating $S$, since it comes from the field $k(S)$.
It is therefore a member of $k(S)=k(Y)$. The strict lifts just
constructed test its order at every strict prime of $Y$, giving one
inclusion. Conversely an effective Cartier divisor represented by a
section on $Y$ has effective pullback and trace on $\mathcal Y'$.
This also proves the assertion for the possibly nondivisible degrees,
by viewing the conditions as inequalities of rational divisors.
Run a divisor MMP for $D^*$ on the Fano type variety $\mathcal Y'$.
Its effective rational class is pseudo-effective, and its transform
is nef semiample. Negativity preserves the divisible section systems
in \eqref{b:scale-auxiliary-sections}. Their Proj is $Y$, so the
resulting model has a morphism to $Y$ inducing the given field
inclusion. This morphism is a contraction: $k(Y)=k(S)$ is relatively
algebraically closed in $K_1$, and normality identifies its Stein
factor with $Y$. The model remains $\Q$-factorial of Fano type, and
the trace of $A_1^+$ remains effective under non-extraction.
Composing with $Y\to V$ supplies the next auxiliary model.
Valuative composition shows that its push of $A_1^+$ is exactly
$A^+_{\rm next}$.
A klt log Fano boundary on this auxiliary model supplies an effective
generalized klt CY datum with b-nef big part: take minus its own
adjoint as the moduli divisor. Apply
Corollary~\ref{b:principal-denominators} to this contraction and the
initial principal-moduli datum $A_1^+$. Its dimension and denominator
$n$ are bounded independently of the level and member. Thus a
bounded integer clears every discrepancy denominator of
$A^+_{\rm next}$. On a smooth descent of its already constructed CY
moduli, the form discrepancies are integral, so the same integer
clears the coefficients of the moduli trace. An integral divisor on
that smooth model is Cartier. This proves the required bounded
b-Cartier multiple, without introducing any fibre-multiplicity factor.
\smallskip\noindent
\emph{Weak essentiality on the base.}
If $P$ is lc for $A^+_{\rm next}$,
choose a lift $T$, with restriction $m_TP$, computing the push of
$(1-t)A^++tC$ for a fixed sufficiently small rational $t>0$.
Lemma~\ref{b:primary-minimization} gives $A^+(T)=0$. The formulas for
$C$, together with $x\asymp s^{-1}$, give discrepancy inequalities
\[
A^+\lesssim C,\qquad x\Lambda\lesssim C.
\]
For example, $C=(1-\gamma)A^++\gamma Q+(a+x)H$, all summands are
nonnegative, $1-\gamma$ is bounded below, and
$x\lambda/\gamma=xs/b$ is bounded when $Q$ is present.
Compare the mixture at $T$ with its value at a lift minimizing $C/m$.
Since $A^+(T)=0$, this proves
\begin{equation}\label{b:scale-essential-lift}
\frac{\Lambda(T)}{m_T}
\le C_4\Lambda_{\rm next}(P).
\end{equation}
The factor $t^{-1}$ is harmless because $t$ was fixed uniformly.
If $H_V'\ge0$ is rationally linearly equivalent to $H_V$, apply its
rational principal change to $H$ through the field inclusion.
Inequality~\eqref{eq:B4} shows on a common resolution that the new
divisor is effective: it is the sum of $H-\pi^*H_V$ and $\pi^*H_V'$.
Pushing down gives an effective representative $H'\sim_{\Q}H$ on $S$.
Its change at $T$ is exactly
$m_T(P(H_V')-P(H_V))$. Apply \eqref{b:scale-essentiality} at $T$
and then \eqref{b:scale-essential-lift}. We obtain
\[
P(H_V')\le P(H_V)+C_4\eta\Lambda_{\rm next}(P).
\]
Replacing $\eta$ by $C_4\eta$ preserves its convergence to zero.
\smallskip\noindent
\emph{Invariant-prime inequalities and strict coefficient gaps.}
For \eqref{eq:B2}, the first two inequalities follow immediately from
valuative minimization and \eqref{eq:B4}. If an invariant prime has
restriction $jP$ at the old level and remains nontrivial at the next,
its full-multiplicity next $\Lambda$-value is at most
\[
x^{-1}j\bigl((1-\gamma)A^+(P)+\gamma Q(P)+(a+x)H(P)\bigr).
\]
When $Q$ is present, \eqref{eq:B2} gives
\[
j\gamma Q(P)\le\gamma+\frac bsC_0h_E.
\]
The constant $\gamma$ cancels the $-\gamma$ in the bound
$(1-\gamma)jA^+(P)\le(1-\gamma)(1-h_E)$. Using $jH(P)\le h_E$ and
$x\asymp s^{-1}$, we obtain a bound
$x^{-1}+C'h_E$, exactly the next third inequality. Initially the
same calculation simply omits the $Q$-term.
It remains to check the strict coefficient gaps and witnesses; they
cannot be inferred solely from the preceding inequalities. For a
strict prime $P_V$ take a strict lift $F$ on $Y$ with restriction
$mP_V$. The effective threshold boundary and the lower bound for
$Q_{\rm next}$ imply
\[
0<\delta_{\rm next}\le Q_{\rm next}(P_V)
\le\frac{(C-x\pi^*H_V)(F)}m\le\frac1m.
\]
Hence $m$ is uniformly bounded. If $H_V(P_V)>0$ and
$A^+_{\rm next}(P_V)>0$, the lift is non-lc for $A^+$. It is therefore
an old strict prime of $S$, lies outside $J$, and satisfies
$mH_V(P_V)=H''(F)\ge c_0$. This proves the absolute positive gap.
If instead $P_V$ is lc and $H_V(P_V)>0$, the nonpivot coefficient bound
for $H''-J''/x$ gives $H_V(P_V)\gtrsim s$, since the pivot is horizontal
and $m$ is bounded. Before any terminal replacement of the scale,
effectivity of the $Q_{\rm next}$-boundary gives
\[
s_{\rm next}\le H_V(P_V)+\lambda_{\rm next}Q_{\rm next}(P_V)
\le H_V(P_V)+\lambda_{\rm next}.
\]
Since $\lambda_{\rm next}\asymp s$, this proves the required lower
bound by a fixed multiple of $s_{\rm next}$.
Finally suppose $H_V(P_V)=0$. The same nonpivot coefficient bound
forces $H''(F)=0$. All extracted primes have positive $H$-value, so
$F$ is an old strict prime. Its old witness has $h_E=0$ and bounded
restriction multiplicity. Restricting that witness to $k(V)$ only
multiplies this multiplicity by the bounded $m$, giving the next
witness. The lower bound for $Q_{\rm next}$ also yields
$s_{\rm next}\gtrsim\lambda_{\rm next}\asymp s$ from the definition of
the scale. All invariants have now been proved.
\end{proof}
\subsection{The terminal level}
At the last level the unmodified scale is bounded away from zero. We use
that lower bound to choose the constants below, while writing $s=1$ as
agreed above. No further field contraction is needed beyond the map to a
point.
\begin{lemma}\label{b:scale-terminal}
At the final level one may take $Y=S$ up to a small
$\Q$-factorialization, with target a point, and choose a fixed small
rational $x>0$ so that
\[
C=A'+xH=a_K+\mathbf M_C,
\qquad \mathbf M_C=\mathbf M'+(x-d')\overline H
\]
is effective and uniformly generalized klt. Its b-nef big part has
a fixed positive $H$-reserve, its boundary dominates a fixed positive
multiple of $H$, and the underlying $Y$ belongs to a bounded family.
For every invariant prime with restriction $E|_{k(S)}=cP\ne0$,
\begin{equation}\label{b:scale-terminal-bound}
cC(P)\le1-\frac{h_E}{2}.
\end{equation}
\end{lemma}
\begin{proof}
Use $s=1$ and the actual lower bound for the unmodified scale to
choose $e$ as in Lemma~\ref{b:scale-bad-primes}. There are no bad
primes, so $A'$, and hence $C\ge A'$, has a uniform positive discrepancy
bound. Take $b\ll a$ and then a fixed small positive $x$.
Now $x-d'=x+a-b>0$ (initially it is $x+a$), giving a fixed positive
$H$-reserve. The boundary of $C$ is
\[
(1-\gamma)B_0+\gamma B_Q-(a+x)H
\ge(1-\gamma-a-x)H,
\]
with the usual initial convention. Make the constants small enough
that the right side is a fixed positive multiple of $H$.
For the displayed estimate, the same constant-term cancellation as
in the inheritance proof gives
\[
\begin{split}
cC(P)&\le (1-\gamma)(1-h_E)
+\gamma+bC_0h_E+(a+x)h_E\\
&=1-(1-\gamma-bC_0-a-x)h_E.
\end{split}
\]
Choose $\gamma+bC_0+a+x<1/2$; this is uniform since
$\gamma=b\lambda$ and $\lambda\to0$, or $\gamma=0$ initially.
This proves \eqref{b:scale-terminal-bound}.
Ordinary approximation with a tiny ample reserve gives an effective
ordinary uniformly positive-lc log Fano pair on $Y$.
Theorem~1.1 of \cite{BirkarBAB} bounds the underlying variety.
All assertions in dimension zero are interpreted vacuously.
\end{proof}
Starting with $S_1=S$, repeat Proposition~\ref{b:scale-inheritance}
whenever the scale tends to zero, and use
Lemma~\ref{b:scale-terminal} otherwise. Each collapsing level strictly
lowers the dimension. The construction therefore terminates after a
bounded number of levels, with contractions
\[
Y_i\longrightarrow S_{i+1},\qquad
k(Y_i)=k(S_i)=K_i,\qquad
S_{f+1}=\Spec\C,
\]
where the $K_i$ form a nested tower of actual function fields.
Passing to a subsequence makes the number of levels and their dimensions
constant. All positive constants fixed during this construction are
uniform along that subsequence.
==== END Bounded-klt-complements-for-Fano-contractions//scale-tower.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//trace-lifting.tex ====
\section{Cartier trace and controlled lifting}
\label{c:trace-section}
The strict inequality from the previous section supplies a nonzero trace
on a geometric generic fibre of a full initial field. To turn it into
the section estimates needed later, we must lift that nonvanishing twice
and keep its exact value formula. This section proves each lifting step,
then checks that the finite spaces used to count all possible outputs
can be retained before reduction.
We write $\mathfrak p$ for the residue characteristic, reserving $p$
for the auxiliary cone dimension used earlier. We write $P_0=\mathfrak p^l$ for a Frobenius
power. All valuations, optimizers, strict discrepancy inequalities and
finite output bounds are chosen in characteristic zero. Reduction transports
the specified finite algebraic data and numerical monomial formulas; it does
not assert the existence of an optimizer with the same properties in positive
characteristic.
The applications seek nonzero traced outputs $F'$ from actual inputs
$G\in V_M$. In the retained reductions, survival of the $w$-initial
trace will give
\[
F'\in V_{m'},\qquad m'\le M/P_0+n,\qquad
w(F')=\frac{w(G)}{P_0}
-\left(1-\frac1{P_0}\right)A_\theta(w).
\]
The corresponding lower value inequality will remain valid when the
positive weights vary in the same open stratum cone. Thus inputs whose
degrees and values are $O(P_0)$ give outputs in degree and value ranges
bounded independently of $P_0$ for this fixed application, even though
the trace power is not known in advance. Independent
traces in the full initial field will be counted inside the span of
actual output initials. The degree and value bounds determine the
possible filtered output spaces; the span inclusion transfers a lower
dimension bound to them. We state the
relative nonvanishing input first, then prove the lifting properties
before returning to its independent proof.
\subsection{A trace supplied by a strict divisorial budget}
If $J$ is a nonempty finite set of elements of a field, $J^a$ denotes the
set of products of $a$ members, with repetitions, and $\min P(J)$ denotes
$\min_{F\in J}P(F)$.
\begin{lemma}[Strict budget and trace]\label{c:trace-existence}
Let $L$ be a finitely generated field over an algebraically closed field $k$
of characteristic zero, let
$\vartheta=\nu/f^{1/n}$ be a fractional rational volume form, and let
$I\subset L^*$ be a nonempty finite system whose ratios generate $L/k$.
Let $k_I$ be a positive integer. Suppose that every prime divisor $P$ over
a proper model of $L/k$ satisfies
\begin{equation}\label{eq:E6}
A_\vartheta(P)>\frac{\min P(I)}{k_I}.
\end{equation}
After spreading these data, in sufficiently general reductions of arbitrarily
large characteristic $\mathfrak p\equiv1\pmod{nk_I}$ there are $l\ge1$ and
$H\in\operatorname{span}(I^{(P_0-1)/k_I})$, where $P_0=\mathfrak p^l$, such that
\begin{equation}\label{eq:E7}
\mathcal C^l_\vartheta(H)
:=\frac{\mathcal C^l\bigl(Hf^{(P_0-1)/n}\nu\bigr)}{\nu}\ne0.
\end{equation}
Here $\mathcal C^l$ is iterated Cartier trace on rational top forms over
perfect constants. One may choose $H$ to be a single product from
$I^{(P_0-1)/k_I}$.
\end{lemma}
The operator in \eqref{eq:E7} is independent of the presentation of
$\vartheta$. Indeed replacing $(\nu,f)$ by $(a\nu,a^nf)$ multiplies the
trace input by $a^{P_0}$ and the denominator by $a$, and Cartier trace is
$\mathfrak p^{-l}$-linear. We will repeatedly use its logarithmic-coordinate formula.
If $x=(x_1,\ldots,x_s)$ is a $\mathfrak p$-basis, then
\[
b=\sum_{0\le\alpha_j<P_0}b_\alpha^{P_0}x^\alpha
\quad\Longrightarrow\quad
\mathcal C^l\left(b\bigwedge_jd\log x_j\right)
=b_0\bigwedge_jd\log x_j.
\]
This is the iterated Cartier isomorphism
\cite[Th\'eor\`eme~1]{Cartier1957}: logarithmic differentials are
fixed, and all the other displayed monomials are killed.
The proof of Lemma~\ref{c:trace-existence} appears in the final
subsection, after the lifting and finite-space estimates. Those steps
explain how its surviving product supplies
the actual degree, value, and dimension bounds just described.
\subsection{Survival of initials and the degree bound}
We now distinguish the actual field $K$ from the full initial field $K_w$
of a real monomial valuation $w$. The latter is the rational function
field in the independent normal variables over the field of the stratum.
We use the angular systems $V_m$ and the fractional form $\theta$ from
\eqref{eq:A13}. The reductions in the following statements retain the
monomial chart, its stratum, its form identities, and separating coordinates.
The simultaneous finite-data construction is given below in
Lemma~\ref{c:trace-finite-data}.
\begin{lemma}[Initial survival]\label{c:trace-initial-survival}
Transport $w$ to a reduction of its monomial chart, using the same positive
normal weights. Let $x$ consist of its normal parameters and a unit
separating tangential basis. Retain the conditions that $x$ is a $\mathfrak p$-basis
of the actual field and that its initials form a $\mathfrak p$-basis of $K_w$.
Write
\[
\theta=\frac{\bigwedge_jd\log x_j}{f_x^{1/n}}.
\]
For $\mathfrak p\equiv1\pmod n$, $P_0=\mathfrak p^l$, and every $G$ in the reduced actual
field,
\begin{equation}\label{eq:E8}
w\bigl(\mathcal C^l_\theta(G)\bigr)
\ge \frac{w(G)}{P_0}
-\left(1-\frac1{P_0}\right)A_\theta(w).
\end{equation}
We set $w(0)=+\infty$. For $G\ne0$, equality holds if and only if
$\mathcal C^l_{\theta_w}(\bar G)\ne0$; in that case
\[
\overline{\mathcal C^l_\theta(G)}
=\mathcal C^l_{\theta_w}(\bar G).
\]
The discrepancy in \eqref{eq:E8} is the characteristic-zero numerical
quantity transported by the chart formula. The same assertions hold
with ordered monomial weights, and the inequality holds under local real
weight variation within the same open stratum cone.
\end{lemma}
\begin{proof}
The $P_0$-basis expansion has the unique form
\[
Gf_x^{(P_0-1)/n}
=\sum_{0\le\alpha_j<P_0}c_\alpha^{P_0}x^\alpha,
\qquad \mathcal C^l_\theta(G)=c_0.
\]
There is no cancellation between the least-valued terms on the right:
the monomials $\bar x^\alpha$ are linearly independent over $K_w^{P_0}$.
Consequently
\[
P_0w(c_0)\ge w(G)+\frac{P_0-1}{n}w(f_x).
\]
The logarithmic form identity \eqref{eq:A6} says
$A_\theta(w)=-w(f_x)/n$, proving \eqref{eq:E8}.
Taking least terms in the same expansion computes
$\bar G\bar f_x^{(P_0-1)/n}$ in the initial $P_0$-basis. Its zero-index
coefficient is nonzero exactly when the inequality is an equality, and
then it is the initial of $c_0$. The leading-form identity in
\eqref{eq:A6} identifies the fractional form in this calculation with
$\theta_w$. This proves the equality criterion as well as the asserted
initial identity. The argument only uses the order, addition of values,
and independence over powers, so it also applies to ordered value groups.
For nearby positive weights in the same chart the coefficient argument
again gives the inequality, with the corresponding numerical discrepancy.
\end{proof}
\begin{lemma}[Angular degree of a traced output]\label{c:trace-degree}
In the retained reductions of the angular algebra, if $G\in V_M$, then
$\mathcal C^l_\theta(G)\in V_{m'}$ for some $m'\in n\Z_{\ge0}$ with
\[
m'\le\frac{M}{P_0}+n.
\]
\end{lemma}
\begin{proof}
Choose $m'\in n\Z_{\ge0}$ with
$P_0m'+P_0-1\ge M$ and $m'\le M/P_0+n$. Recall the cone notation
\[
\Omega=t\,\omega\wedge d\log t,\qquad
h=t^nf_h^W,\qquad \theta=\omega/(f_h^W)^{1/n}.
\]
Since $\mathfrak p\equiv1\pmod n$, the exponent in
\[
b=G h^{(P_0m'+P_0-1)/n}
\]
is integral; $b$ is regular on $W$ by $G\in V_M$ and the displayed
inequality. Apply $b\mapsto\mathcal C^l(b\Omega)/\Omega$.
The log-$t$ formula and Frobenius-linearity give
\[
\frac{\mathcal C^l(b\Omega)}{\Omega}
=h^{m'/n}\mathcal C^l_\theta(G).
\]
Indeed the $t$ exponent in $b\Omega$ is $P_0(m'+1)$, which trace divides
by $P_0$, and the remaining coefficient is precisely the defining
coefficient of $\mathcal C^l_\theta$.
The output is regular in codimension one, because $\Omega$ is a regular
canonical generator there and Cartier trace sends regular top forms to
regular top forms. It is therefore regular everywhere by normality.
Its physical degree is $m'$, so division by $h^{m'/n}$ places the output
in $V_{m'}$. The normal graded algebra of $W$, its field and its
codimension-one canonical generator are retained in the spread. This
argument requires no new discrepancy assertion in positive characteristic.
\end{proof}
\subsection{From a relative trace to the full initial field}
Fix independent homogeneous probes $g=(g_1,\ldots,g_{r_g})$ in $K_w$ and
use the relative fractional form obtained by dividing $\theta_w$ by the
wedge of the \emph{ordinary} differentials $dg_j$. To describe its
geometric generic field without confusing constants and functions, first
introduce independent constants $\xi_1,\ldots,\xi_{r_g}$ and put
$k'=\overline{k(\xi_1,\ldots,\xi_{r_g})}$. Extend the full field model by
these constants; the functions $g_j$ remain independent over $k'$ on this
total model. At the generic point $\eta$ of a chosen component of
$g=\xi$, the residue field is the chosen geometric generic field for
$K_w/k(g)$. Every nonzero rational function from $K_w$ before this
constant extension is a unit at $\eta$, and its residue is its prescribed
relative image, since $\eta$ maps to the original generic point.
Choose a smooth model on which the map $g$ is smooth at $\eta$. Then
\[
u_j'=g_j-\xi_j\quad(1\le j\le r_g)
\]
are a regular system of parameters at $\eta$. Choose unit supplementary
coordinates $s$ whose residues form a separating transcendence basis of
the relative field. They may be chosen among pre-extension rational
functions. Spread the total model, this component, its relative model,
and all these coordinate and restriction data together. After reduction,
$u',s$ are a $\mathfrak p$-basis of the total field, and the Taylor initials of these
coordinates form a $\mathfrak p$-basis of the Taylor initial field. The relative
trace below is taken in the field of this transported component. Its
existence is supplied by Lemma~\ref{c:trace-existence} applied before
reduction to the fixed characteristic-zero pair.
\begin{lemma}[Taylor flags]\label{c:trace-taylor-flags}
Suppose a product of restrictions of homogeneous initials gives a
nonzero relative trace as in \eqref{eq:E7}, and let $H$ denote the lift
of that product by the same factors to the full initial field. Then
$\mathcal C^l_{\theta_w}$ is nonzero on
\begin{equation}\label{eq:E9}
H\prod_{j=1}^{r_g}(u_j')^{P_0-1}.
\end{equation}
More precisely, give $u_j'$ the standard positive basis values in
lexicographically ordered $\Z^{r_g}$, and use the resulting Taylor valuation
at $\eta$. If a shift input $B$ has Taylor order $P_0a$, with
$a\in\Z^{r_g}$, and has a nonzero scalar leading coefficient, then
\[
\operatorname{ord}_{\mathrm T}\left(
\mathcal C^l_{\theta_w}\left(
B H\prod_j(u_j')^{P_0-1}\right)\right)=a.
\]
For $r_g=0$ the empty product is one and the statement is the original
relative nonvanishing assertion.
\end{lemma}
\begin{proof}
Write $\bar u_j'$ for the normal variables of the Taylor initial field;
this bar refers to Taylor initial, rather than the original $w$-initial.
In the logarithmic basis for $u',s$, write the denominator of $\theta_w$
as $f_{u',s}$. Its Taylor initial is
\[
\operatorname{in}_{\mathrm T}(f_{u',s})
=\prod_j(\bar u_j')^{-n}f_{\mathrm{rel}},
\]
where $f_{\mathrm{rel}}$ is the denominator of
$\theta_{w,\mathrm{rel}}$ in the logarithmic basis of the residue
coordinates $s$. This is precisely the effect of first dividing by
$\bigwedge_jdg_j=\bigwedge_jdu_j'$ and only then changing from ordinary
to logarithmic normal differentials. The unit coefficient identities
are among the retained data at the fiber.
The factors of $H$ are units at $\eta$ and restrict to the factors of
the nonzero relative trace. If the scalar leading coefficient of $B$
is $c\ne0$, the Taylor initial of the coefficient to which Cartier trace
is applied is consequently
\[
c(\bar u')^{P_0a}
H|_\eta\, f_{\mathrm{rel}}^{(P_0-1)/n}.
\]
The normal factor $\prod_j(\bar u_j')^{-(P_0-1)}$ from the denominator
has been canceled exactly by the flags in \eqref{eq:E9}. Extraction in
the initial $P_0$-basis therefore gives
\[
c^{1/P_0}(\bar u')^a
\mathcal C^l_{\theta_{w,\mathrm{rel}}}(H|_\eta),
\]
which is nonzero. The ordered-weight version of
Lemma~\ref{c:trace-initial-survival} identifies it with the Taylor initial
of the full trace, proving the asserted order and nonvanishing.
\end{proof}
There is one more lifting step. Expand an input in
Lemma~\ref{c:trace-taylor-flags}, including any shift, as a linear
combination of products of homogeneous angular initials over the enlarged
constants. Suppose that the factors in this expansion have actual lifts
with the prescribed degree and value bounds. A homogeneous summand with
nonzero trace \emph{in the full initial field} then has an actual lift
$G$ for which Lemma~\ref{c:trace-initial-survival} gives equality in
\eqref{eq:E8} and identifies its traced initial. Additivity and perfectness
of the constants show that the full initial-field outputs of all these
inputs belong to the span of such actual output initials. In particular,
a family of independent shifted outputs is spanned in this way; there
is no need to select a single actual summand equal to each full output.
Degree and value budgets are checked for every expanded summand. Thus
cancellation in the expansion is not needed for the bounds.
The order of the argument is essential: relative survival first gives
full initial-field survival by Lemma~\ref{c:trace-taylor-flags}, and only
then does Lemma~\ref{c:trace-initial-survival} give actual survival and
its equality formula. Mere nonvanishing of an actual trace would not
give that equality.
Figure~\ref{fig:trace-survival} summarizes these two lifting steps. The
output degree is controlled term by term, while independent full-field
outputs are counted inside the span of actual output initials.
\input{figures/trace-survival}
\subsection{Finite data retained before choosing the characteristic}
The trace power and the surviving products are chosen only after
reduction. Their bounds must therefore hold for every possible output
in the prescribed degrees. We retain the relevant finite-cutoff
filtrations of the whole modules $V_m$, including their high-cost
parts, before making those choices.
\begin{lemma}[Simultaneous finite-data reduction]\label{c:trace-finite-data}
For each single application of the preceding trace constructions one may
retain simultaneously the following data on a sufficiently general spread:
the specified actual and initial monomial charts and fractional forms;
the total and relative charts for the geometric generic component;
finitely many functions, field identities and initial identifications;
finitely many Taylor leading terms; and all specified finite-degree,
finite-cutoff comparisons between the angular filtrations of $v$ and $w$.
The last assertion concerns every element of each tested space, including
elements with arbitrary coefficients in the extended reduced constants.
A strict lower $w$-bound on the part of high $v$-cost also persists locally
under positive weight variation in the specified chart.
These data, and all required output degrees and cutoffs, are fixed before
$\mathfrak p$ and $P_0$. They may depend on the sequence member, the optimizer,
the parameter, and the particular application. Products formed later
from the retained finite lists, with multiplicities depending on $P_0$,
require no new spreading assertion.
\end{lemma}
\begin{proof}
\emph{Models, coordinates, and forms.}
Descend the finite algebraic data to a finitely generated integral base
over $\Z$. When geometric generic constants $k'$ are used, first extend
the actual and full initial models by these constants as well, and include
the finitely many constants occurring in the constructions in the spread.
Spread generators, relations, maps, their inverse identities on birational
opens, and rational forms. Shrinking the base retains flatness, geometric
integrality, and normality where used. It also retains the smooth charts,
chosen transverse integral strata, normal parameters and \'etale differential
or supplementary coordinates. On $W$ retain the graded algebra, its
birational field description, and the fact that $\Omega$ generates in
codimension one: its generating smooth open has complement of codimension
at least two, and the same remains true after shrinking. Thus the
$\mathfrak p$-bases and their initial-field identifications used above coexist with
the angular algebra. They may also be imposed together with the finite
spreading conditions in the deferred proof of
Lemma~\ref{c:trace-existence}, by a common shrinking of the spread.
\emph{Finite initial expressions.}
Consider a regular numerator or denominator at one of the stratum points.
For sufficiently large ordinary normal degree $N$, successive jets express
it modulo the $N$-th power of the normal ideal as a finite sum of distinct
normal monomials with lifted unit coefficients. Retain their nonzero
residue coefficients and the congruence, whose remainder is a combination
of degree-$N$ monomials with regular coefficients. Near fixed positive
normal weights, all terms in that remainder have weight greater than the
weights under consideration. Consequently the finite expansion determines
the value and the initial expression there. Spread this finite expansion
and its unit and stratum data. Its minimum and homogeneous initial
expression then agree in characteristic zero and in the identified reduced
stratum/normal-variable field. Applying this to numerators and denominators
handles the specified rational functions. For a finite list it also
retains the piecewise leading rules for locally varying positive weights.
Finite initial linear independences are retained by a nonzero evaluation
determinant, after adjoining the finitely many points or constants needed
to witness it. Finite algebraic independences in characteristic zero are
retained by nonzero differential wedges. Include the full initial-field
model itself through its stratum/normal-variable identification. This
ensures that preservation of the initials is a statement in the full
initial field, rather than only a list of numerical equalities.
\emph{The geometric generic component and Taylor data.}
Spread the smooth component chart at $g=\xi$ as a subvariety of the total
initial-field model over the enlarged constants. Retain geometric
integrality of the selected component, the field identifications, the
local normal equations $u'=g-\xi$, the differential conditions, and the
restrictions of the specified functions and forms to that component.
For a specified Taylor leading exponent, the monomial ideal of exponents
strictly above it is finitely generated. The required leading expansion
is a finite congruence modulo this ideal. A congruence first obtained in
the completion contracts to the local ring, by faithful flatness of
completion. Thus it too can be spread, together with any required nonzero
scalar leading coefficient. Apply this to a numerator and a denominator
if necessary. Later shift products only multiply these predetermined
leading expressions. Their possibly $P_0$-dependent multiplicities cause
no new condition on the spread. This retains the relative form, the
Taylor flag formula, and the actual-to-initial identification simultaneously.
\emph{Entire spaces at finite cutoffs.}
The angular space $V_m$ is a finite module over $k[Z]$. At each of the
finitely many tested degrees, choose module generators $e_{m,t}$ and
choose affine algebra generators $a_1,\ldots,a_s$ of $k[Z]$ vanishing at
$z$. Both $v$ and $w$ are positive on these $a_j$, by the given barriers
and the initial construction. Retain the module generation and these
values after constant extension and reduction. There is a common
$\delta>0$ that is a lower bound for all their $v$- and $w$-values;
for $w$ it remains a lower bound, after a harmless decrease, on a small
neighborhood of the chosen positive weights. If
$\mathfrak a=(a_1,\ldots,a_s)$, every element of
$\mathfrak a^NV_m$ has both values at least
\[
N\delta+\min_t\{v(e_{m,t}),w(e_{m,t})\}.
\]
Choose $N$ so this exceeds every relevant upper cutoff. The same estimate
holds after reduction, and locally for $w$. Modulo this tail, the finite
span of $a^\alpha e_{m,t}$ with $|\alpha|<N$ represents all of $V_m$.
Choose bases of that finite span adapted separately to the two valuation
filtrations. Retain the relations, change-of-basis matrices, values, and
linear independence of initials in each same-value slice. The last
condition prevents cancellation of a least-valued slice for any nonzero
coefficient vector over the extended constants. Therefore the retained
finite spans have exactly the same filtration dimensions, gaps, and
specified cutoff inclusions. Combined with the tail estimate, these
statements hold for every element of $V_m$ through the specified cutoffs.
For the high-$v$-cost subspace use its retained finite basis. A lower
$w$-bound with strict margin persists under a small change of positive
weights by continuity of each basis value; the tail remains above the
same bound. This proves also the local assertion. Only these finite
comparisons are transported; no global optimization or infinite ratio
test is asserted in the reduction.
\end{proof}
\subsection{Applying the strict budget and accounting for all factors}
\begin{proposition}[Budgeted actual outputs]\label{c:trace-budget}
Apply the strict budget \eqref{eq:E5} on the geometric generic field of
$K_w$ over the independent probes $g$, choosing the optional $\bar U_j$
to include generators of the relative field. Assume each probe has a fixed
angular lift $g_j=\overline{G_j}$ with $G_j\in V_{d_j}$. Any prescribed
shift is assumed to satisfy the Taylor leading-term hypothesis of
Lemma~\ref{c:trace-taylor-flags} and to have an expansion in fixed homogeneous
factors with angular lifts and bounded factor counts divided by $P_0$.
Clear all the rational exponents
by one positive integer $k_I$. The trace product may be chosen so that
the multiplicities of the active factors in its lift $H$ are
\begin{equation}\label{eq:E10}
(P_0-1)\tau_i\alpha_i,\qquad
\tau_i\ge0,\qquad
\sum_i\tau_i=1
\quad\text{or}\quad \sum_i\tau_i\le1\ \text{if the cap is active}.
\end{equation}
The multiplicity of a tiny factor $\bar U_j$ is at most
$(P_0-1)e_j$. Expanding the Taylor flags adds between zero and $P_0-1$
copies of each probe. Every surviving homogeneous initial summand has
an actual traced output $F'$ of degree at most its total physical input
degree divided by $P_0$, plus $n$. At the chosen $w$ its value satisfies
equality in \eqref{eq:E8}; on a local affine continuation in the same
open stratum cone the lower inequality remains available.
Common mandatory factors included in every generator of the input
system remain in every product and every expanded summand. All output
spaces and finite cutoff comparisons needed for this conclusion may be
chosen before $\mathfrak p$ and $P_0$, from bounds for the normalized factor counts.
\end{proposition}
\begin{proof}
Choose $k_I$ so that $k_I\alpha_i$ and $k_Ie_j$ are positive integers.
On the relative field take the system consisting of
\[
\bar F_i^{k_I\alpha_i}\prod_j\bar U_j^{l_j},
\qquad 0\le l_j\le k_Ie_j,
\]
for each active index $i$; if the cap is active include also the same
products with initial active factor $1$. The minimum of a valuation on
this system, divided by $k_I$, is exactly
\[
\min_{i\ \mathrm{active}}\alpha_iP(\bar F_i)
+\sum_je_j\min\{0,P(\bar U_j)\},
\]
with zero included in the minimum for the active cap. Its ratios include
each $\bar U_j$, by comparing the exponents $0$ and $1$ while holding
the other choices fixed. The field-generation provision in
\eqref{eq:E5} therefore makes the system birational. Equation
\eqref{eq:E5} is exactly the hypothesis \eqref{eq:E6} for this fixed
system.
Apply Lemma~\ref{c:trace-existence} and choose a surviving product of
$N_I=(P_0-1)/k_I$ generators. If $N_i$ occurrences use the active index
$i$, put $\tau_i=N_i/N_I$. Then its $\bar F_i$ exponent is
$N_ik_I\alpha_i=(P_0-1)\tau_i\alpha_i$, proving \eqref{eq:E10}.
A cap occurrence accounts for the possible deficit in $\sum_i\tau_i$.
The bound for the $\bar U_j$ exponents follows from
$N_Ik_Ie_j=(P_0-1)e_j$.
Lift this product using the same full initial factors and multiply by
the Taylor flags of \eqref{eq:E9}, and by any shift prescribed in the
application. Each flag expands over the extended constants as a linear
combination of $g_j^a$ with $0\le a\le P_0-1$. More generally, expand
every prescribed shift in its fixed list of homogeneous factors, with
the Taylor leading term required by Lemma~\ref{c:trace-taylor-flags}.
Lemma~\ref{c:trace-taylor-flags} supplies full initial-field survival.
For every homogeneous summand whose initial trace survives,
Lemma~\ref{c:trace-initial-survival} produces the actual trace and its
initial with equality in \eqref{eq:E8}. Multiplying actual lifts puts
the input in $V_M$, where $M$ is the sum of their physical degrees, and
Lemma~\ref{c:trace-degree} gives $m'\le M/P_0+n$.
These surviving initials span the full initial-field output, as explained
after Lemma~\ref{c:trace-taylor-flags}.
If, in a later application, every system generator is multiplied by
$\prod_a\bar B_a^{k_I\beta_a}$, with nonnegative rational $\beta_a$
and integral $k_I\beta_a$, then every selected product contains
$\prod_a\bar B_a^{(P_0-1)\beta_a}$. These factors also remain in each
term after expanding flags and shifts. Selection of a surviving term
therefore retains all mandatory factors, including their degree and
value contributions.
Finally fix the finite lists of factors and their actual lifts before
reduction. Their physical degrees, values at the tested valuations and,
when needed, Lipschitz bounds on the given continuation are now known.
For any additional shifts, include their separately verified bounds on
normalized factor counts among these data. Each multiplicity is then
bounded by a fixed constant times $P_0$. Summing and dividing by $P_0$,
the degree estimate and equality at $w$ give predetermined output degree
and $w$-value bounds; the rounding term $n$ is also fixed. The required
$v$-filtration cutoffs are chosen before reduction as well. The same
accounting is applied separately to every term in
the expansions of flags and shifts. Any output spaces, their
$v$-filtration cutoffs, and high-$v$-to-high-$w$ inclusions used later
are retained in their entirety by Lemma~\ref{c:trace-finite-data}, before
$\mathfrak p$ and $P_0$ are chosen. Thus they cover every possible later output
coefficient vector, without having to lift that particular vector to
characteristic zero.
The normalized contributions of the tiny factors are bounded by the
fixed finite data times $e_j$. Those from replacing $\kappa/b_i$ by
$\alpha_i$ are bounded by the same finite data times the approximation
errors, since the $\tau_i$ lie in the simplex of \eqref{eq:E10}.
As in the construction of \eqref{eq:E5}, first choose the positive
$e_j$ sufficiently small and then approximate sufficiently closely to
respect any prescribed finite tolerances. This controls physical degrees,
values and Lipschitz changes on the already specified continuation.
The strict budget is kept during these choices. Neither uniform
comparison for varying pairs nor a global characteristic-$\mathfrak p$ optimizer
is used.
\end{proof}
\subsection{Proof of relative trace existence}
\begin{proof}[Proof of Lemma~\ref{c:trace-existence}]
\emph{The cone and its sub-klt boundary.}
Introduce an indeterminate $u$, put $S=k[Iu]$, and let $R_c$ be the
normalization of $S$ in its fraction field. The ratios of $I$ generate $L$,
so $\Frac(S)=L(u)$. Finite normalization gives a nonnegatively graded
normal affine domain $R_c$ with $(R_c)_0=k$. The conductor is a nonzero
homogeneous ideal; multiplying a nonzero homogeneous conductor element by
a positive-degree element if necessary, choose $d_c$ of positive degree
with
\[
0\ne d_c\in R_c,\qquad d_cR_c\subset S.
\]
Set
\[
\Theta=\nu\wedge d\log u,\qquad
K^\Theta=\Div_{R_c}(\Theta),\qquad
\Delta_c=-K^\Theta+\frac1n\Div(f)-\frac1{k_I}\Div(u).
\]
In particular $K^\Theta+\Delta_c$ is rational-principal.
We verify that $(\Spec R_c,\Delta_c)$ is sub-klt. An invariant prime
valuation $D$ centered on $\Spec R_c$ is Gaussian in $u$ over $L$.
Its restriction to $L$ is either trivial or a positive multiple of a prime
divisorial valuation $P$. For the latter assertion, the restriction has
rational rank one, and the residue-transcendence-degree inequality for
$L(u)/L$, together with Abhyankar equality for $D$, forces Abhyankar
equality for the restriction. We allow the positive multiple in the
notation $P$; \eqref{eq:E6} is homogeneous and therefore applies to it.
The logarithmic differential formula \eqref{eq:A6} gives
\[
A_{R_c,\Delta_c}(D)=A_\vartheta(P)+\frac{D(u)}{k_I}.
\]
Since $D(Fu)\ge0$ for every $F\in I$, we have
$D(u)\ge-\min P(I)$, and the displayed discrepancy is positive by
\eqref{eq:E6}. If the restriction is trivial, set $A_\vartheta(0)=0$.
Then $D(u)\ge0$, and equality would make the Gaussian valuation $D$
trivial; hence again the discrepancy is positive. These tests suffice:
an equivariant log resolution of the homogeneous divisor data has invariant
divisor components, because a connected torus cannot permute a finite set
nontrivially. All the divisors on this resolution consequently have
positive log discrepancy, which is the sub-klt criterion.
\emph{An effective boundary and multiplier-ideal membership.}
Choose a homogeneous $0\ne h_c\in R_c$ such that
\[
\Delta'=\Delta_c+\Div(h_c)\ge0.
\]
We record a construction that also makes every coefficient of $\Delta'$
at most one. Each coefficient of $\Delta_c$ on $\Spec R_c$ is strictly
less than one. If they are all nonnegative, take $h_c=1$. In particular,
when $L=k$, the ring is $k[u]$ and the sole boundary coefficient is
$1-1/k_I$. Otherwise $\dim R_c\ge2$. At the finitely many negative
prime components $Q_i$ write
$a_i=\lceil-\operatorname{coeff}_{Q_i}\Delta_c\rceil$, and use the
homogeneous ideal
\[
\mathfrak a=\bigcap_i Q_i^{(a_i)}.
\]
Its minimum order is $a_i$ at $Q_i$ and zero at every other height-one
prime, as follows by localization. Choose homogeneous generators of
$\mathfrak a$. Multiply each by all products of the degree-one generators
$Iu$ that bring it to one sufficiently large common degree. The resulting
finite system retains the same minimum order at every height-one prime.
Indeed no such prime contains all of $Iu$: by the finite extension from
$S$, the inverse image of the vertex has codimension $\dim R_c\ge2$.
A general combination attains the prescribed minimum orders at the finite
support of $\Delta_c$. Away from that support the system has no fixed
prime divisor, and Bertini on the smooth basepoint-free open makes any
new divisorial zeros reduced. This gives the asserted $h_c$ and boundary.
Write $\mathcal J$ for the multiplier ideal, allowing a fractional ideal
for a non-effective sub-pair. Sub-klt gives
\[
R_c\subseteq\mathcal J(R_c,\Delta_c).
\]
For example, this containment follows on a log resolution by rounding up
the relative discrepancy divisor, whose coefficients are greater than
$-1$. Cartier translation therefore yields
\[
\mathcal J(R_c,\Delta')
=h_c\mathcal J(R_c,\Delta_c)\supseteq h_cR_c.
\]
Only the membership of $h_c$ is needed. The fractional multiplier ideal
of a non-effective sub-klt boundary need not equal the structure sheaf.
\emph{Comparison and the full family of Cartier maps.}
Apply the fixed-pair multiplier/test-ideal comparison to the effective pair
$(\Spec R_c,\Delta')$. Its hypotheses hold because
\[
K^\Theta+\Delta'
=\frac1n\Div(f)-\frac1{k_I}\Div(u)+\Div(h_c)
\]
is rational-principal, with index dividing $nk_I$; neither klt of
$\Delta'$ nor an independent $\Q$-Gorenstein hypothesis on $R_c$ is
required. The comparison is \cite[Theorem~3.2]{Takagi}; the dense-open
form with the big test ideal is recalled in
\cite[Theorem~11]{DFDTT}, applied with
$Z=0$. Thus, on a sufficiently general reduction to a finite residue
field of characteristic $\mathfrak p\equiv1\pmod{nk_I}$, the image of $h_c$ belongs
to the big test ideal $\tau_b(R_c,\Delta')$.
We use the following characterization: on an $F$-finite normal domain the
big test ideal of an effective rational divisor pair is the smallest
nonzero ideal preserved by all restricted maps
\[
\Hom_{R_c}\!\left(
F_*^eR_c\bigl(\lceil(\mathfrak p^e-1)\Delta'\rceil\bigr),R_c\right),
\qquad e\ge1.
\]
This is \cite[Definitions~3.15--3.16 and Theorem~3.18]{SchwedeNonQG},
with the divisor-pair identification in its Remarks~3.10 and~3.17.
The log-$\Q$-Cartier comparison with the usual test ideal is also recorded
in \cite[Theorem~2.8(2)]{Takagi} and
\cite[Remark~6.1]{SchwedeNonQG}.
In the reduction define
\[
T^e(b)=\frac{\mathcal C^e(b\Theta)}{\Theta},\qquad
L_e=R_c\bigl((1-\mathfrak p^e)(K^\Theta+\Delta')\bigr).
\]
We use the convention
$R_c(D)=\{b\in\Frac(R_c):\Div(b)+D\ge0\}$, so explicitly
\[
L_e=f^{(\mathfrak p^e-1)/n}u^{-(\mathfrak p^e-1)/k_I}h_c^{\mathfrak p^e-1}R_c.
\]
All the allowed maps of degree $e$ are exactly $T^e(r\,\cdot)$,
$r\in L_e$. Here every $(\mathfrak p^e-1)\Delta'$ is integral, because
$\mathfrak p-1$ is divisible by $nk_I$; thus this statement describes the full
allowed family in every Frobenius degree, with no rounding or omitted
degrees. This is the divisor--Frobenius-map correspondence
\cite[Corollary~3.10 and Theorems~3.11, 3.13]{SchwedeFAdjunction}.
We check it in the chosen trivialization to fix the normalization.
On the fraction field a nonzero
Frobenius-linear functional generates all such functionals by input
multiplication. At a smooth codimension-one point, Cartier trace for a
regular canonical generator in ordinary coordinates extracts the monomial
with all exponents $\mathfrak p^e-1$. Multiplication and trace give the perfect
pairing on the Frobenius basis, also after an \'etale change of coordinates.
Changing the canonical generator to $\Theta$ twists the multiplier by
the $(1-\mathfrak p^e)$-th power of its coefficient. Allowing the poles prescribed
by $(\mathfrak p^e-1)\Delta'$ gives precisely $L_e$ at that height-one point.
Normality makes these codimension-one conditions sufficient globally.
\emph{A compatible ideal and homogeneous extraction.}
Set $T^0=\id$, $L_0=R_c$, and
\[
J=\sum_{l\ge0}T^l(d_ch_cL_l)\subset R_c.
\]
This is an ideal: ordinary multiplication can be moved inside $T^l$ by
$aT^l(b)=T^l(a^{\mathfrak p^l}b)$. It is nonzero, since its degree-zero Frobenius
summand is $d_ch_cR_c$. It is preserved by every allowed map. Indeed,
if $a_s\in L_s$ and $a_l\in L_l$, composition gives
\[
T^s\bigl(a_sT^l(d_ch_ca_l)\bigr)
=T^{s+l}\bigl(d_ch_ca_s^{\mathfrak p^l}a_l\bigr),\qquad
L_s^{\mathfrak p^l}L_l\subseteq L_{s+l}.
\]
In particular the conductor factor stays present. The minimality
characterization therefore gives $\tau_b(R_c,\Delta')\subseteq J$.
A comparison localized at the homogeneous vertex would already suffice:
compatibility localizes, because Hom localizes and denominators move
inside trace after taking Frobenius powers.
The ideal $J$ is homogeneous. Indeed log-$u$ trace kills a homogeneous
input whose degree is not divisible by $\mathfrak p^l$, and otherwise divides its
degree by $\mathfrak p^l$. Thus local membership of the homogeneous element $h_c$
also gives global membership: multiply by a vertex unit, and project the
result to degree $\deg h_c$; the unit's nonzero constant part is the only
contribution in that degree. We conclude $h_c\in J$.
For $P_0=\mathfrak p^l$ the input factor $h_cL_l$ contains $h_c^{P_0}$.
Dividing the finite trace identity for $h_c\in J$ by $h_c$, using
Frobenius-linearity in every summand, gives
\[
1\in\sum_{l\ge0}T^l\left(
f^{(\mathfrak p^l-1)/n}u^{-(\mathfrak p^l-1)/k_I}d_cR_c\right).
\]
Project to homogeneous degree zero. The term with $l=0$ contributes
nothing, since $d_c$ has positive degree and $R_c$ is nonnegatively
graded. At least one term with $l\ge1$ has nonzero degree-zero output.
It arises from a homogeneous element
\[
H u^{(P_0-1)/k_I}\in d_cR_c\subset k[Iu].
\]
Consequently $H\in\operatorname{span}(I^{(P_0-1)/k_I})$.
The log-$u$ coefficient formula identifies its nonzero output with
\eqref{eq:E7}. Additivity and perfectness of the constants let us choose
a single product in that span with nonzero output.
\emph{The reduction choices.}
Spread the finite normalization algebra, its generators and inclusions,
its birational field identification, the conductor, the forms, and all
the boundary data together. Shrinking the base preserves geometric
normality, grading, codimension-one orders, effectivity of $\Delta'$,
and the displayed rational-principal log canonical relation. For the
form and divisor identities, spread their rational sections on smooth
opens, or on the chosen resolving model, and retain codimension at least
two for the complements used in codimension-one tests downstairs.
The fixed-pair comparison above then holds on a sufficiently general
open of this spread. These requirements are compatible with any finite
additional open conditions. A nonempty open of a finite-type integral
base of characteristic zero has nonempty fibers at all sufficiently
large primes; choose arbitrarily large primes
$\mathfrak p\equiv1\pmod{nk_I}$ and closed points in those fibers. Their residue
fields are finite and hence $F$-finite. After the trace identity has been
obtained, extension to algebraically closed perfect constants preserves
it, and preserves its particular nonzero output, by the coefficient
formula and faithful scalar extension. This proves the lemma.
\end{proof}
The relative nonvanishing input is now established. Together with the
lifting and finite-space arguments, it proves the budgeted-output
conclusions of Proposition~\ref{c:trace-budget}, which we apply next to
move the cheap probe values to zero and then to bound all angular values.
==== END Bounded-klt-complements-for-Fano-contractions//trace-lifting.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//trace-survival.tex ====
\begin{figure}[tbp]
\centering
\begin{minipage}{0.94\textwidth}
\centering
\fbox{\parbox{0.94\linewidth}{\centering
\textbf{Geometric generic fibre of $K_w/k(g)$}\\[2pt]
A strict divisorial budget gives a product $H|_\eta$ with
nonzero relative Cartier trace.}}
\par\smallskip
$\Big\downarrow$\quad
\parbox{0.76\linewidth}{Taylor flags
$\prod_j(g_j-\xi_j)^{P_0-1}$\\
Lemma~\ref{c:trace-taylor-flags}}
\par\smallskip
\fbox{\parbox{0.94\linewidth}{\centering
\textbf{Full initial field $K_w$}\\[2pt]
The flagged product has nonzero trace. Expand it in homogeneous
angular factors and retain the summands whose traces survive.
Independent full-field outputs lie in the span of the resulting
actual output initials.}}
\par\smallskip
$\Big\downarrow$\quad
\parbox{0.76\linewidth}{Lift the same factors to the original field\\
Lemmas~\ref{c:trace-initial-survival} and~\ref{c:trace-degree}}
\par\smallskip
\fbox{\parbox{0.94\linewidth}{\centering
\textbf{Actual function field $K$}\\[2pt]
Each surviving summand gives an output $F'$ attaining equality
in~\eqref{eq:E8}, with controlled physical degree.}}
\end{minipage}
\caption{The order of trace lifting. The downward steps use the retained
reductions over the extended perfect constants. Whole output spaces and their
filtration bounds are fixed in characteristic zero before the
characteristic and the Frobenius power are chosen.}
\label{fig:trace-survival}
\end{figure}
==== END Bounded-klt-complements-for-Fano-contractions//trace-survival.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//valuation-optimization.tex ====
\section{Real valuations and a finite constrained optimization}
\label{c:optimization-section}
The section estimates are now in place. We seek a valuation whose
discrepancy is small, whose values on all sections satisfy a lower bound,
and whose selected initial coordinates remain independent. The first task
is to make this optimization finite and attain its minimum on the original
function field. The second is to extract a strict divisorial inequality
from optimality; that inequality is the input for Cartier trace in the
next section.
We return to the actual function field
\(K=k(X)=k(Z)\), the fractional form \(\theta\), the valuation \(v\),
and the nested multiplicative systems \(V_m\) constructed above.
All physical degrees in this and the following sections belong to
\(n\Z_{\ge0}\). If \(0\ne F\in V_m\), its \emph{assigned cost} is
\[
b(F)=m+v(F).
\]
The assigned degree is part of this notation. We have
\(V_0=k[Z]\) and \(\Frac(V_0)=K\). The only zero-cost elements
are the degree-zero functions invertible at \(z\). The preceding
weight estimates and \eqref{eq:A7} give
\[
m\le Cb(F)\quad\text{when }b(F)>0,
\]
and a uniform positive lower bound for positive costs along the
chosen subsequence. Constants may depend on the fixed subsequence
bounds already selected.
The earlier normalized-volume minimization supplied the graded model
underlying these section estimates. In the present optimization,
\(v\) remains fixed and determines the costs \(b(F)\), while the
candidate valuation varies on the actual field \(K\). The objective
balances its angular discrepancy against a common lower bound for
section values, with prescribed independent initial coordinates.
\subsection{Constants and full initial fields}
We may, and do, take \(k\) countable and algebraically closed of
characteristic zero. Here is the descent needed for this choice.
Descend the countably many members of the sequence and all their finite
data simultaneously from \(\C\): the points \(z\), the sections \(h\),
the graded algebra presentations and generators with their lifts,
the monomial charts for \(v,\widetilde v\), the models and contractions,
the lattice and field identifications, the forms and divisors, and the
chosen probes. Retain the numerical valuation weights. These data
are defined over the algebraic closure of a countably generated
subfield of \(\C\), which is countable.
We explain why the assertions used below survive this descent.
On an independent-normal-weight chart as in \eqref{eq:A6}, the full
initial field is the stratum field with independent normal variables.
The leading-monomial congruences and their nonzero residue coefficients
are preserved on extending constants. For any finite-dimensional
space of functions over \(k\), there are only finitely many values:
elements of distinct values are linearly independent. Taking bases
on its successive value slices gives a basis with independent initials.
These initials remain independent after extension to \(\C\), since
the stratum is geometrically integral and its initial coordinates
extend injectively. This computes the whole filtered space after
extension as well. Sections of the reflexive powers defining the
graded affine algebra commute with constant extension by flat base
change; alternatively compute them on the smooth locus. Taking
finite spans therefore proves that the associated graded algebras,
the prescribed lifts, and the field identifications extend as required.
In particular the leading-space dimensions, widths, and nonemptiness
assertions descend. Retain \(S_1,H_1,A_1^+\), \eqref{eq:A9}, and the
terminal field \(K_f\). A divisorial test over \(k\) extends from a
smooth divisor chart to \(\C\), with its orders and discrepancies
unchanged, so \eqref{eq:C4} continues to hold. If its restriction to
\(K_f\) is trivial, the residue embedding of \(K_f\) stays injective
after tensoring and taking fractions, so this condition is preserved
too. The \(\epsilon\)-lc condition on \(X\) and the lc condition
defined by \(h\) likewise descend. A contradiction over \(k\)
will consequently suffice. We shall enlarge constants again when
forming geometric generic fibres.
All valuations are trivial on their stated ground field. For a real
Abhyankar valuation \(w\) of \(K\), including the zero valuation,
write \(K_w\) for the fraction field of its associated graded ring,
and \(\overline F\) for a homogeneous initial. Thus \(K_w\) is the
\emph{full initial field}, not merely the residue field. Write \(E\)
for the residue field of \(w\). A valuation is \emph{Gaussian} on a
list if the initials of that list are algebraically independent over
the constants. Equivalently, every nonzero polynomial in the list
has the termwise minimum value.
\subsection{The barrier and its finite test}
Fix one sequence member, positive real numbers
\(B_*,\kappa,q_*,q_{\rm bar}\) with \(q_{\rm bar}<q_*\), and an
algebraically independent list \(y=(y_j)\) of rational functions,
possibly empty. The symbol \(B_*\) is a budget scalar, not a
boundary. Define
\begin{equation}\label{eq:E3}
q(w)=\min\left\{q_*,
\inf_{\substack{m,\;0\ne F\in V_m\\ b(F)>0}}
\frac{w(F)+B_*m}{b(F)}\right\},
\qquad
\mathcal L(w)=A_\theta(w)-\kappa q(w).
\end{equation}
For a real parameter vector \(t\), minimize \(\mathcal L\) over
real quasi-monomial \(w\) with \(q(w)\ge q_{\rm bar}\), Gaussian
on \(y\), and \(w(y)=t\). A feasible valuation is centred over
\(z\) on \(X\): its bound on \(V_0=k[Z]\) gives positivity on the
maximal ideal of \(z\), zero on functions invertible there, and
properness supplies the centre on \(X\). The lc boundary in
\eqref{eq:A13} then gives \(A_\theta(w)\ge0\).
\begin{lemma}[Finite barrier test]\label{c:finite-barrier}
There are finitely many pairs \((m_i,F_i)\), with
\(b_i=m_i+v(F_i)>0\), such that whenever their minimum ratio,
including \(q_*\), is at least \(q_{\rm bar}\), it equals \(q(w)\).
An index attaining the minimum is called active; the cap is an
additional active entry when \(q(w)=q_*\).
\end{lemma}
\begin{proof}
Take the degree-\(n\) Veronese \(\mathcal A'\) of the actual section
algebra \(\mathcal A\). Write \(o'\) for the image of \(o\) in
\(\Spec\mathcal A'\). The initial algebra of \(\mathcal A'\)
at \(\widetilde v\) is the corresponding Veronese of \(R\):
\(\widetilde v\) is degree-torus invariant, and linear combinations
of torus translates separate the physical degree characters.
This algebra is finitely generated, with value-zero part consisting
only of constants. Choose lifts
\(s_i=h^{m_i/n}F_i\) of homogeneous initial generators and include
ordinary homogeneous algebra generators of \(\mathcal A'\).
In degree zero subtract their value at the point \(o'\), so all
these generators vanish there and have positive cost.
Every homogeneous \(s=h^{m/n}F\) vanishing at \(o'\) has an exact
polynomial representative in these generators, all of whose terms
have physical degree \(m\) and cost at least
\(b=\widetilde v(s)\). To prove exactness, subduct by the initial
generators, retaining physical degree, to arbitrarily high cost.
The residual value strictly increases; only finitely many costs
occur below a fixed bound, by positivity and finite graded
generation. The fixed-germ log Izumi estimate at the bare klt
cone point \(o\) then puts a sufficiently high residual in any
prescribed power of its maximal ideal. The powers are primary
also on the affine ring, so this assertion is global. It may
equally be checked after extension to \(\C\) and contracted.
The same conclusion holds in \(\mathcal A'\). Indeed the radical
of its point ideal extended to \(\mathcal A\) is the point ideal
of \(o\), as seen by taking degree powers; moreover the Veronese
inclusion splits as \(\mathcal A'\)-modules. Thus a sufficiently
high residual belongs to any prescribed power of the point ideal
in \(\mathcal A'\). Such a power has representatives by
sufficiently long monomials in the ordinary generators, and hence
by terms of high cost. Project to physical degree \(m\).
This finishes subduction by an exact finite representative.
Divide by \(h^{m/n}\). If the finite ratios are bounded below
by a positive number \(q\), every term satisfies
\(w(\text{term})+B_*m\ge q\,(\text{its cost})\ge qb(F)\).
Taking the minimum gives the same bound for \(F\).
Zero-cost units cause no additional condition. This proves the
asserted finite test.
\end{proof}
\subsection{Monomial models and variations}
The finite test replaces the section inequalities by finitely many
actual functions. We still need to control discrepancy while
retracting a valuation to a finite skeleton, and to construct variations
with controlled probe values and preserved Gaussianity. The
following monomialization and differential arguments provide these
two kinds of control.
We will also use ordered-group valuations satisfying Abhyankar
equality. The relative Abhyankar inequality says that, for an
extension of valued function fields, the increment of rational rank
plus the increment of residue transcendence degree is at most the
transcendence degree of the extension. Indeed choose values
independent modulo the old values over \(\Q\), and value-zero
elements with residues independent over the old residue field.
Their monomials are independent, by first comparing values and then
residues. It follows that Abhyankar equality passes by restriction
to subfunction-fields. The transcendence degree of the full initial
field is rational rank plus residue transcendence degree: choose
homogeneous elements whose values rationally span the value group;
all remaining homogeneous elements become algebraic after taking
monomial ratios.
\begin{lemma}[Ordered monomialization]\label{c:ordered-monomialization}
An ordered-group valuation of a characteristic-zero function field
satisfying Abhyankar equality is monomial on a smooth projective
model, which may dominate a prescribed model and resolve prescribed
finite divisor data. Its centre is the generic point of a transverse
stratum; the positive normal-parameter values are rationally independent.
Its full initial field is the stratum field with independent normal
variables.
\end{lemma}
\begin{proof}
Choose functions with rationally independent values spanning the
rational value group, and value-zero functions with residues forming
a complementary transcendence basis. Resolve the divisors of the
first functions and the maps to \(\PP^1\) of the second functions,
together with the prescribed finite data. At the centre, the number
of crossing components is at least the rational rank: the chosen
functions are units times Laurent monomials in their local equations.
The closure of the centre has dimension at least the residue
transcendence degree, because the chosen residue functions are regular
and invertible there. Equality of their sum with the field dimension
forces the centre to be generic in a transverse stratum and the normal
values to be rationally independent.
Complete the regular local ring at that generic point and expand over
a coefficient field. The support of a nonzero series has finitely
many componentwise-minimal exponents. Since all normal weights are
positive, its least weight is attained among them; rational
independence makes the least exponent unique. This also works in the
ordered group. The ideal of monomials of strictly higher weight is
finitely generated. Faithful flatness contracts the congruence modulo
this ideal to the local ring. Hence the function equals its least
monomial times a residue-coefficient lift, modulo strictly higher
weight. This proves the value formula and the initial-field
description. Independence of the normal values also identifies the
degree-zero field with the stratum field.
\end{proof}
For a real \(w\), Lemma~\ref{c:ordered-monomialization} gives
\[
K_w=E(\mathbf z),
\]
where the homogeneous Laurent variables \(\mathbf z\) have degrees
forming a basis of the value lattice. This lattice is free of finite
rank by the same chart. Any homogeneous elements of those basis
degrees can serve as the variables: correct characters to degree zero
and use independence of distinct degrees. Fractional forms are
interpreted by taking their \(n\)-th powers and dividing form orders
by \(n\). In particular \(\theta_w^n\) is homogeneous for this value
torus, by the leading-form rule \eqref{eq:A6}.
\begin{lemma}[Logarithmic determinant]\label{c:log-determinant}
If \(a_1,\ldots,a_d\) are a transcendence basis and \(w\) is a real
Abhyankar valuation in characteristic zero, then
\[
A_{\bigwedge_i d\log a_i}(w)\ge0.
\]
Equality holds exactly when the initials of the \(a_i\) are
algebraically independent.
\end{lemma}
\begin{proof}
Include their divisors in the monomialization. Use logarithmic
differentials of the normal parameters and of unit lifts of a
separating basis on the stratum. Their wedge has form discrepancy
zero. Each \(a_i\) is a unit times a normal Laurent monomial.
The logarithmic differential matrix has entries of nonnegative
value, and its reduction is the corresponding matrix of initial
logarithmic differentials. Logarithmic normal derivatives of a unit
reduce to zero; tangential derivatives differentiate the residue
coefficient. The determinant therefore has value zero exactly
when the initial differentials are independent, equivalently when
the initials are algebraically independent in characteristic zero.
\end{proof}
\begin{lemma}[Invariantization]\label{c:invariantization}
Let \(Q^*\) be a real Abhyankar valuation of \(K_w=E(\mathbf z)\).
Put \(P=Q^*|_E\), and let \(Q\) be the Gauss valuation with restriction
\(P\) and the same values as \(Q^*\) on \(\mathbf z\). Then \(Q\)
is Abhyankar, agrees with \(Q^*\) on every homogeneous element, and
\begin{equation}\label{eq:E1}
A_{\theta_w}(Q^*)\ge A_{\theta_w}(Q).
\end{equation}
Equality implies \(Q^*=Q\). Gaussianity of \(Q^*\) on a list of
homogeneous probes is preserved by \(Q\).
\end{lemma}
\begin{proof}
The restriction \(P\) is Abhyankar by the relative inequality.
The Gauss extension is Abhyankar because its variable initials
are independent over the full field \(E_P\). Choose functions
\(x\) in \(E\) with full independent initials at \(P\).
The coefficient of \(\theta_w^n\) in the \(n\)-th power of
\(d\log x\wedge d\log\mathbf z\) is homogeneous, so its value is
the same for \(Q\) and \(Q^*\). The logarithmic differential
contribution is zero at \(Q\), and is nonnegative at \(Q^*\) by
Lemma~\ref{c:log-determinant}. Equality requires independent
initials of \(x,\mathbf z\). As \(E_P\) is algebraic over the
field generated by the \(x\)-initials, the \(\mathbf z\)-initials
are then independent even over \(E_P\). This gives the termwise
Gauss rule at \(Q^*\), proving equality of valuations.
For the last assertion, group a polynomial in the probes by its
torus characters. The two valuations agree on each homogeneous
part, and \(Q\) takes their minimum. Gaussianity at \(Q^*\)
therefore gives the same polynomial minimum at \(Q\).
\end{proof}
\begin{lemma}[Flattening a secondary valuation]\label{c:flattening}
For the invariant valuation \(Q\) of
Lemma~\ref{c:invariantization}, there are real monomial valuations
\(w_\lambda\), for small \(\lambda>0\), such that
\begin{equation}\label{eq:E2}
\begin{split}
w_\lambda(F)&=w(F)+\lambda Q(\overline F),\\
A_\theta(w_\lambda)&=A_\theta(w)+\lambda A_{\theta_w}(Q).
\end{split}
\end{equation}
For each fixed nonzero rational function the first identity holds
for all sufficiently small \(\lambda\), with the interval allowed
to depend on that function. If \(w\) is Gaussian on a prescribed
finite list \(y\), and \(Q\) is Gaussian on \(\overline y\), then
\(w_\lambda\) may be kept Gaussian on \(y\).
\end{lemma}
\begin{proof}
Give \(F\) the lexicographically ordered value
\((w(F),Q(\overline F))\). Grouping tied first values proves the
valuation inequality. Its initials identify with double initials:
on each first homogeneous slice the second comparison is exactly
the definition, while distinct first degrees remain separated by
their \(\mathbf z\)-characters, with coefficients in \(E_P\).
Multiplication is preserved. Lifts to \(K\) of the variables
\(\mathbf z\) and of the functions \(x\) used above have full
independent lex initials. Such lifts exist for all homogeneous
elements, including degree-zero residue elements. The lex valuation
therefore satisfies Abhyankar equality.
Monomialize it by Lemma~\ref{c:ordered-monomialization}. Replace each
normal weight by its first coordinate plus \(\lambda\) times its
second coordinate. These weights are positive for small
\(\lambda>0\). For any fixed numerator and denominator, the finitely
many componentwise-minimal support exponents show that its former
least exponent remains least eventually. The resulting initial
with its stratum coefficient and normal variables is unchanged.
The normal initial variables remain independent even if the new
real values have ties, since a monomial valuation retains all
least-weight terms. This proves the value assertion and preserves
independence of the lifted full probes. Double initials likewise
preserve independence of \(y\) under the two stated Gaussian
conditions.
Express \(\theta^n\) in the logarithmic basis of those full probes.
Its coefficient computes the discrepancy at \(w\) by
\eqref{eq:A6}, and its initial is the coefficient of \(\theta_w^n\).
The logarithmic basis contributes zero at \(w_\lambda\) and at \(Q\).
Apply the value identity also to this one fixed coefficient and
divide by \(n\); the second identity follows.
\end{proof}
\subsection{Retraction and compact optimization}
We now apply these valuation tools to the finite barrier data.
Retraction will restrict the valuations to finitely many skeleton
cones; the comparison with \(A_X\) in \eqref{eq:A13} will exclude
escape to infinity on an objective sublevel.
\begin{lemma}[Retraction and strict discrepancy drop]
\label{c:skeleton}
Resolve on a smooth projective model the divisor supports of
\(\theta^n,F_i,y_j\). Retraction to its normal crossing skeleton
preserves the finite barrier data, probe values and probe
Gaussianity, and does not increase \(A_\theta\). The discrepancy
decreases strictly for a real Abhyankar valuation outside the
skeleton. On every closed skeleton cone, each fixed rational
function has a continuous finite piecewise-linear value, with
a finite Lipschitz constant.
\end{lemma}
\begin{proof}
At a centre, keep the weights on the crossing normal equations
and move to the generic stratum; if there are no such equations,
the retraction is zero. Retraction can only decrease the value
of a regular function. To see this with the required control at
faces, let \(x\) be the normal parameters. A monomial ideal of
positive threshold contracts from the generic stratum to the
local ring. In its completed quotient there is a finite free
module over the formal series in the other parameters. A
function outside the stratum prime acts injectively: its reduction
in the other parameters is nonzero, and the remaining normal part
acts nilpotently. The monomial ideal is therefore primary to
the stratum prime, and faithful flatness gives contraction.
Membership in this ideal characterizes the threshold at the
generic stratum and bounds the value at the original centre.
On an open cone, normal expansion computes values by the minima
over finitely many componentwise-minimal support exponents of
regular numerators and denominators. At a face apply the same
primary-ideal argument to just the remaining positive-weight
parameters. It proves that these formulas extend to the face;
at the origin both values are zero. This gives continuity, finite
piecewise linearity and a Lipschitz bound. Divisors included in
the resolution have linear values on each cone. So does
\(A_\theta\), using the local logarithmic top generator.
The retraction preserves all resolved divisor values. For a
polynomial in \(y\), clear poles by a normal monomial, since each
\(y_j\) is a unit times a normal Laurent monomial. Its retracted
value is at most its old value and at least its termwise minimum.
If \(w\) is Gaussian these agree, proving preservation.
Within one open cone the initial formulas of \(y\) in stratum
coefficients and independent normal variables do not depend on
the positive weights. Gaussianity thus selects whole open cones.
It persists on their faces by continuity of all polynomial values.
For the discrepancy comparison, choose supplementary regular
functions \(z'\) with étale differentials at the centre, including
a separating residue basis, and put
\(\lambda_0=d\log x\wedge dz'\). The coefficient of \(\theta^n\)
relative to \(\lambda_0^n\) is a unit-monomial. The difference
from the retracted discrepancy is \(A_{\lambda_0}(w)\).
Switching the regular supplementary factors to logarithmic ones
and applying Lemma~\ref{c:log-determinant} makes this nonnegative.
If the centre is proper in its stratum, a supplementary parameter
vanishing there has positive value, making the difference strict.
If the centre is generic in the stratum, use unit lifts of a
separating stratum basis. Equality in the determinant test makes
those initials and the normal initials independent. The normal
initials are independent even over the embedded stratum field,
which is algebraic over that basis field. Least normal terms
therefore cannot cancel; contracting their congruence modulo
strictly higher monomial weight proves that \(w\) itself is the
monomial retraction. This proves strictness outside the skeleton.
The proof is unchanged over other algebraically closed
characteristic-zero constant fields.
\end{proof}
\begin{proposition}[Compact optimization and continuation]
\label{c:optimization}
The optimization in \eqref{eq:E3} is a finite polyhedral
optimization with compact sublevels, and attains its minimum
whenever feasible. Every optimizer lies on the fixed skeleton.
At any parameter admitting an optimizer with \(q>q_{\rm bar}\),
the minimum is locally finite and continuous. On full-dimensional
generic parameter cells it is affine. Through every optimizer at
such a generic parameter there is a local affine continuation by
optimizers within its same open stratum cone and with the same
active mask.
\end{proposition}
\begin{proof}
By Lemmas~\ref{c:finite-barrier} and \ref{c:skeleton} we may optimize
on a finite union of closed polyhedral pieces, with affine
projection to \(t\) and piecewise affine objective. Strict
retraction excludes every optimizer outside them.
Suppose a sublevel, allowing \(t\) to vary, were unbounded.
Normalize an escaping sequence in one closed cone by the norms
of its weights and take a limit of norm one, defining a nonzero
monomial valuation \(w'\). The barrier and continuity give
\(w'(F)\ge0\) for every \(0\ne F\in V_m\), including degree-zero
units. The bounded cap in \eqref{eq:E3} gives
\(A_\theta(w')\le0\). Nonnegativity on \(k[Z]\), and properness
over the affine base, give a centre on \(X\).
Equation~\eqref{eq:A13}, valid at such centres by freeness of
sufficiently divisible multiples over the affine base, implies
\(A_X(w')\le0\). This is impossible: on a monomializing resolution
a nonzero quasi-monomial valuation centred on a klt variety has
strictly positive discrepancy. The sublevels are compact, proving
attainment.
For continuity, let \(w\) be an optimizer with strict barrier
slack. Give \(k(\overline y)\) arbitrary rational-direction
Gauss weights. Extend this valuation to \(K_w\) by completing
to a transcendence basis with freely weighted variables and then
extending over the finite algebraic extension. The relative
Abhyankar inequality gives equality, and the extended value
group is contained in the rational span of the starting real
weights, so the extension remains real. Invariantize by
Lemma~\ref{c:invariantization} and flatten by
Lemma~\ref{c:flattening}. The finite test makes a small segment
in the prescribed direction feasible, with cost tending to the
optimum. Retraction can only improve it.
A compact skeleton sublevel cut strictly above that optimum
projects to a finite union of closed polytopes. The tangent
cones at the parameter contain all rational directions and hence,
by closedness and density, all directions. Near the point only
the active inequalities of the incident polytopes matter.
Its projection therefore contains a neighbourhood. Applying
this for sublevels arbitrarily close above the optimum proves
local finiteness and upper continuity. Compactness proves lower
continuity.
Finally subdivide a compact sublevel into finitely many polyhedra
with affine cost and fixed active mask and stratum on each
relative interior. Projection to parameter and cost gives the
piecewise-affine minimum. Avoid its walls and all
lower-dimensional parameter projections of domain faces.
The affine hull of the relative interior containing any
optimizer then projects with full rank. Choose a local affine
right inverse through that optimizer, staying in the same
relative interior. Its cost minus the optimal value is affine,
nonnegative nearby and zero centrally, hence identically zero.
This is the required continuation. In the applications below,
strict objective bounds maintain strict barrier slack; compactness
and continuity then also control limits along nongeneric segments.
\end{proof}
\subsection{A strict budget on a geometric generic fibre}
Compactness and continuation allow us to choose and vary optimizers.
We next examine a second valuation of their full initial field. If it
preserves the old initial coordinates, flattening gives an admissible
variation on the original field. Optimality controls its discrepancy;
a tie rule will make the resulting inequality strict after adding
finitely many small auxiliary terms.
Fix parameters admitting an optimizer with \(q>q_{\rm bar}\),
and let \(\mathcal M\) be the compact set of all optimizers.
We first maximize \(q\) on \(\mathcal M\). The finite barrier
test makes \(q\) continuous there, so its maximum \(q_{\max}\)
is attained and satisfies \(q_{\max}>q_{\rm bar}\). The locus
\[
\mathcal M_{\max}=\{u\in\mathcal M:q(u)=q_{\max}\}
\]
is therefore nonempty and compact.
Choose the coefficients \(c_F>0\), indexed by the countable nonzero
angular functions, so rapidly decreasing that
\[
\Psi(u)=\sum_F c_Fu(F)
\]
converges absolutely and uniformly on \(\mathcal M\), and that
the weighted Lipschitz constants are summable on each of the
finitely many closed cones. This is possible by
Lemma~\ref{c:skeleton}.
We then choose \(w\in\mathcal M_{\max}\) maximizing \(\Psi\)
there. In particular \(q(w)=q_{\max}>q_{\rm bar}\). We call
this an optimizer with the stated tie rule.
Take independent homogeneous probes
\(g=(\overline y,g_{\rm add})\) in \(K_w\). The optional
\(g_{\rm add}\) are not optimization constraints. Let \(L\)
be any geometric generic fibre field over \(k(g)\): choose a
component of the compositum with
\(k'=\overline{k(g)}\) and work over \(k'\). Divide \(\theta_w\)
by the ordinary base wedge \(\bigwedge_j dg_j\), interpreting
fractional forms through powers, to obtain
\(\theta_{w,\mathrm{rel}}\).
\begin{proposition}[Relative optimality and its equality case]
\label{c:relative-budget}
For every nontrivial real Abhyankar valuation \(Q^\dagger\) of
\(L/k'\),
\begin{equation}\label{eq:E4}
A_{\theta_{w,\mathrm{rel}}}(Q^\dagger)
\ge \kappa\min_{i\ \mathrm{active}}
\frac{Q^\dagger(\overline F_i)}{b_i}.
\end{equation}
An active cap contributes the entry zero. If equality holds,
then \(Q^\dagger(\overline F)<0\) for some actual angular
function \(F\).
\end{proposition}
\begin{proof}
Restrict \(Q^\dagger\) to a valuation \(Q^*\) of \(K_w\).
The extension \(L/K_w\) is algebraic, and its enlarged constants
contribute \(|g|\) to residue transcendence degree over \(k\).
The relative inequality thus shows that \(Q^*\) has Abhyankar
equality. The \(g_j\) have value zero and independent initials.
Complete them, using functions of \(K_w\), to a list with full
independent initials. Their supplementary initials remain
independent over \(k'\) at \(Q^\dagger\): the graded-field
embedding is injective, and the constant extension is algebraic
over \(k(g)\). Expressing the form in \(dg\) wedged with
logarithmic supplementary differentials and using \eqref{eq:A6}
gives
\[
A_{\theta_{w,\mathrm{rel}}}(Q^\dagger)
=A_{\theta_w}(Q^*).
\]
The coefficient is unchanged on dividing by the base wedge;
changing \(dg_j\) to \(d\log g_j\) contributes only value-zero
factors.
Invariantize \(Q^*\) to \(Q\). It agrees on the homogeneous
active entries and preserves zero-value Gaussianity on the old
probes \(\overline y\). Equation~\eqref{eq:E2} gives a path
\(w_\lambda\) preserving the old Gaussian parameters. No
preservation of additional constraints is being required. Let \(\mu\) be the
active minimum in \eqref{eq:E4}, including the zero entry if the
cap is active. For small \(\lambda>0\), the finitely many test
ratios and the cap have minimum \(q(w)+\lambda\mu\).
Since \(q(w)>q_{\rm bar}\), this stays above the barrier;
Lemma~\ref{c:finite-barrier} identifies it with \(q(w_\lambda)\)
and makes the path feasible for the original optimization.
Original objective optimality and \eqref{eq:E1} prove the inequality.
Equality forces equality in invariantization, hence \(Q^*=Q\),
and an exact path of original optimizers \(w_\lambda\).
Suppose every angular initial had nonnegative \(Q^\dagger\)-value.
Then \(\mu\ge0\), including the possible cap entry.
Since the path consists of original optimizers and
\(q(w)=q_{\max}\), maximality of \(q\) forces \(\mu=0\).
Thus this equality path lies in \(\mathcal M_{\max}\).
Some angular value is positive:
otherwise all homogeneous initials, obtained as ratios of initials
from \(V_0\), have value zero, and the invariant Gauss rule makes
\(Q\) trivial. Algebraicity would then make \(Q^\dagger\)
trivial as well.
This positive value contradicts the tie rule by differentiation
of \(\Psi\). The interchange of sum and derivative is justified
as follows. Strict retraction puts every small \(w_\lambda\)
on the fixed test skeleton. Monomialize the lex valuation on a
model dominating that skeleton. The flattened positive normal
weights have fixed centre for small positive \(\lambda\), so
their images use one skeleton cone. Values of its finitely
many normal equations are affine in \(\lambda\) on a common
interval by \eqref{eq:E2}. Their face limit is \(w\), by the
pointwise identities and closed-cone continuity. Thus the
coordinate distance is \(O(\lambda)\). The summable weighted
Lipschitz constants dominate all difference quotients; termwise
differentiation follows even though the eventual interval in
\eqref{eq:E2} depends on \(F\). Every derivative is nonnegative
and at least one is positive, contradicting maximality of
\(\Psi\) on \(\mathcal M_{\max}\).
\end{proof}
\begin{corollary}[Strict rational budget]\label{c:strict-budget}
On the same geometric generic fibre, one can choose positive
rational numbers \(\alpha_i\) arbitrarily close to
\(\kappa/b_i\), a finite list of additional actual angular
functions \(U_j\), and arbitrarily small positive rational
\(e_j\), so that every prime divisor \(P\) over a proper model
satisfies
\begin{equation}\label{eq:E5}
A_{\theta_{w,\mathrm{rel}}}(P)>
\min_{i\ \mathrm{active}}\alpha_i P(\overline F_i)
+\sum_j e_j\min\{0,P(\overline U_j)\}.
\end{equation}
The same zero-entry convention applies to an active cap.
The list may include field generators over \(k'\).
The added \(e_j\)-weighted terms and the errors from approximating
\(\kappa/b_i\) may meet any prescribed finite degree, value and
Lipschitz tolerances along a specified local affine optimizer
continuation.
\end{corollary}
\begin{proof}
Resolve the finite form and active-divisor data on a proper
relative model. By strict retraction, the nonzero equality
directions in \eqref{eq:E4} lie on its normal crossing skeleton:
retraction preserves the minimum term and strictly lowers
discrepancy off the skeleton. A nonzero valuation retracted
to zero has strictly positive discrepancy gap and is not an
equality direction. Normalize the sum of the nonnegative
normal weights to one. The equality locus is closed in the
finite compact union of these simplices.
Proposition~\ref{c:relative-budget} supplies an open cover of
this locus by conditions \(P(\overline U)<0\). Choose a finite
subcover. Add field generators: initials from \(V_0\) generate
\(K_w\), since \(K=\Frac(V_0)\). Adding the indicated
two-element minimum systems with any positive small coefficients
makes the inequality strict everywhere, first with coefficients
\(\kappa/b_i\).
When a local affine continuation is specified, fix it before
choosing these coefficients. It is a continuation by optimizers
of the original problem through \(w\); the tie rule is imposed
only at \(w\). For each added or active actual function \(F\),
fix its assigned degree. On a sufficiently small compact
parameter neighbourhood, its absolute values and Lipschitz
constant are bounded. Let \(N(F)\) bound the sum of these
quantities and its assigned degree. The added terms meet any
prescribed positive tolerance \(\eta\) by choosing the \(e_j\)
so small that \(\sum_j e_jN(U_j)<\eta\). With no continuation
specified, retain just the degree and value bounds needed.
Resolve also the added functions. The strict gap has positive
minimum on the normalized enlarged skeleton. Each of the
finitely many active evaluations is bounded there, and
\[
|\min_i a_i-\min_i b_i|\le\max_i|a_i-b_i|.
\]
Consequently a single sufficiently close rational perturbation
of the \(\kappa/b_i\) preserves strictness, including where
the minimizing index switches. The cap remains a fixed zero
entry. Retraction deals with every remaining direction,
including zero retraction. Having chosen the \(e_j\) first,
make the rational perturbation still closer so that
\(\sum_i|\alpha_i-\kappa/b_i|N(F_i)<\eta\) and the resulting
strict margin is retained. Finitely many prescribed tolerances
can be met simultaneously by taking \(\eta\) sufficiently small.
After clearing denominators, realize each term
\(e_j\min\{0,P(\overline U_j)\}\) by product powers of the
two-element system \(\{1,\overline U_j\}\).
The ratio \(\overline U_j\), rather than only one of its powers,
remains among ratios of the combined system: keep all other
factors fixed and replace one \(1\) by \(\overline U_j\).
This also supplies its asserted birationality.
\end{proof}
==== END Bounded-klt-complements-for-Fano-contractions//valuation-optimization.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//width-estimates.tex ====
\section{Uniform estimates and widths}
\label{b:width-estimates}
The tower controls discrepancies one relative field at a time. We now
translate this control into estimates for sections: upper bounds on the
dimensions of their initial spaces, together with independent sections
of comparable cost. These two directions are both used in the later
valuation argument. Divisor orders provide the upper bounds; opposite
torus twists and birational systems provide the sections.
Index the tower by \(i=1,\ldots,f\), with
\[
k(S_i)=k(Y_i)=K_i,\qquad
Y_i\longrightarrow S_{i+1},\qquad
S_{f+1}=\mathrm{pt},\qquad K_{f+1}=k=\C.
\]
The first field is the residual field in \eqref{eq:A9}.
After a subsequence all dimensions and ranks are constant. Put
\[
p_i=\trdeg_{K_{i+1}}K_i,\qquad
a_i=s_iC_i\asymp s_iA_i^++\Lambda_i,\qquad
s_f=1,\quad s_i\longrightarrow0\quad(i<f).
\]
Indeed the tower gives, with the constants chosen at level \(i\),
\[
a_i=(1-\gamma_i)s_iA_i^+
+b\lambda_iQ_i+s_i(a+x_i)H_i,
\]
omitting the \(Q_1\)-term. All terms have nonnegative values.
Moreover
\[
\lambda_{i+1}=1/x_i\asymp s_i,\qquad s_{i+1}\gtrsim s_i.
\]
Constants in this section may depend on the previously fixed
sequence bounds and dimensions, but are uniform in the functions,
divisors, weights and cutoffs subsequently tested.
The b-nef Calabi--Yau moduli of \(a_i\) contain simultaneous
positive reserves of \(\mathbf H_i=\overline{H_i}\) and, for \(i>1\),
of \(s_{i-1}\mathbf M_{Q_i}\). The first coefficient is
\(s_i(x_i-d_i')\gtrsim1\); the second follows from
\(s_i\gamma_i=b\lambda_i\).
For an invariant prime \(E\) on \(\Spec R\), write
\(E|_{K_i}=j_iP_i\) when nontrivial, with \(P_i\) normalized. Then
\begin{equation}\label{b:scaled-witness}
j_ia_i(P_i)\le s_i+C_0h_E.
\end{equation}
To see the exact constant, substitute \eqref{eq:B2} in the displayed
formula for \(a_i\). The constant contributions are
\((1-\gamma_i)s_i+b\lambda_i=s_i\); the remaining terms are bounded
multiples of \(h_E\), using \(j_iH_i(P_i)\le h_E\).
The unscaled $C_i$ have uniform positive horizontal discrepancy bounds.
The scaled functions $a_i=s_iC_i$ measure the smaller divisor orders
that correspond to width $1/s_i$; their positive lower bounds are not
uniform at collapsing levels. All restrictions and discrepancy functions
are evaluated homogeneously. At a trivial restriction their value is defined to
be zero.
\subsection{Horizontal orders and errors of nef traces}
Let \(\mathcal F_i\) be the arithmetic generic fiber of
\(Y_i\to S_{i+1}\), over \(K_{i+1}\); for \(i=f\) this is \(Y_f\).
The models \(Y_i\) are \(\Q\)-factorial. A horizontal prime means
a prime-divisor valuation on \(K_i\) trivial on \(K_{i+1}\).
Horizontally \(C_i=a_{K_i}+\mathbf M_{C_i}\) has effective uniformly
positive-lc generalized Calabi--Yau data with big nef part.
For \(i<f\), use the transformed threshold discrepancy
\(C_i-x_i\pi^*H_{i+1}\); for \(i=f\), use the last-step data.
The trace boundary \(B_{0,i}''\) of
\(A_i^+=a_{K_i}+\mathbf M_{0,i}\) is effective and dominates the
trace \(H_i''\) of \(\mathbf H_i\).
The normal projective arithmetic fibers are geometrically integral
and normal, by characteristic zero and the contraction property.
Their arithmetic Weil divisors are rationally Cartier, by taking
closures on \(Y_i\). We do not require geometric
\(\Q\)-factoriality. Divide the chosen volume form by a base volume
form generating canonical on a smooth base open. At horizontal
generic points, smoothness of the generic fiber identifies the
resulting relative canonical coefficients, including on resolving
models. The dimension formula identifies normalized horizontal
tests with prime-divisor tests of the relative field.
Such a test extends after algebraic closure of \(K_{i+1}\) with
unchanged normalization on arithmetic functions: take a component
of its extension on a smooth birational chart, whose generic
ramification index is one by separability. Conversely snc
resolution transfers the arithmetic discrepancy inequalities to all
geometric tests. In particular an snc boundary with coefficients at
most \(1-\delta\), \(0<\delta\le1\), has discrepancy at least
\(\delta\) on every normalized prime: subtract the reduced-support
discrepancy and use integrality of the remaining orders.
Lemma~\ref{b:ordinary-approximation}, with an ample reserve, makes
the geometric fibers klt of Fano type and puts them in a bounded
family by \cite[Theorem 1.1 and Corollary 1.4]{BirkarBAB}.
Use a bounded very ample Cartier polarization \(D\) to measure
degrees. These bounded-family assertions over \(\C\) apply here:
the algebraic closure of a finitely generated function field over
\(\C\) is individually isomorphic to \(\C\).
The ordinary effective approximation \(\Delta\) can be chosen with
\(K+\Delta\sim_\Q0\), a uniform positive-lc bound, and ordinary
discrepancies no larger than \(C_i\). This approximation may first
be made on the whole threshold model and then restricted
geometrically. Dimension-zero fibers need none of the following
divisor estimates.
\begin{lemma}\label{b:horizontal-estimates}
For every effective rational \(\Q\)-Cartier divisor \(L'\) on
\(\mathcal F_i\) and every horizontal test \(P\),
\begin{equation}\label{eq:C1}
P(L')\le C_0\deg(L')C_i(P).
\end{equation}
If \(\mathbf U\) is rational b-Cartier and b-nef, with trace
\(U_{\rm tr}\) on \(Y_i\), then
\begin{equation}\label{eq:C2}
0\le P(\overline{U_{\rm tr}}-\mathbf U)
\le C_0\deg(U_{\rm tr})C_i(P).
\end{equation}
\end{lemma}
\begin{proof}
Put \(p=p_i>0\). The bare geometric fiber is klt, so the cone
theorem and its length bound \cite[Theorem~1.1(5)]{FujinoFundamental}
give \(K+2pD\) nef. Thus \(A=(2p+1)D\) satisfies
\(A-\Delta\sim_\Q A+K\) ample. Its degree is bounded, and the
polarized threshold theorem \cite[Theorem 1.8]{BirkarBAB}
gives a uniform positive lower threshold for the
\(\R\)-linear system of \(A\) against the ordinary positive-lc pair.
The same cone inequality bounds \(\deg(-K)\).
Every geometric prime of degree \(e\) lies in a Cartier section of
\(|qD|\) for some \(1\le q\le e\). Indeed a general basepoint-free
projection to \(\PP^p\) is finite, since its pulled-back hyperplane
is ample. The image of the prime is a hypersurface of degree at
most \(e\), and its pullback contains the prime. Consequently a
nonzero \(L'\) is dominated geometrically by a rational weighted
sum of such sections, of rational class \(q'D\) with
\(q'\le\deg L'\). The difference is effective rationally Cartier,
even when the individual geometric primes are not rationally
Cartier. Scale this sum to the system of \(A\), apply the threshold
bound, and compare the ordinary discrepancy with \(C_i\).
Extension of the arithmetic test gives \eqref{eq:C1}.
Negativity makes \(\overline{U_{\rm tr}}-\mathbf U\) b-effective.
On a high projective descent add a positive rational ample amount
tending to zero. The error of the ample addition is again
nonnegative, so this can only increase the error being bounded.
The perturbed nef class is ample and has, after a rational
principal adjustment, an effective representative upstairs.
That adjustment changes neither error. Its nonnegative orders
bound the new error by the pullback order of its effective trace.
Apply \eqref{eq:C1} and let the added amount tend to zero.
The traces in question are arithmetically rationally Cartier
and geometrically pseudo-effective; their degrees are
nonnegative. This proves \eqref{eq:C2}.
\end{proof}
\subsection{A pure upper estimate}
\begin{proposition}\label{b:pure-upper}
For \(D_1\ge0\), \(D_1\sim_\Q H_1\) on \(S_1\), and every divisorial
test \(P_1\), with restriction multiplicities included in
\(P_i=P_1|_{K_i}\), one has
\begin{equation}\label{eq:C3}
D_1(P_1)\le C_0\sum_i a_i(P_i).
\end{equation}
For tests trivial on \(K_f\), this implies
\begin{equation}\label{eq:C4}
D_1(P_1)\le C_0\bigl(H_1(P_1)+s_{\exp}A_1^+(P_1)\bigr),
\qquad s_{\exp}:=\max_{i<f}s_i\longrightarrow0,
\end{equation}
where an empty maximum is zero.
\end{proposition}
\begin{proof}
Horizontally on \(Y_1\), \(\deg D_{1,\rm tr}\lesssim s_1\).
For a nonlast level, \(x_1H_1''=J''\le B_{0,1}''\);
at the last level use \(H_1''\le B_{0,1}''\).
The Calabi--Yau trace identity and nonnegative moduli degree bound
\(\deg B_{0,1}''\) by \(\deg(-K)\).
The b-nef divisor \(D_1\) has orders at most those of its effective
trace closure. Hence \eqref{eq:C1} gives
\(D_1(P)\le C_0a_1(P)\) on horizontal tests.
Subtract \(\tau_1D_1\) from \(a_1\) for a small uniform positive
rational \(\tau_1\). The moduli stay nef and big by the
\(\mathbf H_1\)-reserve, and the horizontal data stay generalized
klt and effective.
Inductively define
\[
D_{i+1}=\pi_\#a_i-\pi_\#(a_i-\tau_iD_i).
\]
The minimum formula makes it nonnegative everywhere.
Proposition~\ref{b:push-presentation} writes it as
\(\mathbf U-\mathbf U'\) with both terms rational b-Cartier
and b-nef. Their canonical coefficients agree, and
\[
\mathbf U=s_i\mathbf M_{Q_{i+1}},
\qquad
\pi_\#C_i=Q_{i+1}+x_iH_{i+1}.
\]
The next level has a positive reserve of this \(\mathbf U\).
Its effective scaled horizontal Calabi--Yau identity therefore
bounds \(\deg U_{\rm tr}\le C_0s_{i+1}\).
Nonnegativity of trace degrees and of the trace of \(D_{i+1}\)
gives the same bound for \(U'_{\rm tr}\) and \(D_{i+1,\rm tr}\).
Apply \eqref{eq:C1} to the latter trace and \eqref{eq:C2} to
the two trace errors. Thus
\[
0\le D_{i+1}(P)\le C_0a_{i+1}(P)
\]
horizontally. For a uniform sufficiently small \(\tau_{i+1}>0\),
subtracting \(\tau_{i+1}D_{i+1}\) preserves nef big moduli:
it subtracts \(\tau_{i+1}\mathbf U\) from its reserve and adds
\(\tau_{i+1}\mathbf U'\). It also preserves horizontal klt
and trace effectivity. A level without horizontal tests needs
no degree estimate.
For a test continuing nontrivially to \(P_{i+1}\),
\[
(a_i-\tau_iD_i)(P_i)
\ge \pi_\#(a_i-\tau_iD_i)(P_{i+1})
\ge -D_{i+1}(P_{i+1}),
\]
because \(a_i\ge0\). Rearranging and using the horizontal estimate
at the last nontrivial restriction proves \eqref{eq:C3}.
For a test trivial on \(K_f\), only \(i<f\) contribute.
The tower gives
\[
A_{i+1}^+(P_{i+1})\le A_i^+(P_i),\qquad
\Lambda_{i+1}(P_{i+1})
\le\lambda_{i+1}C_i(P_i)\lesssim a_i(P_i).
\]
Recursively use \(a_i\asymp s_iA_i^++\Lambda_i\) and
\(\Lambda_1=H_1\); the tower length is bounded.
This gives \eqref{eq:C4}.
\end{proof}
For tests trivial on the terminal field $K_f$, \eqref{eq:C4} involves
only the scales that tend to zero. The uniformly bounded residual
directions have disappeared from that estimate.
\subsection{Integral classes for opposite characters}
Recall \(M_T=\Z\chi\oplus M_0\), with \(M_0=\ker D\), and the
normalized dual polytope \(Q_T\). On \(M_{0,\R}\) set
\[
\|\beta\|=\max_{\rho\in Q_T}|\rho(\beta)|.
\]
This is a norm by the occurring-cone and interiority properties.
For an occurring weight \(\mu=m\chi+\beta\) of positive cost
\(B'=\widetilde v(\mu)\), the comparison on \(Q_T\) following
\eqref{eq:A12} gives
\[
0\le m\lesssim B',\qquad
\|\beta\|\lesssim B'.
\]
Indeed \(D\in Q_T\) and \(0\le\rho(\mu)\le CB'\) for all
\(\rho\in Q_T\).
To see how the pure estimate will control a general weight space, write
$R_\mu=z^\mu U$ with coefficient space $U\subset K_1$.
Finite generation supplies an opposite section of weight
$d(m'\chi-\beta)$ for a suitable $m'>\|\beta\|$ and some positive
clearing integer $d$. If its coefficient is $g'$, then for every
$g\in U\setminus\{0\}$ the product
$z^{d(m+m')\chi}g^dg'$ is a pure section. Both its degree and the
exponent of $g$ have been multiplied by $d$, so this factor cancels
when \eqref{eq:C3} is used to bound the order of $g$. We carry out the
resulting coefficient count below.
Actual probes require more: their degrees and costs must be bounded by
a uniform multiple of $\max\{1,\|\beta\|\}$, so their clearing factor
must be bounded as well.
The following integral class data make that possible on the bounded
fibres. They will also control the divisor classes in the marked families
of Section~\ref{b:bounded-axes}.
\begin{lemma}[Integral class data]\label{b:class-lattices}
For projective normal klt varieties of Fano type occurring in a
bounded family with bounded polarizations, the groups
\(\Cl/\mathrm{torsion}\) can be identified with Euclidean lattices
of finitely many types, so that
\begin{equation}\label{b:class-norm}
\|[G]\|_{\rm cl}\le C_0\deg(G)
\end{equation}
for effective rational Weil divisors and for pseudo-effective
rationally Cartier rational classes. The orders of \(\Cl\)-torsion
are bounded. For the \(\Q\)-factorial members, a single bounded
positive multiple of every integral Weil divisor is Cartier.
A numerically trivial rational Cartier divisor is rationally
linearly trivial.
\end{lemma}
\begin{proof}
The bounded family need only contain the varieties under
consideration; its other members need not satisfy positivity.
After finite stratification and finite base extensions, take
simultaneous projective smooth resolutions \(b_t:W_t\to X_t\)
with labeled exceptional prime divisors. To construct them,
resolve the geometric generic fiber
\cite[Theorems~1.0.1--1.0.3]{WlodarczykResolution}, spread after a
finite extension, shrink, and apply noetherian induction to the
complement. Keep an isomorphism on dense opens and spread the
components of the complements. Stabilizing their fiber and image
dimensions identifies all exceptional prime families and excludes
additional exceptional divisors. Smoothness of \(W_t\), normality
and geometric integrality of the fibers in use, and geometric
integrality of the prime families hold after the corresponding
open restrictions. Fixed labeled prime or Cartier data can be
included in this process.
On the selected fibers, Kawamata--Viehweg vanishing for a klt log
Fano boundary \cite[Theorem~3.2]{FujinoFundamental} and rationality
of klt singularities \cite[Theorem~0.1]{FujinoRational} give
\(H^1(\OO)=H^2(\OO)=0\), both on \(X_t\) and on \(W_t\).
The exponential sequence and GAGA
\cite[Section~12, Theorems~1--3; Section~20, Proposition~18]{SerreGAGA}
therefore identify \(\Pic\) with integral \(H^2\) on both. Strict transform and pushforward give the diagram
\[
H^2(X_t,\Z)\ \xrightarrow{\,b_t^*\,}\ H^2(W_t,\Z)
\ \longrightarrow\
H^2(W_t,\Z)/\langle[E_1],\ldots,[E_q]\rangle=\Cl(X_t).
\]
The image of the first group in the quotient is \(\Pic(X_t)\).
These are locally constant integral group data, including torsion,
after stratification: proper topological constructibility gives
the integral cohomology local systems and functorial pullback,
while relative line bundles give the labeled Chern classes.
Here is the finite stratification argument, including compatibility
with the markings. For either projective family \(\mathcal V\to T\),
let \(Z_1,\ldots,Z_s\subset\mathcal V\) be its finitely many closed
marked subsets. Apply \cite[Theorem~3.2.3]{DeCataldoMiglioriniHodge}
to the finite map
\[
\mathcal V\sqcup\bigsqcup_{i=1}^s Z_i\longrightarrow\mathcal V
\]
given by the identity and the inclusions. Split the source strata by
the open-and-closed summands and their connected components. For each
connected target stratum \(S\), the subset \(Z_i\cap S\) is closed.
It is also open, since it is a union of images of submersive source
strata. Hence it is either empty or all of \(S\), so the finite target
Whitney stratification is compatible with every marking.
Work over each irreducible base piece and restrict to a smooth dense
open. Remove the closures of the
images of all nondominating strata and of the critical loci of the
dominating strata. These closures are proper by generic smoothness.
The remaining strata map submersively to the base, and the whole
family remains proper. Embed the family relatively in a smooth
projective ambient space. Thom's first isotopy lemma
\cite[Proposition~11.1]{MatherTopological} gives local topological
triviality. Intersect the opens for the finitely many families in use.
Noetherian induction treats the remaining base pieces by reapplying
this construction to their reduced base-changed families and closed
markings; no Whitney assertion for arbitrary closed restrictions is
needed.
In trivializations over a small contractible base open, the fiber
maps vary continuously and hence induce constant cohomology maps.
No simultaneous trivialization of the resolution morphism, or trivial
monodromy, is required. Monodromy preserves the isomorphism type of
this group diagram.
There are consequently finitely many group-diagram types.
Their cokernels bound the torsion of \(\Cl\); when
\(\Cl/\Pic\) is finite, its exponent is bounded as well.
The latter finiteness is exactly rational factoriality here,
so it gives the asserted Cartier multiple.
For the norm assertion in dimension \(p\ge2\), fix a relative very
ample \(D_W\) on each resolved stratum. After further finite
extension and shrinking, choose relative line bundles
\(T_1,\ldots,T_r\) spanning \(H^2(W_t,\Q)\), with a common integer
\(l>0\) such that \(lD_W\pm T_j\) are ample. We justify this choice,
which contains more information than the abstract local systems.
Shrink where the coherent vanishings hold and the Betti ranks are
constant. Descend the finite family data to a countable
algebraically closed subfield of \(\C\), and realize its geometric
generic fiber as a complex fiber generic over that smaller field.
The relevant algebraic closures and \(\C\) are isomorphic over the
embedded defining parameter field, since their remaining
transcendence degrees have the same infinite cardinality.
On that fiber \(\Pic=H^2(\Z)\). Hard Lefschetz and the
Hodge--Riemann bilinear relations
\cite[Theorem~3.1.2(a)--(c)]{DeCataldoMiglioriniHodge} make the
pairing \(TT'D_W^{p-2}\) nondegenerate on \(H^2(\Q)\).
Indeed \(H^2=P^2\oplus D_WH^0\) is an orthogonal decomposition:
the primitive summand is nondegenerate by polarization, and the
ample summand has nonzero self-pairing \(D_W^p\).
Choose spanning line bundles there, spread them after a finite
extension, and retain their nonsingular intersection matrix and
ample comparisons on an open. Betti-rank constancy proves they
continue to span. Noetherian induction treats the other strata.
Use the coordinates
\[
\bigl(G_W\cdot T_j\cdot D_W^{p-2}\bigr)_j.
\]
For an effective divisor their absolute values are at most
\(l\deg_{D_W}(G_W)\), by the ample comparisons.
The integral free lattice embeds in \(\Z^r\) and contains the
fixed full-rank subgroup generated by the \(T_j\)-coordinates.
Thus only finitely many intermediate lattices occur. The
exceptional classes have fixed coordinates too. Orthogonal
projection modulo their real span gives lattices of finitely
many types for the free class groups downstairs.
If \(G_W\) is the strict transform of an effective prime \(G\),
the containing hypersurface used in the proof of \eqref{eq:C1}
pulls back to an effective Cartier divisor dominating \(G_W\).
The fixed family intersection degrees give
\(\deg_{D_W}(G_W)\le C\deg(G)\).
Linearity and the triangle inequality prove \eqref{b:class-norm}
for effective rational Weil divisors. In dimension one use
ordinary degree and the Picard description above; dimension
zero is vacuous.
A numerically trivial rational Cartier divisor pulls back to
a numerically trivial class upstairs. The same nondegenerate
pairing and \(\Pic(W_t)=H^2(W_t,\Z)\) make its rational class zero.
Pushforward proves rational linear triviality downstairs.
Finally a pseudo-effective rational Cartier class becomes big
after adding any small positive rational ample amount, and a
big rational Cartier class is rational-linearly effective.
Apply the effective bound and pass to the limit.
\end{proof}
\begin{lemma}[Arithmetic principal witnesses]\label{b:principal-descent}
An integral Weil divisor defined on a proper geometrically integral
normal arithmetic fiber which becomes principal geometrically is
already principal over the arithmetic field. No extension-degree
factor is needed. The same holds for a rational divisor after
clearing its specified denominator.
\end{lemma}
\begin{proof}
A witnessing function is defined over a finite normal extension
\(L/K\). For \(\sigma\in\operatorname{Gal}(L/K)\), its conjugate
quotient has zero divisor, hence is an invertible global regular
function on the proper normal fiber. Geometric integrality makes
it a constant in \(L^*\). The resulting scalar cocycle is trivial
by Hilbert 90. Rescale the function by the corresponding scalar;
its divisor is unchanged and the function is Galois invariant.
\end{proof}
\subsection{Saturation of opposite twists}
For a character $\beta$ with $\|\beta\|\le L$ and $L\ge1$, we now use
the integral class data to replace arbitrary monoid clearing by a bounded
factor. On each fibre for which $Ls_i\to0$, discreteness
of divisor classes forces the obstruction to be a multiple of the pivot
class. A rational principal correction removes it, and the discrepancy
bounds allow us to pass the corrected inequalities to the next field.
At the first scale with $Ls_i\gtrsim1$, a section construction and
integrality close the argument.
\begin{proposition}\label{b:twist-saturation}
If \(\beta\in M_0\), \(\|\beta\|\le L\), and \(L\ge1\), then
there are \(m\in n\Z_{\ge0}\) and a bounded positive integer
\(d_0\), with \(m\lesssim L\), such that
\begin{equation}\label{eq:C5}
R_{m\chi+d_0\beta}\ne0,\qquad
R_{m\chi-d_0\beta}\ne0 .
\end{equation}
\end{proposition}
\begin{proof}
We may test the assertion on a contrary sequence, allowing $\beta$ and
$L$ to vary with its member, and take subsequences throughout.
The first clearing factor $d$ below may be arbitrary; only the eventual
factor $d_0$ is required to be uniformly bounded.
At each individual member, finite
generation of the weight monoid gives the assertion with
member-dependent constants: rational cone points enter the monoid
after a common multiple, by decomposition into simplicial
generator cones. For \(m_0\in n\Z\) a little larger than \(L\),
both \(m_0\chi\pm\beta\) lie in the cone by its normalized dual
inequalities. Thus \(m_0\lesssim L\), and actual opposite sections
exist after a possibly arbitrary common clearing factor \(d>0\):
\[
\bar h^{dm_0/n}z^{\pm d\beta}r_\pm,
\qquad r_\pm\in K_1^* .
\]
We can take late members of a contrary sequence and round \(L\)
up to an integer. Start with
\[
u_\pm=-\Div(r_\pm)/d,\qquad
\ell_E=E(z^\beta),\qquad \mu=0.
\]
The arbitrary \(d\) occurs in \(u_\pm\); the initial
\(\ell_E\) are integral.
After subsequences, stop at the first stage with \(Ls_i\gtrsim1\);
at all preceding stages \(Ls_i\to0\). A stop exists because
\(s_f=1\). At entry to each reached stage we maintain
\(\mu\ge0\), \(\mu=o(1)\), \(\mu\lesssim L\lambda_i\), and
\begin{equation}\label{eq:C6}
\begin{aligned}
u_\pm&=\mu a_{K_i}+\mathbf U_\pm,
&\mathbf U_\pm&\text{ rational b-Cartier and b-nef},\\
u_++u_-&\le C_0L\Lambda_i,\\
j_iu_\pm(P_i)&\le\pm\ell_E+C_0Lh_E+\mu,
&j_iP_i&=E|_{K_i}.
\end{aligned}
\end{equation}
The first inequality is on every prime and is homogeneous.
The last line has left side zero at a trivial restriction.
We allow \(\ell_E\) to change by orders of fractional rational
functions from tower fields, with bounded clearing denominators.
Such fractional functions are notation for rational principal
b-divisors, not field extensions.
Initially the product of the two sections and \eqref{eq:A9} give
\[
\Div(r_+r_-)+2dm_0H_1\ge0,
\]
which is the sum bound; their regularity gives the last line.
\smallskip\noindent
\emph{Removing the class obstruction before the stopping scale.}
Consider a stage before the stop, so $Ls_i\to0$. On its arithmetic fiber every strict
prime other than the pivot \(G_*\) comes from a strict prime of
\(S_i\) with \(H_i(P)=0\): the pivot alone among the positive
support survives horizontally, and every extracted prime had
positive \(H_i\). Choose a witness \(j_iP=E|_{K_i}\) from
\eqref{eq:B2} for each such prime, with bounded positive \(j_i\)
and \(h_E=0\), and set
\[
c_P=\ell_E/j_i,\qquad
\Delta_{\rm col}=\sum_{P\ne G_*}c_PP.
\]
This arithmetic divisor has finite support and bounded denominator.
Indeed \(\ell_E\) is initially an integral character order, and
subsequently only the orders of previously bounded fractional
gauges are added. There are dimension-many stages and bounded
restriction integers. The original arbitrary \(d\) never enters
\(\Delta_{\rm col}\). A rational function has nonzero orders at
only finitely many strict primes of \(\Spec R\), proving the
support assertion.
Write \(B_0''=B_{0,i}''\), \(\mathbf M_0=\mathbf M_{0,i}\), and
\(c_\pm^*=u_\pm(G_*)\). At nonpivot strict primes,
\[
(\mathbf U_\pm-\mu\mathbf M_0)(P)
=u_\pm(P)-\mu A_i^+(P)
\le\pm c_P+\mu B_0''(P).
\]
At the pivot \(A_i^+(G_*)=0\), while
\(c_+^*+c_-^*\le C_0Ls_i\). Hence the divisors
\[
W_\pm=\pm\Delta_{\rm col}+c_\pm^*G_*+\mu B_0''
-(U_\pm-\mu M_0)_{\rm tr}
\]
are effective on the arithmetic fiber. Their degrees and those
of the pseudo-effective traces \(U_{\pm,\rm tr}\) are \(O(Ls_i)\):
add the two identities to obtain
\[
W_++W_-+(U_++U_-)_{\rm tr}
=(c_+^*+c_-^*)G_*+2\mu(B_0''+M_{0,\rm tr}).
\]
All terms on the left have nonnegative degree.
The degrees of \(B_0''\), \(M_{0,\rm tr}\) and \(G_*\) are bounded
by Calabi--Yau effectivity; the pivot has coefficient one in
\(B_0''\). Also \(\mu\lesssim Ls_i\).
On the geometric fiber, Lemma~\ref{b:class-lattices} applied to
these identities gives
\[
\|\pm[\Delta_{\rm col}]+c_\pm^*[G_*]\|_{\rm cl}
\le C_1Ls_i .
\]
The nonzero integral vector \([G_*]\) has bounded length; it is
nonzero already by its positive degree. Bounded-denominator
lattice vectors projected modulo its line are uniformly discrete,
because there are finitely many lattice types and finitely many
possible bounded pivot vectors. As \(Ls_i\to0\), we obtain the
exact equality
\[
[\Delta_{\rm col}]=q[G_*],\qquad
|c_\pm^*\pm q|\lesssim Ls_i .
\]
The denominator of \(q\) is bounded. In fact if
\([G_*]=h g\) with \(g\) primitive, bounded length bounds \(h\);
if \(Q[\Delta_{\rm col}]\) is integral, then \(Qqh\in\Z\).
After also killing the bounded torsion, a bounded positive
multiple of \(-\Delta_{\rm col}+qG_*\) is principal geometrically.
Lemma~\ref{b:principal-descent} descends the witness without an
extension-degree multiplier. Let \(\phi\) be a fractional function
in \(K_i\), with bounded clearing denominator, having this divisor
on the fiber.
Replace \(u_\pm\) by \(u_\pm\pm\Div(\phi)\), and
\(\ell_E\) by \(\ell_E+E(\phi)\). These principal changes preserve
\eqref{eq:C6}. They give \(u_\pm(P)\le\mu\) on strict nonpivot
primes and \(u_\pm(G_*)\le C_1Ls_i\). The new signed trace of
\(\mathbf U_\pm-\mu\mathbf M_0\) is
\[
(c_\pm^*\pm q)G_*+\mu B_0''-W_\pm.
\]
Its positive and negative parts have degrees \(O(Ls_i)\), and
the trace degrees of \(\mathbf U_\pm\) are unchanged. All these
arithmetic traces and parts are rationally Cartier.
Apply \eqref{eq:C1} to the signed trace and \eqref{eq:C2} to
the errors of \(\mathbf U_\pm\) and \(\mathbf M_0\), in the identity
\[
u_\pm=\mu A_i^++\mathbf U_\pm-\mu\mathbf M_0.
\]
Since \(A_i^+\ge0\), this proves, on every horizontal test,
\[
u_\pm\ge-C'La_i.
\]
\smallskip\noindent
\emph{Pushing the corrected inequalities.}
The horizontal lower bound just proved will give klt discrepancies after
adding a large multiple of $La_i$. The strict-prime upper bounds give
effective horizontal trace. An additional multiple of $Ls_iA_i^+$
supplies the needed allowance at the pivot.
For sufficiently large fixed positive rational \(K',T'\), push
\[
u_\pm+K'La_i+T'Ls_iA_i^+
\]
to the next field and use these pushes as the new \(u_\pm\).
Their canonical coefficient is
\(\mu_{\rm next}=\mu+(K'+T')Ls_i>0\).
They have big nef moduli and horizontal generalized klt
discrepancies by the preceding lower bound.
They are effective in horizontal trace as well:
on strict nonpivot primes use \(u_\pm\le\mu\) and
\(C_i,A_i^+\le1\); at the pivot use \(u_\pm\le C_1Ls_i\),
\(A_i^+=0\), and the extra slack \(T'Ls_i\).
Proposition~\ref{b:push-presentation} therefore applies.
We have \(\mu_{\rm next}=o(1)\) and
\(\mu_{\rm next}\lesssim L\lambda_{i+1}\).
For the sum bound, evaluate both signs on one common lift
minimizing \(C_i\). Use
\(\Lambda_i+s_iA_i^+\lesssim a_i\) and
\(s_i\pi_\#C_i\lesssim\Lambda_{i+1}\).
For the witness bound, if the projection remains nontrivial,
evaluate on the old \(j_iP_i\). By \eqref{b:scaled-witness}
and \eqref{eq:B2}, the newly added constants are exactly
\((K'+T')Ls_i\), while the other terms are \(O(Lh_E)\).
They are therefore precisely absorbed in \(\mu_{\rm next}\).
If the projection becomes trivial, the expression just pushed
has positive value at the old horizontal test, so the same upper
bound holds with left side zero. An already trivial projection
continues to satisfy the bound. This proves the induction.
\smallskip\noindent
\emph{Completing the canonical coefficient and obtaining regular sections.}
At the stopping stage \(Ls_i\gtrsim1\), let \(d_0\) be a bounded
positive integer clearing all accumulated gauges. There are only
boundedly many stages, and eventually \(d_0\mu<1\). The positive
coefficient $1-d_0\mu$ will complete the canonical coefficient to one;
the same equality will give the one-unit allowance in the final integral
regularity test. For each sign
we claim that some \(f_\pm\in K_i^*\) satisfies, at every prime,
\begin{equation}\label{eq:C6a}
P(f_\pm)+d_0u_\pm(P)+(1-d_0\mu)A_i^+(P)+C_2LH_i(P)>0 .
\end{equation}
If \(S_i\) is a point, take \(f_\pm=1\). Otherwise the middle
terms have the presentation
\[
a_{K_i}+\mathbf N,\qquad
\mathbf N=d_0\mathbf U_\pm+(1-d_0\mu)\mathbf M_{0,i},
\]
with b-nef \(\mathbf N\).
On a high smooth descent, \eqref{eq:B3} provides a big free
integral Cartier divisor \(D'\le kH_i\) in actual orders,
with \(k\lesssim1/s_i\lesssim L\).
Indeed choose a list \(1,f_1,\ldots,f_t\) with poles bounded by
\(kH_i\) giving the birational system, resolve its projective map,
and pull back its first coordinate hyperplane. Freeness makes
the order of this divisor \(-\min(0,P(f_1),\ldots,P(f_t))\),
which is at most \(kH_i(P)\).
Resolve the fractional support of \(N=\mathbf N_{\rm tr}\) too,
and use the canonical divisor \(K\) of the same chosen form.
For \(j=1,\ldots,\dim S_i+1\), Kawamata--Viehweg vanishing
\cite[Theorem~3.1]{FujinoFundamental} applies to
\[
K+\lceil N\rceil+jD',
\]
with snc klt boundary \(\lceil N\rceil-N\) and nef big adjoint
difference \(N+jD'\). Riemann--Roch makes its Euler
characteristic a polynomial in \(j\) of degree \(\dim S_i\)
with positive leading coefficient. It cannot vanish at all
these integers; vanishing identifies its values with section
dimensions. Choose a nonzero section at one of them.
The associated function \(f_\pm\) makes the left side of
\eqref{eq:C6a}, with \(jD'\) in place of \(C_2LH_i\),
the sum of the positive discrepancy of the snc pair and the
order of the section's effective divisor. This proves the claim.
For a nontrivial \(j_iP_i=E|_{K_i}\), multiply
\eqref{eq:C6a} by \(j_i\) and substitute \eqref{eq:C6},
\(j_iA_i^+(P_i)\le1-h_E\), and \(j_iH_i(P_i)\le h_E\).
The constant allowance is exactly
\[
d_0\mu+(1-d_0\mu)=1.
\]
Consequently, for \(m\in n\Z_{\ge0}\) sufficiently large with
\(m\lesssim L\), both signs satisfy
\[
E(f_\pm)\pm d_0\ell_E+mh_E>-1.
\]
At trivial restrictions this follows directly from
\eqref{eq:C6} and \(d_0\mu<1\).
Multiply \(f_\pm\) by \(z^{\pm d_0\beta}\), by the corresponding
cleared gauge functions, and by \(\bar h^{m/n}\).
The displayed orders are integers, hence nonnegative.
These are nonzero eigenfunctions. Any polar divisor of an
eigenfunction is torus invariant, so the invariant-prime tests
and normality make them regular. Their weights are
\(m\chi\pm d_0\beta\), proving \eqref{eq:C5} along every
alleged contrary subsequence and therefore uniformly.
\end{proof}
\subsection{Axes, coefficient spaces, and actual probes}
We now combine the residual tower and the torus directions into one list
of widths. We first complete the coefficient count described above and
sum it over the torus characters at each fixed physical degree. That
upper bound uses only arbitrary monoid clearing. The bounded clearing
just proved supplies the matching actual probes and their independent
initials.
Choose a lattice basis \(\beta_1,\ldots,\beta_{\rank T-1}\)
of \(M_0\) and positive widths \(L_j\) so that
\begin{equation}\label{b:torus-axis-norm}
\Big\|\sum k_j\beta_j\Big\|
\asymp\sum_j|k_j|L_j.
\end{equation}
Here is a uniform construction in bounded dimension.
Replace the norm by an equivalent Euclidean norm using an
ellipsoid, then choose successive shortest nonzero vectors
in the orthogonal projected lattices. Each is primitive, so
the next projection is again a lattice. Choose size-reduced
lifts, subtracting earlier vectors in reverse order until
their earlier Gram--Schmidt coefficients have absolute
value at most \(1/2\). Let \(L_j\) be their Gram--Schmidt
lengths. A size-reduced one-step lift of the next projected shortest vector has
component along the preceding shortest vector at most half its length.
The lift cannot be shorter than that preceding shortest vector, so
\[
L_j^2\le L_{j+1}^2+(L_j/2)^2.
\]
Thus the next projected length is at least $\sqrt3/2$ times the preceding
shortest length. Hence the normalized basis columns in
orthonormal Gram--Schmidt coordinates form a triangular
matrix with diagonal one and bounded entries; its inverse
is bounded in fixed dimension. This proves
\eqref{b:torus-axis-norm}.
Adjoin \(p_i\) residual axes of width \(1/s_i\) for each level.
Denote all widths by \(W_j\). There are
\[
d=(\rank T-1)+\trdeg_kK_1
\]
axes in total.
\begin{proposition}\label{b:coefficient-dimension}
For every occurring weight $\mu=m\chi+\beta$ of positive cost
$B'=\widetilde v(\mu)$,
\begin{equation}\label{eq:C7}
\dim R_\mu\lesssim
\prod_i(1+B's_i)^{p_i}.
\end{equation}
At each fixed physical degree \(m\), the dimension through
cost \(S'\ge0\), including constants if present, is at most
\begin{equation}\label{eq:C8}
C_3\prod_j(1+S'/W_j).
\end{equation}
For \(m\in n\Z_{\ge0}\), the same expression bounds the
truncated associated graded dimension of \(V_m\).
\end{proposition}
\begin{proof}
Write \(R_\mu=z^\mu U\) for a finite-dimensional coefficient
space \(U\subset K_1\); positive finite generation ensures
finiteness. Choose rational \(m'>\|\beta\|\) with
\(m'\lesssim B'\), and a nonzero opposite section of weight
\(d'(m'\chi-\beta)\), where \(d'>0\) is an arbitrary
clearing factor as at the start of the twist proof.
Increase \(d'\) so that \(M=d'(m+m')\in n\Z_{>0}\).
If \(g'\) is its coefficient, every nonzero \(g\in U\) gives
\(z^{M\chi}g^{d'}g'\in R\). By \eqref{eq:A9},
\[
D_1=H_1+\frac1M\Div_{S_1}
\bigl(g^{d'}g'/f_h^{M/n}\bigr)\ge0.
\]
Assume \(\dim S_1>0\). Construct compatible smooth projective
high models with morphisms down the tower, resolving rational
maps successively from the bottom, and include descents for
all displayed data. Choose compatible general closed points
where the morphisms are smooth. Lift relative parameters to
nested Taylor systems with \(p_i\) parameters at level \(i\).
Avoid the supports of the fixed forms, boundary, moduli and
\(H_1\). Make \(g',f_h\) units and a fixed basis of \(U\)
regular there. Give the parameters at level \(i\) rational
weights \(1/s_i\). This monomial test \(P_1\) is divisorial
up to scale: it is discrete up to scaling, and ratios of
equal-weight monomials give the required residue
transcendence degree. Its restriction \(P_i\) uses exactly
the parameters at levels \(\ell\ge i\).
At the chosen points the smooth monomial formula gives
\[
a_i(P_i)=s_i\sum_{\ell\ge i}p_\ell/s_\ell
\lesssim1,
\]
using the bounded tower length and \(s_{\ell}\gtrsim s_i\).
Proposition~\ref{b:pure-upper} therefore implies
\[
P_1(g)\le CM/d'\le C'B'
\]
for every nonzero \(g\in U\).
Taylor truncation at this weighted bound is injective on \(U\);
the number of allowed monomials is at most
\(C\prod_i(1+B's_i)^{p_i}\).
If \(\dim S_1=0\), use \(K_1=k\). This proves \eqref{eq:C7}.
At fixed physical degree, \eqref{b:torus-axis-norm} bounds each
integer coordinate by \(|k_j|\le CS'/L_j\).
Counting these weights and using \eqref{eq:C7} proves
\eqref{eq:C8}. Weight zero contributes only constants.
The angular translation following \eqref{eq:A12} identifies
this count with the associated graded truncation of \(V_m\).
There are only finitely many costs below a fixed cutoff,
so it is also the dimension modulo higher costs.
\end{proof}
\begin{proposition}\label{b:actual-probes}
Each residual level has \(p_i\) actual probes in some \(V_m\),
of degree and cost \(O(1/s_i)\), whose initials are in \(K_i\)
and algebraically independent over \(K_{i+1}\).
Each torus axis has opposite angular probes in $V_{m_j}$ with initial
weights $\pm d_j\beta_j$ in the full initial field; their corresponding
degree-$m_j$ sections have weights $m_j\chi\pm d_j\beta_j$.
Here \(d_j\) are bounded positive integers,
\(m_j\in n\Z_{\ge0}\), and degrees and costs are \(O(1+L_j)\). The residual and positive torus initials together
are algebraically independent, and their monomials give box
lower bounds with widths \(\max(1,W_j)\).
\end{proposition}
\begin{proof}
The birational system in \eqref{eq:B3}, containing \(1\), gives
\(p_i\) functions algebraically independent over \(K_{i+1}\)
whose poles are bounded by \(kH_i\), \(k\lesssim1/s_i\).
For any one of them, \eqref{eq:B4} and pullback orders give
\[
\Div_{S_1}(g)+kH_1\ge0.
\]
A trivial restriction evaluates both the lower function and
lower divisor at zero. Increase \(k\) to
\(m\in n\Z_{\ge0}\) with \(m\lesssim1/s_i\).
Then \eqref{eq:A9} gives \(g\bar h^{m/n}\in R_{m\chi}\).
Its angular lift lies in \(V_m\) with initial \(g\) and cost \(m\).
For a torus axis use \eqref{eq:C5} with
\(L=C\max(1,L_j)\), where the fixed uniform constant \(C\ge1\)
is chosen using \eqref{b:torus-axis-norm} so that \(\|\beta_j\|\le L\).
The resulting opposite sections lift to
actual angular functions after dividing their initials by
\(\bar h^{m_j/n}\). Their costs satisfy the asserted bound
because \(\widetilde v\in Q_T\).
Tower independence first makes the residual initials
algebraically independent. The positive torus initials then
have independent Laurent charges over \(K_1\), so adjoining
them preserves algebraic independence. For \(S'\ge1\),
take monomial exponents through
\[
\left\lfloor
\frac{cS'}{\max(1,W_j)}
\right\rfloor
\]
at each axis, for sufficiently small fixed \(c>0\).
Their degrees sum to at most \(S'\); nest the products into
a common allowed degree, rounding up to a multiple of \(n\).
Their costs are \(O(S')\), and their independent initials
ensure the same upper cost bound for nonzero combinations.
This proves the probe and box assertions.
\end{proof}
==== END Bounded-klt-complements-for-Fano-contractions//width-estimates.tex ====
==== BEGIN Bounded-klt-complements-for-Fano-contractions//references.bib ====
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doi = {10.1137/17M1160148},
eprint = {1610.00298v4},
archivePrefix = {arXiv},
primaryClass = {math.AG},
url = {https://arxiv.org/abs/1610.00298v4},
note = {All numbered references refer to arXiv:1610.00298v4}
}
@article{FilipazziGeneralized,
author = {Filipazzi, Stefano},
title = {On a generalized canonical bundle formula and generalized adjunction},
journal = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze},
series = {5},
volume = {21},
year = {2020},
pages = {1187--1221},
doi = {10.2422/2036-2145.201810_001},
eprint = {1807.04847v3},
archivePrefix = {arXiv},
primaryClass = {math.AG},
url = {https://journals.sns.it/index.php/annaliscienze/article/view/934},
note = {All numbered references refer to arXiv:1807.04847v3}
}
@article{Cartier1957,
author = {Cartier, Pierre},
title = {Une nouvelle op{\'e}ration sur les formes diff{\'e}rentielles},
journal = {C. R. Acad. Sci. Paris},
volume = {244},
year = {1957},
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url = {https://www.ihes.fr/wp-content/uploads/2024/09/CartierCRAS57.pdf}
}
@article{SchwedeFAdjunction,
author = {Schwede, Karl},
title = {{F}-adjunction},
journal = {Algebra Number Theory},
volume = {3},
number = {8},
year = {2009},
pages = {907--950},
eprint = {0901.1154},
archivePrefix = {arXiv},
primaryClass = {math.AG},
url = {https://msp.org/ant/2009/3-8/ant-v3-n8-p03-s.pdf},
note = {Theorem and corollary numbering refers to the published version}
}
==== END Bounded-klt-complements-for-Fano-contractions//references.bib ====