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UNIFORM CARTIER SECTIONS FOR FANO TYPE CONTRACTIONS
A plain-English companion version, written for curious high school readers
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PURPOSE OF THIS DOCUMENT
------------------------
The original paper (fano_cartier_sections_original.txt, about 415 KB) is a
research paper in contemporary algebraic geometry. It proves a conjecture of
Birkar and Shokurov that had been open for more than twenty years. The audience
of that paper is other algebraic geometers.
This companion version has a different audience: a reader who can follow
high-school algebra and is willing to think carefully about geometric
intuition, but who has not seen modern algebraic geometry. It is meant to
be read alongside the original. Where the original proves a technical
lemma, this version tries to say WHY the lemma is true and WHAT it means,
and then points at the original for the full proof.
The document has these parts:
Part I (this file, lines up to "PART II") -- what the theorem says,
why anyone cares, and the four families of tools.
Part II -- the structural theorems and their plain-English meanings.
Part III -- how a high school reader can genuinely test one small case
with pen and paper or a computer.
Part IV -- a glossary of every specialized word the paper uses.
Part V -- the statement of every lemma and theorem from the original,
lightly rephrased but mathemtically unchanged.
"More concise than the original" and "more complete than the original"
sound like opposites. They are not. The original is long because it needs
to justify every step to a professional reader who will check. This one
is short because it only needs to explain the shape of the argument to
someone who is not going to referee it. Nothing is dropped; some things
are compressed and some things are spelled out that the original takes
for granted.
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PART I. WHAT THIS PAPER IS ABOUT
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1. THE QUESTION IN ONE SENTENCE
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Given an "e-lc Fano type contraction" with a specified point z on its
base, can you always find a Cartier hypersurface through z whose pullback,
added with some small coefficient tau, keeps the whole pair log canonical?
And can tau be chosen to depend only on the total dimension and the
discrepancy gap e, not on any tricky measurement of the particular
example?
The answer this paper gives is: yes. That is Theorem A (the "main"
theorem). The paper also proves the birational klt complement theorem
(Theorem B) which is a stronger statement in a narrower setting, and
which was the missing input in Bingyi Chen's recent reduction.
If the words in the previous paragraph mean nothing yet, that is fine.
The rest of Part I fills them in, in the order a high schooler would want.
2. START FROM THE SIMPLEST CASE
-------------------------------
Take the simplest interesting case: the base Z is a curve, say the complex
plane (or any smooth curve). Above each point sits a fibre, a curve or
higher-dimensional object. A "Cartier hypersurface" on Z through a point
z is just the zero set of a single equation: in two variables x, y, the
set {x = 0} writes as x = 0; {x - y^3 = 0} also writes as one equation; a
curve through the origin is a "Cartier hypersurface" if it is the zero of
a single polynomial. That is the model to keep in mind.
Pulling back through f means: take that curve on Z and look at its
preimage inside X. The preimage is a hypersurface in X (a curve or
higher), and we want to know how singular it is.
3. WHAT "LOG CANONICAL" MEANS (PLAIN LANGUAGE)
----------------------------------------------
A divisor is an integer sum of codimension-one pieces. A typical piece on
a smooth surface is a curve. A "boundary divisor" B is a sum of such
pieces with rational coefficients between 0 and 1. A pair (X, B) is a
variety plus a boundary.
The number "log discrepancy a(E; X, B)" measures how singular the pair is
at any already-blown-up curve E. For a prime divisor E on a smooth X with
empty boundary, a(E) = 1. For a pair with a boundary B, a(E) is reduced by
the coefficient of E in B.
You can always invent new blow-ups, and E ranges over all pairs
(blowup, curve on the blowup). The pair is
- "log canonical" if a(E) >= 0 for every E;
- "klt" if a(E) > 0 for every E (strictly log canonical, from
Kawamata Log Terminal);
- "e-lc" if a(E) >= e for every E and the e is a positive constant.
Intuition: log canonical = "as singular as we allow"; klt = "a bit less
singular"; e-lc = "strictly less singular, in a measurable way". The
constant a(E) is small when E sits in a very singular place. So log
canonical is the promise "no E has negative discrepancy".
4. WHY FANO TYPE AND WHAT IS A CONTRACTION
-----------------------------------------
A "contraction" is a proper surjective map f: X -> Z with connected
fibres and the property that functions on X that are constant on each
fibre are exactly pullbacks of functions on Z. The simplest example is
the map C^2 -> C that forgets the second coordinate.
The word "Fano" is the geometric analogue of "negatively curved" in the
complex-direction sense: -K_X is ample. "Fano type" allows a klt
boundary C: -K_X - C is ample. Fano varieties include P^n and any smooth
quadric; they are the building blocks of higher-dimensional algebraic
geometry the way spheres are building blocks of topology.
The contraction f: X -> Z is "of Fano type" if on X there is such a
Fano type pair whose -K is relatively ample over Z. This is a natural
setup: it says "X is Fano-ish, and Z parametrizes how."
5. WHAT A LOG CANONICAL THRESHOLD IS
------------------------------------
Given a pair (X, B) and an effective divisor D on X, define
lct(X, B, D) = sup{ t > 0 : (X, B + t D) is log canonical }.
That is: how much can you add of D before the pair stops being log
canonical? For the pair (smooth point 0 in C, 0), D = m * [0], the
threshold is 1/m. For the pair (smooth curve, 0), D = [0] (a single point
with coefficient 1), the threshold is 1. So the threshold goes to 0 as
the divisor becomes more tangled.
The main theorem says: however bad the example, lct(X, B, f^*D) for a
well-chosen D through a prescribed base point is bounded below by a
tau that depends only on dim(X) and the e in e-lc. It does NOT depend
on dimension of Z, indices, denominators, or anything pathological
about the specific example.
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PART II. EXACT STATEMENTS OF THE MAIN THEOREMS
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We now state the two central theorems with no changes from the original.
"Effective rational boundary" means B_eff with rational coefficients.
"Rationally Cartier" means a positive multiple is a Cartier divisor.
THEOREM A (Uniform Cartier sections for Fano type contractions), original
Theorem 1. For every integer d > 0 and every real epsilon > 0 there is
a real tau = tau(d, epsilon) > 0 with the following property. Let (X, B)
be an epsilon-lc rational pair of dimension d, and let f : X -> Z be a
Fano type contraction of normal integral quasi-projective varieties with
dim Z > 0. Suppose -(K_X + B) is nef over Z. For every closed point z in
Z there are an open neighbourhood U of z and a nonzero effective Cartier
divisor D on U, with z in Supp(D), such that
(f^{-1}(U), B|_{f^{-1}(U)} + tau * f^* D) is log canonical.
THEOREM B (Birational klt complements), original Theorem 2. For every
integer d > 0 and real epsilon > 0 there is a positive integer
n = n(d, epsilon) such that the following holds. Let X be an epsilon-lc
rationally Gorenstein variety of dimension d, and let f : X -> Z be a
birational Fano type contraction with -K_X nef over Z. For every closed
z in Z there is an effective rational divisor Delta over a neighbourhood
of z such that (X, Delta) is klt there and n(K_X + Delta) is Cartier and
linearly trivial there, after shrinking the neighbourhood.
COROLLARY (Real boundaries, original Corollary 3). Over C, the
conclusion of Theorem A holds for effective real boundaries B with
K_X + B real Cartier, with the same uniform dependence on d and epsilon.
Plain-English form of Theorem A: "There is always SOME hypersurface
through the prescribed point of the base whose pullback singularities
are mild enough that a fixed tiny coefficient of it (tau) can be added
without breaking log canonicity. And tau only needs to know the total
dimension d and the discrepancy gap e."
Plain-English form of Theorem B: "In the birational case you can always
find a boundary Delta that makes the pair klt and whose index n(K_X + Delta)
is bounded by a number n that depends only on d and epsilon."
Theorem B implies Theorem A through the recent reduction of Bingyi Chen
(2025): boundedness of klt complements on birational Fano contractions
is EQUIVALENT to bounded Cartier sections on Fano contractions. That
equivalence is why the paper only has to prove Theorem B.
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PART III. THE SHAPE OF THE PROOF (NO DETAILS)
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Here is the whole proof of Theorem B in ten short steps. Each step is
indexed lemma-by-lemma in Part VIII of this companion and proved in
full in the original.
Step 1. Suppose the theorem fails. Then there is an infinite sequence of
examples, of bounded dimension, where the smallest n required to get a
klt complement grows without bound: n_1 < n_2 < n_3 < ... -> infinity.
Step 2. Using Birkar's bounded lc complements (2019), reduce each
failing example to a nice local situation: a klt germ x in S with a
finite cyclic group Gamma acting quasi-etale, a regular function h of
the same character as a canonical generator, and a centered valuation
T with A_S(T) = T(h) = 1. This is Section H of the original, done via
"canonical covers" (in the isomorphism case) and "signed anticanonical
torsors" (in the general birational case).
Step 3. Failure of complement index translates into a growth condition
on T: for every fixed m, eventually every regular eigenfunction s of
character chi^m has T(s) >= m. Call this the "saturation hypothesis".
It is the reason the problem is about character degrees, not just
geometry.
Step 4. Take the ratio Y = s / h^m. Then Y is Gamma-invariant and
T(Y) = T(s) - m >= 0. So we are forced into a "common slice"
Q = {valuations q : A_S(q) = q(h) = 1, q(Y) >= 0 for every such Y}.
That is the object Section E of the original optimizes over.
Step 5. On Q, minimize a numerical functional: the normalized volume
of a valuation, with a "height" that lets you add extra constraints
(tiltings) without destroying positivity. The minimization problem on
a non-compact, non-discrete set is hard, but Blum / Xu / Li-Xu /
Xu-Zhuang theory gives existence, uniqueness, and quasi-monomiality
for nearby problems. Section E sets it up.
Step 6. A key trick: build an ordinary klt pair whose minimizer lies
EXACTLY on the just-described slice, by adding a small rational
perturbation of the boundary (the exact-optimization theorem of
Section E in the original). Then the
existing theory applies. So "constrained minimization" reduces to
"unconstrained minimization" by smuggling constraints into the
boundary equation, at the cost of a single positive rational parameter
lambda (fixed for all competitors, not shrinking to zero).
Step 7. Pass to the central cone (stable degeneration): the associated
graded ring of the minimizer is finitely generated, normal, and klt,
as an affine cone over the same combinatorics. This is the "central"
object in which a fresh torus acts, and it makes the "which valuations
are Gauss extensions" question tractable.
Step 8. On the central cone, use the character-of-homogeneous-functions
description: letters of a character in degree k correspond to sections
of a torsor over an integer ray. This dictionary is Section D.
Step 9. Optimize again on the central cone. Repeat Steps 5-8 as long as
you can extract a new algebraically independent variable from the
data. Each stage either terminates in a contradiction (a quotient prime
with negative-but-tiny discrepancy, which the e-lc hypothesis excludes
by a strict gap) or produces a new algebraically independent variable.
Dimension bounds the number of new variables, so eventually one of the
two alternatives must fire. This is Section G's "pinning".
Step 10. Contradiction. The failure sequence cannot exist. Hence
Theorem B holds. Apply Chen's equivalence to conclude Theorem A.
The phrase "character by character" is important. The proof is not
looking just at the total space and saying "make these sections positive"
once. It is looking at each fixed degree of a Z-graded ring and showing
that a bounded finite number of character degrees can be made positive
simultaneously. Then the equivalence with complements is degree-by-
degree, not global.
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PART IV. GLOSSARY
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Every word listed below appears in the paper. Plain definitions, with
the occasional "do not confuse with".
a(E; X, B) log discrepancy of a divisor E over the pair (X, B).
Nonnegative for lc, strictly positive for klt.
ample "positive" in the projective sense: a cartier divisor
whose multiples separate points and give embeddings.
anticanonical -K_X, where K_X is a canonical divisor.
b-divisor a divisor that lives on "some sufficiently high model";
technically a compatible family of divisors across
a directed system of blowups.
base codomain of a contraction: f: X -> Z.
b-nef nef as a b-divisor (i.e. nef on every sufficiently
high determination).
canonical divisor a divisor K_X representing the canonical class, i.e.
one whose section spaces carry the top holomorphic
n-forms.
Cartier locally cut out by a single equation.
coeff coefficient (of a prime divisor in a rational divisor).
contraction proper surjective morphism with connected fibres and
O_Z -> f_* O_X an isomorphism.
dlt, lc, klt discrepancy conditions: dlt is "divisorial log terminal
after descending to the snc case"; lc is >= 0; klt is
> 0.
effective all coefficients nonnegative.
e-lc every log discrepancy is at least e.
Fano -K_X ample.
Fano type there exists a klt boundary C with -(K_X + C) ample.
In the relative version, ample over Z.
fibre preimage f^{-1}(z) for a point z in Z.
gamma, chi a finite cyclic group and a character; both are part
of the reduction in Section H.
Gauss valuation the valuation on a polynomial ring given by a weight
vector: v(sum a_m z^m) = min{ weight(m) : a_m != 0 }.
The simplest nontrivial example besides divisorials.
graded, Veronese an algebraic ring with a Z or N grading. A Veronese
subring is a subring spanned by degrees divisible by
some fixed integer d.
germ neighbourhood of a point, sharpened to "infinitely thin
neighbourhood".
h^0(X, O_X(D)) dimension of the space of global sections of D.
height a linear functional on a cone of weights, used to tilt
the optimizer in a controlled direction.
index, Cartier the smallest positive integer m with mD Cartier.
invariant fixed by a group action.
klt, log terminal see "lc, klt".
linear equivalence D1 ~ D2 iff D1 - D2 = div(f), a principal divisor.
log pullback pullback adjusted by the discrepancy of the map.
mmp minimal model programme; the iteration of
"program of steps that simplify a pair".
moduli part in the canonical bundle formula, the denominator
tower in K_X + B; the "b-nef" part.
multiplicity order of vanishing in codimension one at a point.
nef "numerically effective": intersects every complete
curve nonnegatively.
normalized volume a volume functional on valuation space, normalized
so that the canonical class gets value 1. Central to
Blum / Xu / Zhuang minimization theory.
perturbation a small change in divisor coefficients.
quasi-etale etale outside codimension two. Finite generically.
rational divisor a divisor whose coefficients are rational numbers.
shokurov, birkar named after V.V. Shokurov and Caucher Birkar.
Birkar proved bounded lc complements in 2019;
Shokurov conjectured klt complements in 2004.
snc simple normal crossings: local model is a union of
coordinate hyperplanes.
torsor a principal homogeneous space; here a line bund-
le with a nowhere-zero section removed.
valuational center the point below a valuation; where it "sits".
volume normalized multiplicity growth of a valuation ideal.
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PART V. THE ABSTRACT, AND THE CONVENTIONS
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In characteristic zero, the Cartier-divisor conjecture of Birkar and
Shokurov holds for rational boundaries. Over C, it holds for real
boundaries. The constants depend only on the total dimension and
the discrepancy gap of the log canonical pair.
For the one-line reading of every lemma in the original, see
PART VIII below. A short list of conventions used throughout:
- A "pair" (X, B) always means X normal, B effective with rational
coefficients, K_X + B rationally Cartier.
- "Bounded dimension" means there is an n_0 such that all examples
have dim <= n_0.
- "Eventually", "for all i >> 0" all mean: for all sufficiently
large indices in the sequence. In practice the proof passes to
subsequences.
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PART VI. WHAT A HIGH SCHOOL READER CAN ACTUALLY DO
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The mathematics of the paper is genuinely beyond high school. But the
SHAPE of the proof is not. Below are a handful of concrete things a
curious reader can compute by hand or on a computer, that touch the
"nerve" of the argument. Each is small and finite.
Exercise 1 (The simplest lct calculation). Let X = C^2 with co-
ordinates (x, y), let z = (0, 0), and let D be defined by g(x, y) = x.
Verify that the pair (X, B = 0 + t D) is log canonical if and only if
t <= 1. Use the blowup of z and check the discrepancy of the
exceptional divisor. This is the single easiest case of Theorem A.
Exercise 2 (Threshold of the cube). Let g(x, y) = xy. Show that the
threshold on C^2 is 1. Let g = x^2 + y^2. Show it is 1. Let g = x^2.
Show it is 1/2. Let g = x^3 + y^3. Show it is 2/3. General pattern:
for g homogeneous of degree d with smooth projective zero locus, the
threshold is min(1, 2 / d). Contrast with the smooth case d = 1 where
it is 1.
Exercise 3 (Ratio q(Y) = q(s) - m q(h)). Take X = C, h = x, s = x^m.
Then A(x = 0) = 1 and h(x=0) = 1. Compute Y = s / h^m = 1; q(Y) = 0.
That is the base case of Step 4. Now take X = C, s = x^{m+1}. Then
Y = x, and every q with q(x) > 0 gives q(Y) > 0. This is the "saturation
hypothesis" in the simplest possible case.
Exercise 4 (A character computation). On C[x, y] with the group
mu_2 acting as (x, y) -> (-x, -y), which monomials are invariant? They
are the even-degree monomials. Which have character chi (the nontrivial
character)? The odd-degree monomials. This is the toy model for the
"character by degree" mechanism in Section D.
Exercise 5 (Sanity check of Lemma C.6). Compute the dimension of the
space of homogeneous degree-d polynomials in 3 variables: it is
(d+1)(d+2)/2. This is your typical "section count". The paper needs
bounds of the form "h^0 <= C v_i n^p" where p is the transcendence
degree of a field.
Exercise 6 (Blowup and discrepancy). Blow up the origin in C^2. The
exceptional divisor E is P^1. Since K_{C^2} is trivial, K_{Bl} =
pi^* K_{C^2} + E = E as a divisor. Using the paper's convention
a(E; X, B) = 1 + coeff_E(K_W - pi^*(K_X + B)), check that
a(E; C^2, 0) = 1 + 1 = 2, so (C^2, 0) is 2-lc. This is the basic
blowup calculation underlying Exercises 1 and 2.
Exercise 7 (Fano examples). The complex projective line P^1 is Fano:
-K_{P^1} is ample. The complex projective plane P^2 is Fano. Blowup
P^2 at k <= 8 points in general position; -K remains ample, giving a
del Pezzo surface with (-K)^2 = 9 - k. At k = 9 general points the
square (-K)^2 = 0 and the surface is Calabi-Yau, not Fano; beyond that
-K is not even nef. Compute (-K_{Bl_1 P^2})^2 and (-K_{Bl_1 P^2}) . (a
line) and check positivity. This is the model to keep in mind when the
paper says "Fano type contraction".
Exercise 8 (Can you prove the 2D case?). For X a smooth surface and
z in X with B = 0, pick D = a smooth curve through z. Then (X, D) is
log canonical because both X and D are smooth, so tau = 1 works for
the identity contraction X -> X. More generally the log canonical
threshold of the maximal ideal m_z on a smooth n-fold is n: blow up
z, check discrepancies, then induct. For a surface this is 2, and
Lemma H.7 reduces a bound on lct(m_z) to a Cartier-divisor bound of
min(that, 1) = 1, matching the construction above. Chen (2025) proves
the surface LOCAL assertion with explicit coefficient e^2/24, and
hence both conjectures for bases of dimension at most two (point and
curve bases). The paper here handles arbitrary base dimension. The
2D smooth case is essentially high school; the gap to the general
case is the substance of the paper.
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PART VII. WHY THIS MATTERS
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Three reasons a high schooler might care beyond "it's math":
(1) The theorem settles a conjecture that has been open for over
twenty years. That is rare and worth noticing.
(2) The proof is a machine. It does not look at any individual Fano
type contraction. It uses boundedness of Fano varieties, of
rationally connected CY varieties, and of nef-and-big divisors
(all themselves recent theorems). The general strategy -- "pass
to the central cone, minimize, extract a character, iterate until
dimension stops you" -- is a template. Variants of it are being
used in ongoing work on stability and moduli.
(3) The result is non-effective. It produces a positive tau, but says
nothing about HOW BIG tau is. Only that it depends on d and e.
This is common in algebraic geometry: the constant exists by
compactness, not by an explicit construction. If you ever want
to run this on a computer, you need extra work.
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PART VIII. FULL LEMMA INDEX OF THE ORIGINAL
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Plain-English description of every named result in the original, in
the order they appear within each section. Numbering below (A.1, A.2,
...) is the reading order in the source, not the compiled subsection
numbers of the printed PDF. For the exact statement and proof of any
item, look it up by name (the names are quoted from the original
headings) in the corresponding section of the original file.
Section A (Discriminant comparison and uniform chambers).
A.1. Comparison along long parameter intervals (Lemma). The
discriminant discrepancies of a bounded generalized CY family
obey the same klt and ratio bounds on the base as on the
higher model, uniformly over the family.
A.2. Phases under pure pullback (Lemma). A signed phase bound for a
rational Cartier divisor on a high model descends uniformly to
the base in the small-discrepancy limit.
A.3. A uniform linear chamber (Lemma). For a linear family of
generalized CY data with bounded-denominator initial data, the
discriminants stay affine in the family parameter on a
uniform positive interval.
A.4. Tools at an lc place (Lemma). At a prescribed lc place, one
can build a tailored ordinary lc complement, preserving the
chosen place, with a bounded complement index.
A.5. Uniform charts (Proposition). Any prescribed place in a Mori
step sits on a toroidal chart with bounded matrices.
Discrepancies of arbitrary generalized CY data are Lipschitz
in the angular coordinates, uniformly in the contact.
A.6. Bounded toroidal charts (Lemma). Toroidalisation produces
these charts with bounded structure, after a bounded extension
of the base field.
A.7. Degrees control angular variation (Lemma). A rational function
with bounded generic pole degree has bounded angular Lipschitz
constant on the chart.
A.8. Horizontal variants (Corollary). The same chart and metric
results hold for horizontal places with analogous bounds.
A.9. Integral approximation (Lemma). Real chart points can be
approximated multiplicatively by integral chart points whose
orders are legitimate prime valuations.
A.10. Quantized chart cells (Claim). When the functions involved
have bounded denominators, their cells of linearity are cut
out by finitely many fixed normals and have bounded
denominators at every vertex.
Section B (Nonvanishing from signed phases).
B.1. Phase nonvanishing (Lemma). If a bounded family of Fano type
varieties carries effective CY data whose polarization has
controlled signed phases at small-discrepancy divisors, then a
bounded multiple of the polarization has a nonzero section.
B.2. Bundle construction and discrepancies (Lemma). A rational
divisor H gives a Seifert-like bundle with two sections whose
discrepancy formula reduces to a |h| term plus the base
discrepancy. This is a dictionary from base sections to
bundle sections.
B.3. Clearing the small denominators (Lemma). After multiplying H
by a bounded integer, the denominator boundary becomes klt log
Fano, so the bundle is of Fano type.
B.4. An unbounded-index obstruction (Lemma). If the anticanonical
index of a Fano type variety with a polarization grows without
bound, then a small-height divisor at infinity with small
discrepancy cannot exist.
B.5. Bounding a principalization order (Lemma). If a rational
divisor P has bounded signed phases at small-discrepancy
divisors and the polarization has effective Fano type data,
then the principalization order of P is bounded.
Section C (Cutoff fields, phase bounds, and section counts).
C.1. Uniform cutoff models and order functions (Lemma). The cutoff
contraction D_W splits into an ample base polarization plus
the small-coefficient floor primes, giving bounded field
generators and a b-nef order function satisfying precise
inequalities.
C.2. Scaled phase bound (Lemma). At tiny multiples of the
polarization, principal orders of any rational function have
bounded phase by a fixed multiple.
C.3. Saturation and section counts (Lemma). After a subsequence,
there is a field plateau K_* in which every bounded-cost
function lies, and its section counts match a common
normalization v_i * n^p.
C.4. Vanishing on a smooth orbifold (Lemma). Kodaira-style
vanishing holds on a smooth Deligne-Mumford stack with klt
boundary.
C.5. Adjoint Taylor exponents (Lemma). For a birational nef and big
integral divisor N, the jet body determines which Taylor
exponents actually occur: any interior-lattice exponent of tK
occurs.
C.6. Uniform count from the jet body (Lemma). The upper and lower
bounds for h^0(N) and for the product's dimension are both
controlled by the volume of the jet body.
C.7. Fixed generators and neutral algebraic extensions
(Corollary). At one eventual index, the whole field generated
by a bounded-cost set lands in K_*, and extensions of
trivially valued subfields remain trivially valued.
Section D (Characters and regularization on affine cones).
D.1. Cone dictionary (Proposition). Homogeneous functions on an
affine cone give a dictionary between self-dual ray weights
and Cartier divisors on the Proj, with discrepancies and
phases expressed monomial-minimally.
D.2. All divisorial lc tests are Gauss (Lemma). Every divisorial
lc place of the cone is a Gauss valuation, so the divisorial
testing set is essentially the monomial one.
D.3. Sections as regular symbols (Lemma). Sections of the cone's
polarization correspond to regular symbols of the form
(rational function) * h^b times a monomial z^weight.
D.4. Sandwich regularization (Proposition). Given bounded phase
information on the small-discrepancy locus, a bounded
multiple of J_m has a nonzero regular symbol.
D.5. Bounded-multiple character regularization (Corollary). For
a weight in a bounded-width strip, a bounded multiple of a
character times h^b is regular.
D.6. Count for a single character and weight (Corollary). The
number of homogeneous invariant symbols in a bounded-width
character-weight box is bounded by a constant times v_i *
n^p.
D.7. Characters for a tilted height (Corollary). A bounded number
of negative-degree character eigenfunctions plus a small tilt
of the polarization gives positive-degree regularization.
Section E (Optimization on common log canonical slices).
E.1. Logarithmic jet estimate (Lemma). On a fixed resolution, the
monomial retraction and the valuation differ on polynomials of
degree at most D by O(D * discrepancy). Iterating gives
uniform control on every regular function, not only polynomials.
E.2. Compactness and uniform centring (Lemma). The common lc slice
is a compact finite polyhedral complex, with uniform lower
volume bounds on centred valuations.
E.3. Simultaneous approximation (Corollary). For a single constant
C, translation to a monomial retraction loses at most (1 + C e)
in every regular function, uniformly over all such valuations.
Thus volume of a centred valuation is controlled by the volume
of its retraction.
E.4. Convex domination (Lemma). Any convex combination of
common-slice valuations is dominated (pointwise on every
regular function) by a single common-slice valuation. The
allowed heights behave well under this domination.
E.5. Opposite ratios (Corollary). Opposite-order ratios of the
form q'/q with q*q' regular interpolate exactly under the
convex-domination step.
E.6. Volume interpolation and a strict gap (Lemma). If two
valuations differ on some regular function by more than a
fixed factor, their midpoint interpolant has strictly smaller
volume (relative to either). Near-minimizers are therefore
multiplicatively close on every regular function.
E.7. A boundary encoding the common slice (Lemma). An effective
lc boundary whose zero-discrepancy rays are exactly the
common slice can be built by adding small pencil members
with positive bounded coefficients.
E.8. Exact common-slice minimizer (Theorem). For a rational
height, minimization over the common slice is realized
exactly by the normalized-volume minimizer of an ordinary
klt pair. One single rational perturbation parameter lambda,
fixed for all competitors, suffices.
E.9. Near-minimizers for a height (Corollary). Two valuations
that are both within o(1) relative error of the minimum
must be multiplicatively 1+o(1) equal on every regular
function.
E.10. Domination by the minimizer (Corollary). Any common-slice
valuation is dominated by 2^n times the minimizer, up to
height normalization.
E.11. One-sided height tilt (Lemma). If a common-slice valuation
has a positive value on a sign function, then a small
one-sided tilt of the height yields a minimizer that also
has positive sign and stays bounded on the slice.
Section F (Degeneration and rank refinement).
F.1. Weighted presentations (Lemma). The associated graded
algebra of a quasi-monomial valuation is a quotient of a
power series ring by the initial ideal; nearby rational
regradings give nearby quasi-monomial valuations.
F.2. Canonical forms under degeneration (Proposition). The
reflexive top form descends to the central fibre with the
same grading weight and group character, and the group
action on the cone remains quasi-etale.
F.3. Lc equality and principal boundaries (Corollary). If the
degenerating valuation is h-lc, the central pair is lc and
the canonical generator has the same group character as the
initial symbol of h.
F.4. Sign transfer (Proposition). An inequality of the form
q(g) <= 0 on the actual common slice is equivalent to the
corresponding inequality on the central common slice, for
any test ratio g, without restriction on its cost.
F.5. Exact tests and pure weights (Corollary). Exact vanishing
of a test throughout the actual slice is equivalent to
exact vanishing of its initial form throughout the central
slice; pure-weight valuations are always h-lc and common.
F.6. Compression (Lemma). Two successive gradings (from two
successive minimizations) can be compressed into a single
actual valuation on the original germ, preserving all lc
equalities and all exact tests and increasing the rational
rank.
F.7. Rank refinement (Proposition). If a sequence of balanced
cones satisfies a fixed balance condition, the refinement
step replaces the valuation by a near-equal one with the
same main properties and satisfying the angular order
condition (C-alpha) with error eta_i -> 0.
Section G (Pinning the gradings).
G.1. Pinning obstruction (Theorem). An infinite sequence of
quasi-etale character-graded klt germs with lc equality and
the saturation hypothesis cannot exist. This is the
contradiction that proves the birational complement theorem.
G.2. Non-cancellation in every degree (Lemma). Among finitely
many bounded-cost monomial products, the least-w initial
term does not cancel at any later working valuation on the
pre-pinning slice, in every polynomial degree.
G.3. Uniform product growth (Lemma). For a finite-dimensional
space V of dimension g with transcendence degree a, N-fold
products of members of V have dimension at least
c_n * g * N^a.
G.4. High-scale pin (Proposition). At a high stage, finitely many
proportion tests of cost O(L) can be added to force the
selected values either to zero or to a rational line,
retaining a centred point and all previous constraints.
G.5. Field and lattice consequences of the pin (Lemma). The pin
automatically puts every lattice direction of width O(L) on
the anchor line, and forces the angular tests to act
trivially on the cutoff field at the next scale.
G.6. Additional row via tilted characters (Claim). If a
projection of the current constrained set is thin, a tilted
character block produces an integral tie at the comparator
that is transverse to the previously imposed rows.
Section H (Reduction to the pinning input).
H.1. Factorial saturation (Proposition). Failure of klt
complements gives a factorial order w whose section orders
dominate k A_X(w) on every prime, uniformly, and reduces
to a geometric generic slice.
H.2. The isomorphic case (Lemma). In the birational-but-Cartier
case, a canonical cover produces the exact pinning input
(torsion, character match, lc equality, saturation).
H.3. The signed anti-canonical torsor (Lemma). In the general
birational case, a signed anti-canonical Z-graded torsor
serves as the model, with both positive and negative slopes
to catch section vanishing.
H.4. Padding and the actual quotient discrepancy (Lemma).
Padding by b-1 extra coordinates makes the canonical
character match b and preserves a fixed quotient
discrepancy gap e.
H.5. The pinning input (Proposition). The reduced data satisfy
every hypothesis of Theorem G.1 (the pinning obstruction).
H.6. Point ideals and change of field (Lemma). Reduction to C
via descent, and the passage between the lc threshold of
the point ideal and existence of a good Cartier divisor
through the point.
================================================================================
PART IX. EXACT STATEMENTS OF EVERY RESULT (VERBATIM, FROM ORIGINAL)
================================================================================
This appendix reproduces every theorem, lemma, proposition, corollary, and
claim statement from the original file verbatim, in source order, organized
by the eight sections A-H and the introduction. Proofs are excluded; those
are in the original. Backslash-refs such as \ref{lem:...} point to labels
visible in the source (and most of them appear in this very appendix); grep
the label without the "lem:" prefix to locate the block you want.
For one-line plain-English summaries of each of these results, indexed by
section and number, see PART VIII above. PART VIII is what a first-time
reader should read; PART IX is what a referee should verify against.
Section A
------------------------------------------------------------
\begin{lemma}[Comparison along long parameter intervals]\label{lem:A-comparison} Let \(X\to S\) be a sequence of Fano type contractions as above.
Suppose \(a_t\) are effective generalized CYs over \(S\), continuous finite rational piecewise affine families of discrepancies (with a finite subdivision of parameter per example, so the pieces can be treated on a common resolution), on \(0\le t\le L_i\) or \(-L_i\le t\le L_i\) with \(L_i\to\infty\).
Pieces and endpoints can be taken rational.
Values at each valuation are concave in \(t\), and \(a_0\) is klt.
Assume the corresponding one of the following (one-sided for \(0\le t\le L_i\); two-sided for \(-L_i\le t\le L_i\)):
\begin{enumerate}[label=\textup{(\roman*)}]
\item (one-sided) the discrepancy functions are nonincreasing, and whenever \(a_0(T)\to0\), \(a_t(T)/a_0(T)\to1\) for every fixed \(t\);
\item (two-sided) for some fixed \(\delta>0\), whenever \(a_0(T)<\delta\), \(a_t(T)/a_0(T)\to1\) for every fixed signed \(t\).
\end{enumerate}
As throughout these lemmas, tests on sequences of valuations include along subsequences.
Then on every fixed bounded parameter range the data are eventually klt
over the generic point of \(S\). Write \(b_t\) for their discriminant
discrepancies on \(S\). In case \textup{(i)}, they are nonincreasing
and satisfy \(b_t(P)/b_0(P)\to1\) for every fixed \(t\) whenever
\(b_0(P)\to0\). In case \textup{(ii)}, some fixed \(\delta'>0\)
gives the same convergence for every fixed signed \(t\) whenever
\(b_0(P)<\delta'\). These conclusions include every subsequence and
primitive prime orders on every birational model of \(S\).
\end{lemma}
\begin{lemma}[Phases under pure pullback]\label{lem:A-pullback-phases} Let \(a\) be a sequence of klt effective generalized CYs over \(S\) in the same Fano type setting, and let \(H\) be a sequence of \(\mathbb Q\)-Cartier \(\mathbb Q\)-divisors on the respective bases \(S\).
Suppose for some fixed positive integer \(m\), for every \(a(T)\to0\) the phase \(v_T(m f^*H)\bmod 1\) has lifts to \(\mathbb R\) of size \(O(a(T))\) bounded below by \(-o(a(T))\).
Then with discriminant \(b\), for some fixed positive multiple \(m'\), \(v_P(m'H)\) satisfies the same phase condition wherever \(b(P)\to0\).
\end{lemma}
\begin{lemma}[A uniform linear chamber]\label{lem:A-linear-chamber} In the same setting suppose \(a\) is an lc effective generalized CY over \(S\), with values in \((1/p)\mathbb Z\) for fixed \(p\), and \(a+t d\) for \(0\le t\le t_0\) are effective generalized CYs, klt for positive \(t\), \(d\ge0\) on all divisorials; take \(L_t\) affine, and \(t_0>0\) fixed rational.
Then eventually there is a uniform smaller \(t_1>0\) on which discriminants equal \(a'+t d'\) with \(a'\) of bounded denominator (i.e., the same \(p'\) works on all models).
There is also the following version with extra data:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(a+\lambda h+s c\) are effective generalized CYs over \(S\) for \(0<\lambda\le\lambda_0,\ 0<s\le\lambda\theta_0\), \(\lambda_0,\theta_0>0\) fixed, \(h\) of bounded denominator, \(c\ge0\);
\item on \(a=0\), \(d\le c\le C d\), \(-\eta_i c\le h\le Cc\) with fixed \(C\) and \(\eta_i\to0^+\).
\end{enumerate}
Then there are uniform \(\lambda_1,\theta_1>0\), with
\(\lambda_1\le\lambda_0\) and \(\theta_1\le\theta_0\), such that
eventually, on the sector
\(0<\lambda\le\lambda_1\), \(0<s\le\lambda\theta_1\), the data
are klt generically over \(S\), their discriminants are exactly
\(a'+\lambda h'+s c'\), and \(h'\) has bounded denominator.
Moreover, eventually there is \emph{no} divisorial \(T\) horizontal
over \(S\) with \(a(T)=0,\ h(T)<0\).
In particular for \(S\) a point exact nonnegativity on \(a=0\) follows even though only the approximate bound \(-\eta_i c\) was assumed.
In this statement the minima are lexicographic: first minimize \(a/e\),
then \(d/e\) on its minimum locus; for the extra data first minimize
\(a/e\), then \(h/e\), then \(c/e\).
Here \(e\ord_P=T|_{k(S)}\).
Thus \(a'\) itself is the discriminant in discrepancy convention even
if the endpoint is not klt generically.
The family \(a'+t d'\), \emph{including its endpoint at zero}, has the generalized effective log data (with nef b-part and total logarithmic divisor \(L_t\)).
\end{lemma}
\begin{lemma}[Tools at an lc place]\label{lem:A-lc-place-tools}
Let \(g:X_1\to W\) be a Mori step as above, and let \(A_*\) be an
effective generalized CY discrepancy on \(X_1\) over \(S\).
\begin{enumerate}[label=\textup{(\roman*)}]
\item Suppose that \(A_*\) is lc over the generic point of \(W\),
and that an effective generalized klt CY over \(S\) is available for
perturbation. Any prescribed horizontal divisorial lc place remains
an lc place of an ordinary boundary \(\Omega\), effective on the
generic fibre \(F\), with \(n(K_F+\Omega_F)\sim0\), for a bounded
positive integer \(n\).
\item For \(P\) on a birational model of \(W\), suppose \(A_*\) is klt generically and \(T\) computes the discriminant \(b_*(P)\).
There is an ordinary \(n\)-CY over the DVR, lc in its b-discrepancies
and effective horizontally on \(F\), with \(n\) bounded, keeping
\(T\) as an lc place. Here \(n\)-CY means that the ordinary total
log canonical divisor, multiplied by \(n\), is linearly trivial
over the trait.
Moreover for the adjusted discrepancy \(A_*-b_*(P)e\), any two minimizing divisorials can be joined by a chain/interpolation of lc places in the following sense: on a log resolution over the generic DVR their strict divisors (or extractions of them) belong to a connected snc floor consisting of vertical coefficient-one divisors with monomial cones at the intersections.
One can include any additional finite discrepancy data in the resolution.
In particular normalized evaluations \(a/e\) of such other data interpolate continuously along the chain, using rational weights to approximate intermediate values by divisorial evaluations.
\end{enumerate}
\end{lemma}
\begin{proposition}[Uniform charts]\label{prop:A-uniform-charts}
At a chosen vertical place in Lemma~\ref{lem:A-lc-place-tools}\textup{(ii)},
after a bounded degree extension of \(k(W)\) and a chosen extension of
the DVR and of the place, there is a toroidal cone representing the
place, with nonnegative coordinates
\begin{equation}\label{eq:A-contact-cone}
(x,e_*),\qquad M x=r e_* .
\end{equation}
Here \(M\) ranges over a finite list of integral matrices of bounded sizes, \(r\) is integral (possibly huge), \(e_*\) is value of the new uniformizer, and original uniformizer value is \(\rho e_*\) with positive bounded \(\rho\).
Nonzero integral solutions on the cone give divisorial valuations (not necessarily primitive after restriction); evaluations of discrepancy data always refer to homogeneous values by restriction.
For horizontal places as in the first bullet one just has an orthant on a bounded generic resolution, without \(e_*\).
For \emph{any} effective generalized CY data \(A\) on \(X_1\) over \(S\), values on these cones are finite continuous homogeneous piecewise linear, and are Lipschitz in \(x\) with uniform constant \emph{at fixed} \(e_*>0\) (uniform homogeneous Lipschitz in the horizontal case).
Constants depend on the boundedness in this Mori step and the complement indices, \emph{not} on \(r\), vertical coefficients, or the nef part index or the chosen data \(A\).
In particular finitely many differences at fixed parameter separations have uniform Lipschitz bounds as well.
\end{proposition}
\begin{lemma}[Bounded toroidal charts]\label{lem:A-toroidal-charts}
For the prescribed place and tailored complement in
Proposition~\ref{prop:A-uniform-charts}, there are charts with its stated
contact equations and integral-valuation property. The prescribed
place is monomial on one of these charts. They have rational functions
\(z_1,\ldots,z_f\), \(f=\dim F\), with uniformly bounded pole degrees on
a bounded embedding of \(F\), such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item their logarithmic differentials form a relative log basis in the
vertical case and an absolute log basis over the generic field in the
horizontal case;
\item their orders are linear functions of the chart coordinates with
uniformly bounded slopes;
\item every rational monomial valuation in the vertical chart with
\(e_*>0\) restricts to a Gauss valuation on
\(K_1(z_1,\ldots,z_f)\), where \(K_1/k(W)\) is the bounded extension
used in constructing the chart. Horizontally the same assertion holds
with the trivial valuation on the generic field.
\end{enumerate}
Here Gauss means that evaluating a polynomial in the \(z_j\) gives the
minimum of its coefficient-adjusted monomial weights, without
cancellation among terms of equal weight.
\end{lemma}
\begin{lemma}[Degrees control angular variation]\label{lem:A-function-slopes}
On a chart of Lemma~\ref{lem:A-toroidal-charts}, the order of a nonzero rational
function is continuous, homogeneous and finite piecewise linear,
including on the faces. If its pole divisor on the embedded generic
fibre has degree at most \(m\), where \(m\ge1\), its Lipschitz constant
in \(x\) at fixed \(e_*>0\) is at most \(Cm\). Horizontally this bound
holds on the entire cone. The constant \(C\) depends only on the fixed
families and chart choices. A function with pole degree less than one
is constant on the generic fibre and has zero angular variation.
\end{lemma}
\begin{corollary}[Horizontal variants]\label{cor:A-horizontal-variants}
The chart and metric conclusions have the following two variants.
\begin{enumerate}[label=\textup{(\roman*)}]
\item
The same tailorable bounded complement and homogeneous chart argument works if we are working over an algebraically closed generic field, with the relevant effective CYs on a \(\mathbb Q\)-factorial model of Fano type, and have boundedness only up to isomorphism in codimension one of these models.
\item
A Cartier rational b-order function \(N\) with effective trace of degree
\(O(\xi)\) on the bounded model, b-linearly \(\mathbb Q\)-equivalent
to a difference of b-nef parts both of trace degrees \(O(\xi)\), has
horizontal Lipschitz constant \(O(\xi)\), including when \(\xi\to0^+\).
\end{enumerate}
\end{corollary}
\begin{lemma}[Integral approximation]\label{lem:A-lattice-approximation}
\begin{enumerate}[label=\textup{(\roman*)}]
\item
Let \(u_i\) lie in a fixed nonnegative orthant and
\(\lvert u_i\rvert\to\infty\). After passing to a subsequence there
are \(s_i\to0^+\), nonzero nonnegative integral vectors \(n_i\), and
a fixed, possibly subsequence-dependent, \(C_*\) such that
\begin{equation}\label{eq:A-horizontal-approximation}
\lvert n_i-s_i u_i\rvert\le C_*s_i,
\qquad s_i\lvert u_i\rvert\longrightarrow\infty .
\end{equation}
\item
Let \(M_i x_i=r_i\), where \(x_i\ge0\), \(r_i\) is integral and
\(M_i\) belongs to a fixed finite list of integral matrices.
If \(\alpha_i>0\) tends to zero, then after passing to a subsequence
there are positive integers \(E_i\), nonnegative integral \(n_i\), and
a fixed \(C_*\) such that
\begin{equation}\label{eq:A-vertical-approximation}
M_i n_i=r_i E_i,\qquad
s_i=E_i\alpha_i\longrightarrow0^+,\qquad
\lvert n_i-E_i x_i\rvert\le C_*s_i .
\end{equation}
\item
For every sufficiently small fixed \(\gamma>0\), the same matching
equations admit \(n_i\ge0\) integral and \(1\le E_i\le J(\gamma)\)
with
\begin{equation}\label{eq:A-bounded-dirichlet}
\lvert n_i-E_i x_i\rvert\le\gamma ,
\end{equation}
where \(J(\gamma)\) is uniform in \(i\) and in the contact vectors
\(r_i\).
\end{enumerate}
\end{lemma}
\begin{claim}[Quantized chart cells]\label{claim:A-quantized-cells}
The bounded-denominator functions \(a,h\), when present, have only a
bounded finite set of possible slopes in a horizontal chart, or in
integral kernel coordinates on a vertical slice \(e_*=1\).
Their vertical affine constant terms belong to a rational grid with
bounded denominator, although their magnitudes need not be bounded.
Their common cells are cut out by a fixed finite set of possible
normal vectors, with offsets of bounded denominator. They are
pointed, their vertices have bounded denominator, and they satisfy a
uniform polyhedral distance bound.
\end{claim}
Section B
------------------------------------------------------------
\begin{lemma}[Phase nonvanishing]\label{lem:B-phase-nonvanishing}
Let $(V_i,H_i)$ be a sequence in bounded dimension, where $V_i$ is
projective of Fano type and $H_i$ is a nef and big $\Q$-Cartier rational
Weil divisor. Suppose that, for rational $t\in[0,L_i]$, where
$L_i\to\infty$, there are generalized data with discrepancies $a_i(t)$
such that
\begin{equation}\label{eq:B-family}
B_i(t)\ge R(H_i),\qquad
K_{V_i}+B_i(t)+\mathbf M_i(t)_{V_i}\sim_{\Q}tH_i,
\qquad \mathbf M_i(t)\text{ is b-nef and $\Q$-Cartier}.
\end{equation}
Assume that $a_i(0)$ is klt and that $a_i(t)$ is nondecreasing in $t$
at every divisorial valuation.
Assume the following condition, also after restriction to any
subsequence. For every choice of rational $t_i\to\infty$ in the
parameter range, one can pass to a further subsequence and choose a
fixed positive integer $q$ such that, for every sequence of primitive
divisorial valuations $T_i$ with $a_i(t_i,T_i)\to0$, the signed phases
$\theta_i(T_i)$ of $qH_i(T_i)$ satisfy
\begin{equation}\label{eq:B-phase-hypothesis}
-o\bigl(a_i(t_i,T_i)\bigr)
\le \theta_i(T_i)
\le O\bigl(a_i(t_i,T_i)\bigr).
\end{equation}
Then there is an integer $M>0$ such that for every $i$ some
$m_i\in\{1,\ldots,M\}$ satisfies
$|\lfloor m_iH_i\rfloor|\ne\varnothing$.
Equivalently, after replacing $M$ by a common multiple, the same
positive degree works for all $i$.
\end{lemma}
\begin{lemma}[Bundle construction and discrepancies]\label{lem:B-seifert}
Let $V$ be projective normal, and let $H$ be a $\Q$-Cartier rational
divisor. There is a projective normal compactified Seifert bundle
$p:Y\to V$ with disjoint $\Q$-Cartier zero and infinity sections
$E_0,D$ and a rational fibre coordinate $z$ such that
\begin{equation}\label{eq:B-bundle-identities}
\divisor(z)=p^*H+E_0-D,
\qquad
K_Y+E_0+D=p^*\bigl(K_V+R(H)\bigr).
\end{equation}
The second equality uses compatible canonical divisors. The divisor
$D$ is relatively ample; if $H$ is nef, then $D$ is nef, and if $H$
is nef and big, then $D+p^*H$ is nef and big.
In these identities, pullback of a rational Weil divisor along the
equidimensional map $p$ is defined at the generic points of its prime
components, using the multiplicities of their inverse images. This
agrees with Cartier pullback whenever the divisor is $\Q$-Cartier.
Orders of $H$, $E_0$, and $D$ on higher models use their Cartier
b-pullbacks.
Suppose $B\ge R(H)$ and $\mathbf M$ is a nef $\Q$-Cartier
b-divisor on $V$. Give $Y$ boundary $p^*(B-R(H))$ and nef part
$p^*\mathbf M$. Write $a$ for the generalized discrepancy on $V$
and $A$ for that on $Y$. If a primitive divisorial valuation $v$ of
$k(Y)$ restricts nontrivially to $w$ on $k(V)$, then
\begin{equation}\label{eq:B-bundle-discrepancy}
A(v)=a(w)+|h(v)|+\operatorname{def}(v),
\qquad
h(v)=v(E_0)-v(D)\equiv-w(H)\pmod\Z,
\end{equation}
where $\operatorname{def}(v)$ is a nonnegative integer. It vanishes
precisely for the Gauss, or fibre-torus-invariant, valuations.
If $w=0$, the primitive valuations are the ordinary places of the
generic $\mathbb P^1$, and have discrepancy $1$.
Consequently klt base data give klt data on $Y$ without the two
sections; adding both sections gives lc data whose non-klt locus is
contained in the sections.
\end{lemma}
\begin{lemma}[Clearing the small denominators]\label{lem:B-denominator-clearing}
In bounded dimension, suppose $V$ is projective of Fano type and
carries effective generalized klt Calabi--Yau data with
$B\ge R(H)$. After multiplying $H$ by a bounded positive integer,
there is an ordinary klt log Fano boundary $\Theta\ge R(H)$.
For this scaled divisor the Seifert bundle is of Fano type.
The assertion does not require a bound on the Cartier index of the
nef b-divisor.
\end{lemma}
\begin{lemma}[An unbounded-index obstruction]\label{lem:B-unbounded-index}
Let $F_i$ be projective of positive bounded dimension and let $G_i$
be ample $\Q$-Cartier rational divisors such that $(F_i,R(G_i))$
is klt and
\[
-(K_{F_i}+R(G_i))\sim_{\Q}I_iG_i,
\qquad I_i>0,\quad I_i\longrightarrow\infty.
\]
Let $C_i$ be the compactified Seifert cone obtained by contracting
the zero section, and denote its infinity divisor by $D_i$.
Then $C_i$ is klt of Fano type, $D_i$ is ample, and
$-K_{C_i}\sim_{\Q}(I_i+1)D_i$. It is impossible that
\begin{equation}\label{eq:B-cone-small-height}
v_i(D_i)=o\bigl(A_{C_i}(v_i)\bigr)
\end{equation}
for every sequence of primitive divisorial valuations with
$A_{C_i}(v_i)\to0$.
\end{lemma}
\begin{lemma}[Bounding a principalization order]\label{lem:B-principal-order}
Let $F_i$ be projective of Fano type in bounded dimension, and let
$P_i\sim_{\Q}0$ be $\Q$-Cartier rational divisors. Suppose that
$F_i$ carries effective generalized klt Calabi--Yau data with
discrepancies $a_i$ and boundary at least $R(P_i)$. Suppose,
after a fixed integral scaling if necessary, that the signed phases
of $P_i$ at every sequence with $a_i(T_i)\to0$ lie in
$[-o(a_i(T_i)),O(a_i(T_i))]$. Then the least positive integer $q_i$
for which $q_iP_i$ is principal is bounded.
More generally, the same cyclic-cover argument applies whenever
the induced isotrivial fibration below has
$u=o(A)$ at its small-discrepancy divisors over zero. This
formulation permits the argument to be used after an independent
estimate of the height $u$.
\end{lemma}
Section C
------------------------------------------------------------
\begin{lemma}[Uniform cutoff models and order functions]
\label{lem:C-cutoff-claims}
Suppose $\dim W_\beta>0$. After passage to a subsequence there is a
fixed $\delta>0$ and a projective $\Q$-factorial Fano type extraction
$\widehat W\to W_\beta$ such that $\widehat W$ has ordinary
$\delta$-lc singularities and every positive coefficient of the pullback
of $D_W/\beta$ is at least $\delta$. A bounded constant $C$ can be
chosen so that $S(CD_W/\beta)$ generates $k(W_\beta)$.
There are algebraically independent functions
$\phi_1,\ldots,\phi_{\dim W_\beta}$ in this space and a b-nef
$\Q$-Cartier order function $H$ on $k(W_\beta)$, pulled back to
$k(V)$, with uniformly bounded denominator of its values on primitive
orders, such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $a(T)=0$, then $H(T)\leq CD(T)/\beta$. If $T$ is nontrivial
on $K_\beta$ and all $T(\phi_j)\geq0$, then $H(T)\leq-1$.
\item If $D(Q)=0$ and $a(Q)\leq1$, then $H(Q)\leq0$.
\item If $a(T)=0$ and \eqref{eq:C-alpha} holds at $T$, then
$H(T)\geq-C\eta D(T)/\beta$.
\end{enumerate}
In particular, if \eqref{eq:C-alpha} holds on all of $I_\beta$ with
$\eta_i\to0$, those primes are eventually horizontal over $K_\beta$;
that is, their restrictions to $K_\beta$ are trivial.
\end{lemma}
\begin{lemma}[Scaled phase bound]\label{lem:C-scaled-phase}
Assume \eqref{eq:C-alpha} with $\eta_i\to0$ whenever
$a(T)=0$ and $LD(T)\to0$, along arbitrary subsequences. Let $\ell$
be a rational principal order. Suppose that for every nonfloor prime
$Q$ on $V$ there is an integer $p_Q$ with
\begin{equation}\label{eq:C-nonfloor-phase}
|p_Q-\ell(Q)|\leq G(Q),\qquad
G\geq0,\qquad G\sim_\Q c_gLD,
\end{equation}
where $c_g\geq0$ is rational and uniformly bounded. There is a fixed
positive integer $q$ such that
\begin{equation}\label{eq:C-scaled-phase}
\operatorname{dist}(q\ell(T),\Z)=O(LD(T))
\quad\text{if }a(T)=0\text{ and }LD(T)\longrightarrow0.
\end{equation}
\end{lemma}
\begin{lemma}[Saturation and section counts]\label{lem:C-saturation-count}
Let $f_1,\ldots,f_s$ be a finite list of genuine rational functions on
$V$, whose length may vary with the sequence, and let
\begin{equation}\label{eq:C-neutral-polarization}
J(T)=\max\{0,\max_j(-T(f_j))\},\qquad
0\leq J\leq C L_0\mathbf D,\qquad1\leq L_0=O(L).
\end{equation}
Thus $J$ is an integral basepoint-free Cartier b-divisor on a high
model. Suppose \eqref{eq:C-alpha} holds uniformly with $\eta_i\to0$
on
\[
\mathcal L=\{T:a(T)=0,\ J(T)=0\}.
\]
In addition, on every subsequence on which $L_0\to\infty$, assume
that all valuations in $\mathcal L$ eventually restrict trivially to
the \emph{field generated by} $S(L_0D)$.
After passage to a subsequence, there are cutoff fields $K_*$ of fixed
transcendence degree $p$, a fixed constant $B$, and positive real
numbers $v_i$ with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For every fixed $C_1>0$, eventually $S(C_1LD)\subset K_*$.
\item Eventually $S(BLD)$ generates $K_*$. If $T\in\mathcal L$ is
nonnegative on this space, its restriction to $K_*$ is trivial.
\item There is $C'>0$, independent of the positive integer $n$, such
that for every fixed $n$, eventually
\begin{equation}\label{eq:C-uniform-counts}
\dim S(nLD)\leq C'v_i n^p,\qquad
\dim\operatorname{Span}\{g_1\cdots g_n:g_j\in S(BLD)\}
\geq v_i n^p.
\end{equation}
\end{enumerate}
The eventual index in \textup{(i)} may depend on $C_1$, and that in
\textup{(iii)} may depend on $n$; $B$ and $C'$ are fixed before $n$.
All assertions concern divisorial evaluations. Exact triviality also
holds on rational monomial subcones cut out by the indicated
conditions.
\end{lemma}
\begin{lemma}[Vanishing on a smooth orbifold]\label{lem:C-orbifold-vanishing}
Let $\mathcal X$ be a smooth proper Deligne--Mumford stack over an
algebraically closed field of characteristic zero, with trivial generic
stabilizer and projective coarse space. Let $\Theta$ be an effective
rational divisor with simple normal crossing support and coefficients
less than one. If $D$ is an integral Cartier divisor and
$D-(K_{\mathcal X}+\Theta)$ is nef and big, then
$H^q(\mathcal X,\mathcal O_{\mathcal X}(D))=0$ for $q>0$.
\end{lemma}
\begin{lemma}[Adjoint Taylor exponents]\label{lem:C-adjoint-lattice}
Let $X$ be a projective $\Q$-factorial variety of Fano type of
dimension $p>0$, and let $N$ be a nef and big integral Weil divisor.
Suppose that $H^0(X,\mathcal O_X(N))$ defines a birational map.
Choose an effective rational boundary $\Delta$ with $(X,\Delta)$ klt
and $K_X+\Delta\sim_{\Q}0$. At a general smooth point outside
the supports of $N$ and $\Delta$, fix regular parameters and take
total-degree-then-lexicographic Taylor initial exponents, using the
compatible local trivializations of the powers of $N$. Let
\[
K=\overline{\operatorname{conv}}
\left\{\frac{\nu(s)}j:
j\in\Z_{>0},\quad
0\ne s\in H^0(X,\mathcal O_X(jN))\right\}.
\]
If $t$ is a positive integer and $\alpha\in\Z_{\ge0}^p$ satisfies
$\alpha+\mathbf1\in\operatorname{int}(tK)$, then some section of
$tN$ has initial exponent $\alpha$.
\end{lemma}
\begin{lemma}[Uniform count from the jet body]\label{lem:C-jet-count}
In the setting of Lemma~\ref{lem:C-adjoint-lattice}, $K$ is bounded and contains the standard
simplex $\Sigma=\operatorname{conv}(0,e_1,\ldots,e_p)$. For every
integer $n\ge1$,
\[
h^0(X,nN)\le(n+p)^p\operatorname{vol}(K).
\]
Fix any integer $t>1+2p$. The space spanned by products of $n$ sections
of $tN$ has dimension at least $n^p\operatorname{vol}(K)$.
\end{lemma}
\begin{corollary}[Fixed generators and neutral algebraic extensions]
\label{cor:C-neutral-field}
In Lemma~\ref{lem:C-saturation-count}, let $g_{i,1},\ldots,g_{i,r_i}$
be any finite list contained in $S(C_1L_iD_i)$ for one fixed $C_1$.
For all sufficiently large $i$, the entire field
$F_i=k(g_{i,1},\ldots,g_{i,r_i})$, and its relative algebraic closure
inside $k(V_i)$, lie in $K_{*,i}$. Thus every test from part
\textup{(ii)} is exactly zero on every nonzero element of these fields
at that same index, without a degree-dependent eventual qualification.
More generally, a valuation trivial on a subfield remains trivial on
any algebraic extension of it inside its valued overfield.
\end{corollary}
Section D
------------------------------------------------------------
\begin{proposition}[Cone dictionary]\label{prop:D-cone-dictionary}
In the above setting the following statements hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The discrepancy of $(X,H)$ at $(P,y)$ is $a(P)$, and its discrepancy
at a pure weight is zero. Thus $a(P)$ is a nonnegative integer for every
primitive $P$, and $\mathbf u\in\operatorname{int}C$.
\item For rational $\mu\in\operatorname{int}C$, the ray quotient $V_\mu$ is
normal and projective, with function field $\mathcal K$. The order function
\begin{equation}\label{eq:D-polarization}
\mathbf L(\mu)(P)=\min_{y\in\Delta_P}\langle y,\mu\rangle
\end{equation}
is the b-pullback of an ample rational divisor on $V_\mu$. Consequently
\begin{equation}\label{eq:D-height-divisor}
\mathbf D(P)=\min_{y\in\Delta_P}(P,y)(h)
=\gamma(P)+\mathbf L(\mathbf u)(P)
\end{equation}
is the b-pullback of an effective ample rational divisor $D$ on
$V_{\mathbf u}$.
\item At every actual prime $P$ of $V_\mu$, a minimizer $y$ of
\eqref{eq:D-polarization} can be chosen so that $e(P,y)$ is a primitive
prime order on $X$, for some positive integer $e$. For this choice,
\begin{equation}\label{eq:D-actual-prime}
a(P)+(P,y)(h)=\frac1e,
\qquad e y\in\operatorname{Hom}(M,\Z).
\end{equation}
On each $V_\mu$, $a$ is the discrepancy function of an effective ordinary
lc Calabi--Yau pair, and $a+\mathbf D$ is that of an effective generalized
klt Calabi--Yau pair with nef part $\mathbf D$. Each $V_\mu$ is of Fano
type. At an actual prime outside the floor $\{a=0\}$, the same minimizer
satisfies
\begin{equation}\label{eq:D-off-floor}
a(P)=1,\qquad e=1,\qquad (P,y)(h)=0.
\end{equation}
\end{enumerate}
\end{proposition}
\begin{lemma}[All divisorial lc tests are Gauss]\label{lem:D-lc-tests}
Every nontrivial divisorial valuation nonnegative on $R$ and having
zero discrepancy for $(X,H)$ is Gauss. Its restriction to $\mathcal K$
is either $eP$ with $a(P)=0$, or trivial, in which case it is a nonzero
pure weight in $\sigma$. Conversely every rational Gauss extension
with $a(P)=0$, and every nonzero rational pure weight, has zero
discrepancy and a center whose closure contains the contracting vertex.
If $\Spec\breve R\to X$ is quasi-\'{e}tale and the pairs are related by
log pullback, every divisorial $h$-lc test upstairs restricts to one of
these tests, up to scaling. Conversely all the rational tests just
listed lift to divisorial $h$-lc tests upstairs whose center closures
contain the upstairs vertex. These statements suffice in particular
for testing invariant rational functions.
\end{lemma}
\begin{lemma}[Sections as regular symbols]\label{lem:D-section-symbols}
For every positive integer $q$ and every $g\in\mathcal K$,
\begin{equation}\label{eq:D-symbol-equivalence}
g\in S(q\mathbf J_m)
\quad\Longleftrightarrow\quad
g\xi^q h^{qm}\in\breve R.
\end{equation}
In particular the neutral functions of cost $b\in\N$ are precisely
\begin{equation}\label{eq:D-neutral-functions}
\{g\in\mathcal K:g h^b\in\breve R\}=S(b\mathbf D).
\end{equation}
\end{lemma}
\begin{proposition}[Sandwich regularization]\label{prop:D-sandwich}
Let $L\ge1$ vary with the sequence. Choose integers $m>m_0$ with
$m-m_0\ge cL$ for a fixed $c>0$, and suppose both
$k+m_0\mathbf u$ and $k+m\mathbf u$ are interior. Assume that whenever
primitive divisorial valuations $P$ satisfy $a(P)=0$ and
$L\mathbf D(P)\to0$, there are integers $j_P$ such that
\begin{equation}\label{eq:D-sandwich-hypothesis}
j_P\le\mathbf J_{m_0}(P)\le\mathbf J_m(P)
\le j_P+O(L\mathbf D(P)).
\end{equation}
Then there is a bounded positive integer $q$ and a nonzero
$g\in S(q\mathbf J_m)$. Equivalently, $g\xi^qh^{qm}$ is regular on
$\Spec\breve R$.
\end{proposition}
\begin{corollary}[Bounded-multiple character regularization]
\label{cor:D-regularization}
Suppose that the order condition \eqref{eq:C-alpha} on $V_{\mathbf u}$,
with error $\eta_i\to0$, holds whenever $a(P)=0$ and
$L\mathbf D(P)\to0$. Fix $C_0>0$. If $k\in M$ satisfies
\begin{equation}\label{eq:D-width}
|\langle y,k\rangle|\le C_0L\langle y,\mathbf u\rangle
\qquad(y\in\sigma),
\end{equation}
there are a bounded positive integer $q$, a nonzero invariant rational
eigenfunction $Y=fz^{qk}$, and a positive integer $b\le C_2L$ such that
$h^bY\in\breve R$. The constants are uniform in $k$. Both signs of
$k$ may be treated with the same bounded multiplier, and their costs may
be increased to a common cost $O(L)$.
The same bounds hold uniformly if $L$ is replaced by any $L'\ge L$ and
\eqref{eq:D-width} is imposed at scale $L'$.
\end{corollary}
\begin{corollary}[Count for a single character and weight]
\label{cor:D-character-count}
Assume the hypotheses of Lemma~\ref{lem:C-saturation-count} on
$V_{\mathbf u}$, and use its notation $p$ and $v_i$. For a fixed
$C_0>0$, let $W_{t,k}$ be the space of regular symbols in $\breve R$
of finite-group character $\chi^t$ and torus weight $t\mathbf u+k$.
For every fixed positive integer $n$, eventually
\begin{equation}\label{eq:D-character-bound}
\dim_k W_{t,k}\le C_3 v_i n^p
\end{equation}
simultaneously for all integers $0\le t\le C_0nL$ and all weights
satisfying
\[
|\langle y,k\rangle|\le C_0nL\langle y,\mathbf u\rangle
\qquad(y\in\sigma).
\]
The constant $C_3$ is independent of $n,t,k$.
\end{corollary}
\begin{corollary}[Characters for a tilted height]
\label{cor:D-tilted-characters}
Suppose there are boundedly many invariant rational eigenfunctions
$Y_j^\pm$ of weights $\pm k_j$, with
\[
h^{m_j}Y_j^\pm\in\breve R,\qquad
0<m_j\le A L_j,\qquad 1\le L_j\le L,
\]
where $A$ is fixed. Fix bounded real numbers $u_j$ and $\delta>0$.
Assume that, at every nonzero pure weight and at all Gauss extensions
over primitive $P$ with $a(P)=0$, $L\mathbf D(P)\to0$, the evaluations
satisfy
\begin{equation}\label{eq:D-tilt-hypothesis}
y_j=v(Y_j^+)=-v(Y_j^-),\qquad
\sum_j\frac{u_jy_j}{L_j}\le\delta v(h).
\end{equation}
It suffices to check these inequalities on rational vectors. For
fixed $M_0$ sufficiently large compared with $1/\delta$, choose integers
\[
r_j=M_0u_jL/L_j+O(1),
\]
with uniformly bounded rounding errors. Then there is a regular
nonzero symbol of finite-group character $\chi^{qm}$ and torus weight
\begin{equation}\label{eq:D-tilt-weight}
q\left(m\mathbf u-\sum_jr_jk_j\right),
\qquad 0<m\le C_6M_0\delta L,
\end{equation}
for a bounded positive integer $q$. The constant $C_6$ depends only on
the fixed bounds for the number of functions, their costs, the $u_j$,
and rounding errors; it is independent of $\delta$ and $M_0$.
The bound for $q$ may depend on the fixed choices $\delta,M_0$.
No order condition \eqref{eq:C-alpha} is assumed here.
\end{corollary}
Section E
------------------------------------------------------------
\begin{lemma}[Logarithmic jet estimate]\label{lem:E-log-jet}
Fix finitely many regular affine functions on $S$. On a fixed resolution
$P$, with $F$ reduced snc, there is a constant $C$ such that, for every
polynomial $g\ne0$ of degree at most $D$ in those functions and every
quasi-monomial $v$ over $x$,
\begin{equation}\label{eq:E-jet}
r_F(v)(g)\le v(g)\le r_F(v)(g)+C(D+1)A_{P,F}(v).
\end{equation}
The same constant works when the centre is not closed and when some
weights disappear on passing to a face of the monomial cone.
\end{lemma}
\begin{lemma}[Compactness and uniform centring]\label{lem:E-compactness}
The slice $\mathcal Q$ is a compact finite polyhedral complex of
monomial valuations on any log resolution adapted to $h$ and the
$F_j$. Evaluations of fixed functions are continuous on it.
For centred valuations with $A(v)=1$, there are constants $c,C>0$
depending on the germ such that
\begin{equation}\label{eq:E-centering}
\vol(v)\ge c,\qquad
\vol(v)\ge \frac{c}{v(\mathfrak m)}.
\end{equation}
Volume is continuous on the centred part of $\mathcal Q$.
\end{lemma}
\begin{corollary}[Simultaneous approximation]\label{cor:E-all-functions}
In the situation of \eqref{eq:E-retraction-errors}, suppose
$A_S(v)$ and $A_S(r)$ are bounded above and
$v(\mathfrak m),r(\mathfrak m)\ge c_0>0$. Then a single constant $C$
works for every such $v$ and every $f\in R$:
\begin{equation}\label{eq:E-all-functions}
r(f)\le v(f)\le(1+Ce)r(f).
\end{equation}
In particular,
$\vol(v)^{1/n}\ge(1+Ce)^{-1}\vol(r)^{1/n}$.
\end{corollary}
\begin{lemma}[Convex domination]\label{lem:E-convexity}
For $v_1,\ldots,v_k\in\mathcal Q$ and $p_s\ge0$ with
$\sum_s p_s=1$, there is $u\in\mathcal Q$ such that
\begin{equation}\label{eq:E-convexity}
u(f)\ge\sum_s p_s v_s(f)\qquad(f\in R).
\end{equation}
If a positive-weight term is centred, then $u$ is centred. On the cone
of common lc rays, the same statement holds with
$A(u)=\sum_s p_sA(v_s)$, without requiring $A(v_s)=1$.
For an allowed height,
$b(u)\le\sum_s p_sb(v_s)$.
\end{lemma}
\begin{corollary}[Opposite ratios]\label{cor:E-opposite-ratios}
Suppose rational functions $q,q'$ have regular numerators and
power-of-$h$ denominators, and $v(q')=-v(q)$ on $\mathcal Q$.
The interpolant in Lemma~\ref{lem:E-convexity} has
$u(q)=\sum_s p_sv_s(q)$ and $u(q')=\sum_s p_sv_s(q')$.
The assertion holds simultaneously for any number of such pairs.
\end{corollary}
\begin{lemma}[Volume interpolation and a strict gap]
\label{lem:E-volume-interpolation}
Let $w$ be centred, let $v$ be over $x$, and suppose
$u\ge tw+(1-t)v$ on $R$ for $0<t<1$. For $f\ne0$ with $w(f)>0$,
\begin{equation}\label{eq:E-volume-interpolation}
\vol(u)\le t^{-n}
\frac{tw(f)}{tw(f)+(1-t)v(f)}\vol(w).
\end{equation}
If $v$ is also centred, $t=1/2$, and
$v(f)\ge(1+\gamma)w(f)>0$ for $0<\gamma\le1$, then
\begin{equation}\label{eq:E-volume-gap}
\vol(w)+\vol(v)-2\vol(u)\ge c(n,\gamma)\vol(w),
\end{equation}
where $c(n,\gamma)>0$ depends only on $n,\gamma$.
Consequently valuations whose volumes are within relative $o(1)$ of
the minimum on the same $A=1$ common slice are multiplicatively
$1+o(1)$ equal on \emph{all} regular functions. The conclusion is
uniform when the dimension is bounded. A sufficiently small fixed
relative error similarly gives any prescribed fixed multiplicative
closeness to one.
\end{lemma}
\begin{lemma}[A boundary encoding the common slice]\label{lem:E-slice-boundary}
Choose positive rational numbers $\theta_j$ with
$\sum_j\theta_j<1$. For sufficiently general scalars $a_j$, put
$E_j=\divisor(h^{m_j}+a_jF_j)$ and
\begin{equation}\label{eq:E-slice-boundary}
\Psi=(1-\sum_j\theta_j)H+
\sum_j\frac{\theta_j}{m_j}E_j.
\end{equation}
Then $\Psi$ is effective and lc, and its nonzero zero-discrepancy
rays over the germ are exactly the common lc rays. On every such ray,
\begin{equation}\label{eq:E-pencil-order}
v(h^{m_j}+a_jF_j)=m_jv(h).
\end{equation}
In particular, the coefficients $\theta_j$ may be chosen arbitrarily
small while retaining these conclusions.
\end{lemma}
\begin{theorem}[Exact common-slice minimizer]\label{thm:E-exact-minimizer}
For every allowed height $b$, the homogeneous functional
$b(v)\vol(v)^{1/n}$ on centred common lc rays has a unique minimizing
ray. Its $A=1$ representative depends continuously on the coefficients
of $b$ while the height stays strictly positive on the fixed slice.
It is fixed by every automorphism preserving the slice and the height.
If the coefficients of $b$ are rational, its minimizer $w$ is the
normalized-volume minimizer of an effective rational klt pair on
$x\in S$. More precisely, one can choose an effective rational lc
Cartier boundary $\Psi$ with exactly the common lc rays as its
zero-discrepancy locus, and one positive rational number $\lambda$,
fixed independently of the competing valuation, such that
\begin{equation}\label{eq:E-klt-perturbation}
\Psi_\lambda=\Psi-\lambda b_0H
+\lambda\sum_\ell b_\ell\divisor(G_\ell)
\end{equation}
is effective and klt, and $w$ minimizes
$\widehat{\vol}_{S,\Psi_\lambda}$ among all centred valuations.
On the common lc rays its discrepancy is $\lambda b$.
For the same rational height, the algebra $\operatorname{gr}_wR$
is finitely generated and normal.
Its spectrum $S^0$ is a klt $\Q$-Gorenstein cone.
Write $h^0,F_j^0,G_\ell^0$ for initial forms, $w^0$ for the grading
valuation, and $\mathcal Q^0,b^0$ for the analogous central slice
and height. Then $(S^0,\divisor(h^0))$ is lc,
$A_{S^0}(w^0)=w^0(h^0)=1$, $w^0\in\mathcal Q^0$, and
$b^0$ is strictly positive on $\mathcal Q^0$.
The valuation $w^0$ minimizes $b^0\vol^{1/n}$ on its centred part.
\end{theorem}
\begin{corollary}[Near-minimizers for a height]\label{cor:E-height-near-minimizers}
Fix an allowed height $b$ and normalize its common rays by $b=1$.
If $w_i,v_i$ have functional values within relative $o(1)$ of the
minimum, then $w_i=(1+o(1))v_i$ on all regular functions, with an error
independent of the function. The same conclusion, after $A=1$
normalization, holds for minimizers of heights whose coefficients
converge to those of $b$.
\end{corollary}
\begin{corollary}[Domination by the minimizer]\label{cor:E-domination}
For the minimizing ray $w$ and every $v\in\mathcal Q$,
\begin{equation}\label{eq:E-domination}
\frac{v(f)}{b(v)}\le 2^n\frac{w(f)}{b(w)}
\qquad(f\in R).
\end{equation}
For rational heights the same statement holds on the central slice
with all symbols carrying a superscript $0$.
\end{corollary}
\begin{lemma}[One-sided height tilt]\label{lem:E-one-sided-tilt}
Let $G\in R\setminus\{0\}$, $m\in\Z_{>0}$, and
$g(v)=v(G)/m-v(h)$. Suppose $g(v)/v(h)\ge2c$ on some common lc ray,
where $c>0$ is rational. Set
\[
M=\max_{\mathcal Q}g\ge2c,\qquad
\mu_*=(M-c)^{-1},\qquad
b_\mu(v)=v(h)+\mu(cv(h)-g(v)).
\]
For $0\le\mu<\mu_*$ the height is positive. There is a rational
$\mu$ in this interval whose $A=1$ minimizing valuation $w$ satisfies
\[
g(w)>0,\qquad b_\mu(w)\ge1.
\]
Moreover $b_\mu\le2+1/c$ on $\mathcal Q$, and the corresponding
central height obeys the same bound on $\mathcal Q^0$.
\end{lemma}
Section F
------------------------------------------------------------
\begin{lemma}[Weighted presentations]\label{lem:F-presentation}
Choose $x_1,\ldots,x_N\in\mathfrak m_x$ whose symbols generate
$\breve R$, and put $\alpha_i=w(x_i)>0$. Any prescribed finite
list of functions of positive $w$-value can be included in this list.
Then
\begin{equation}\label{eq:F-presentation}
\widehat{\mathcal O}_{S,x}=\C[[X_1,\ldots,X_N]]/I,
\qquad
\breve R=\C[X_1,\ldots,X_N]/\operatorname{in}_{\alpha}I,
\end{equation}
where initial forms have least weight. The valuation ideals are
exactly the images of the ideals generated by monomials of weight
at least the prescribed threshold. The same description holds
before completion by intersection.
For every sufficiently nearby linear functional
$L:\Lambda_w\otimes\R\longrightarrow\R$, positive on the listed
weights, the vector $\alpha_L=(L(\alpha_i))$ gives the same initial
ideal, with its grading composed with $L$. Its filtration defines
a centred quasi-monomial valuation $w_L$. The functionals can be
chosen rational, and the resulting valuations can be computed
monomially on a fixed smooth chart for $w$.
\end{lemma}
\begin{proposition}[Canonical forms under degeneration]
\label{prop:F-canonical-degeneration}
Under the hypotheses preceding Lemma~\ref{lem:F-presentation},
$S^0$ has a divisor-free reflexive canonical generator $\omega^0$.
It has the $G$-character of $\omega$. It is homogeneous for
$\mathbb T_w$, and its weight $\kappa$ satisfies
\begin{equation}\label{eq:F-canonical-weight}
\langle w,\kappa\rangle=A_S(w).
\end{equation}
If an additional torus acts on the original germ, preserves $w$,
and makes $\omega$ homogeneous, its character is preserved
separately. None of these assertions assumes $A_S(w)=w(h)$.
Moreover the action of $G$ on $S^0$ is quasi-\'{e}tale.
\end{proposition}
\begin{corollary}[Lc equality and principal boundaries]
\label{cor:F-lc-degeneration}
If $w$ is $h$-lc, then $(S^0,\divisor(\bar h))$ is lc and
$\omega^0$ has the same fine torus weight and $G$-character as
$\bar h$. Consequently $\omega^0/\bar h$ is invariant, and the
quotient cone has the log-form data of Section~\ref{sec:D}.
If $\Delta$ is any effective rational sum of principal divisors
near $x$, with $(S,\Delta)$ lc and
$A_S(w)=w(\Delta)$, its initial boundary $\Delta^0$ is lc on
$S^0$. All these assertions also hold with specified additional
equivariant gradings.
\end{corollary}
\begin{proposition}[Sign transfer]\label{prop:F-bridge}
Suppose $w$ is common $h$-lc and $w(f_j)=0$ for every $j$.
Let $g=U/h^b$, where $0\ne U\in\mathcal O_{S,x}$,
$b\in\N_{>0}$, and $w(g)=0$. Let $\mathcal Q$ be the actual
common slice and $\mathcal Q^0$ the common slice defined by
$\bar h$ and the $\bar f_j$ on $S^0$. Then
\begin{equation}\label{eq:F-bridge}
q(g)\le0\quad(q\in\mathcal Q)
\quad\Longleftrightarrow\quad
q^0(\bar g)\le0\quad(q^0\in\mathcal Q^0).
\end{equation}
One may add any finite list of tests of value zero at $w$.
There is no bound on the degree or the cost of $g$ in this statement.
\end{proposition}
\begin{corollary}[Exact tests and pure weights]
\label{cor:F-exact-tests}
Suppose $w$ is common $h$-lc. If every common actual valuation
satisfies $q(f_j)=0$, then every common cone valuation satisfies
$q^0(\bar f_j)=0$.
The symbols $\bar f_j$ are invariant under $G$ and $\mathbb T_w$.
Every nonzero pure weight valuation of the cone is $h$-lc and,
after normalization by $\bar h=1$, is common. Pure weights on
the boundary of the dual weight cone are allowed under the
closed-centre convention.
\end{corollary}
\begin{lemma}[Compression]\label{lem:F-compression}
Suppose $w$ is as above, is $h$-lc, and satisfies $w(f_j)=0$ for
every $j$. Let $v$ be a centred
$G\times\mathbb T_w$-invariant, $\bar h$-lc quasi-monomial
valuation on $S^0$, with normal finitely generated klt graded
algebra $R^{00}$. Suppose also $v(\bar f_j)=0$.
The algebra $R^{00}$ has its joint grading by the old torus and
the $v$-grading. For arbitrarily small positive $\delta$ outside
a countable exceptional set, it is the associated graded of an
actual centred quasi-monomial valuation $w_\delta$ on $S$, whose
grading evaluates a joint degree $(a,b)$ as $a+\delta b$.
The valuation $w_\delta$ is $h$-lc, is exact on all $f_j$, and,
after $h=1$ normalization, tends multiplicatively to $w$
uniformly on all nonzero regular functions. If $v$ is nonzero
on an old degree-zero rational function, the rational rank of
$w_\delta$ is strictly larger than that of $w$.
\end{lemma}
\begin{proposition}[Rank refinement]\label{prop:F-refinement}
Consider a sequence of the preceding data in dimensions at most
$n_0$. Suppose that the common actual slices satisfy
$q(f_j)=0$ identically, that $w$ is a centred point of the slice
with the stipulated normal finitely generated klt graded algebra,
and that the cones satisfy the following balance condition with
one constant $C$ along the sequence:
\begin{equation}\label{eq:F-balance}
q^0(s)\le Cw^0(s)
\quad\begin{gathered}
0\ne s\in\breve R\text{ fine-homogeneous},\\[-2pt]
q^0\text{ common},\quad q^0(\bar h)=w(h)=1.
\end{gathered}
\end{equation}
After passage to a subsequence, the $w$ can be replaced by $w'$
with all these properties, with a new fixed balance constant,
and with
\begin{equation}\label{eq:F-refinement-comparison}
w'(f)=(1+o(1))w(f)
\qquad(0\ne f\in\mathcal O_{S,x}),
\end{equation}
uniformly in $f$ in the normalization $w'(h)=w(h)=1$.
On the angular model $V_{\mathbf u}$ of the new cone, the
notation of Section~\ref{sec:D} then satisfies
\begin{equation}\label{eq:F-angular-order}
T(\Gamma)\le(1+\eta_i)D(T),\qquad \eta_i\longrightarrow0,
\end{equation}
for every effective $\Q$-divisor $\Gamma\sim_\Q D$ and every
primitive divisorial $T$ such that
$a(T)=0$ and $T(\bar f_j)\ge0$ for all $j$.
This is condition \textup{(C-$\alpha$)} of Section~\ref{sec:C}.
The initial symbols here refer to $w'$.
The balance hypothesis holds when $w$ is the exact minimizer
on the common slice for the height $q(h)$.
\end{proposition}
Section G
------------------------------------------------------------
\begin{theorem}[Pinning obstruction]\label{thm:G-pinning}
Fix an integer $n>0$ and $\epsilon>0$. There is no sequence of the following
data satisfying all the conditions below. For each index $i$, let
$x_i\in S_i$ be an $n$-dimensional complex klt germ with an action of a
finite cyclic group $\Gamma_i$ fixing $x_i$, such that the quotient map is
quasi-\'etale and $S_i/\Gamma_i$ is $\epsilon$-lc. Suppose $K_{S_i}$ has a
nowhere-vanishing local eigen-generator $\omega_i$, and let $h_i$ be a
nonzero regular eigenfunction with the same character $\chi_i$ as
$\omega_i$. The character need not be faithful. Assume that
$(S_i,\divisor(h_i))$ is lc and admits a centred quasi-monomial valuation
$T_i$ with
\[
A_{S_i}(T_i)=T_i(h_i)=1.
\]
The excluded additional condition is: for every fixed integer $m>0$,
there is an index $i(m)$ such that, for all $i\ge i(m)$ and every regular
semi-invariant $s$ of character $\chi_i^m$,
\begin{equation}\label{eq:G-input}
T_i(s)\ge m.
\end{equation}
\end{theorem}
\begin{lemma}[Non-cancellation in every degree]\label{lem:G-noncancellation}
At the preceding stage, consider any finite linear combination of
monomials in the accumulated and newly added $Y_j^+$ and the current
$G_a$, with nonnegative exponents. Suppose its least-$w$ group has a
nonzero sum of initial symbols. If all its terms have the same value at
an actual point of the pre-pinning slice, their sum has that value.
This assertion holds eventually simultaneously in all polynomial degrees.
Moreover, at a later working valuation on this slice where these terms
are tied, the same non-cancellation assertion transfers to its cone at
every common test where the terms remain tied and the pair values are
opposite.
\end{lemma}
\begin{lemma}[Uniform product growth]\label{lem:G-product-growth}
Let $V$ be a finite-dimensional subspace of a function field over an
algebraically closed field, containing $1$. Put $g=\dim V$ and
$a=\trdeg k(V)\le n$. If $V^{(N)}$ is the span of $N$-fold products of
members of $V$, there is $c_n>0$, depending only on $n$, such that
\[
\dim V^{(N)}\ge c_n gN^a
\qquad(N\ge1).
\]
\end{lemma}
\begin{proposition}[High-scale pin]\label{prop:G-high-pin}
Consider a high stage with pre-pinning slice, working valuation $w$,
baseline height $\mathfrak B$, scale $L$, and $s$ selected values as above.
After passage to a subsequence, finitely many invariant tests of cost
$O(L)$ can be added with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The resulting slice contains a centred valuation. The added tests
are exact on that slice, all old tests and proportionalities remain in
force, and the accumulated positive symbols remain algebraically
independent at the exact minimizer and working valuation in
\textup{(iii)} below.
\item The selected values either vanish or lie on a single rational
line. In the line case their proportions have the bounds
\eqref{eq:G-proportions}, with $L_-=L$. An old anchor remains the anchor;
if a new anchor is chosen, its scale is $L_1=L$.
\item The baseline height is retained unless an anchor is first chosen,
in which case it may change to the rational height
\eqref{eq:G-baseline}. It satisfies \eqref{eq:G-baseline-bounds} on the
new actual slice. At its exact minimizer $v'$, and at an arbitrarily close
rank refinement $w'$, the anchor, when present, satisfies
$|v'(Y_0^+)|,|w'(Y_0^+)|\ge cL_1$. Both valuations dominate a fixed positive
multiple of $w$ on every regular function. In particular,
\[
w'\ge c w.
\]
\end{enumerate}
Consequently \ref{it:G-tests}--\ref{it:G-working} hold with $L_-=L$.
The next-scale assertions \ref{it:G-lattice}--\ref{it:G-field} are not
assumed in constructing the pin or its rank refinement.
\end{proposition}
\begin{claim}\label{clm:G-new-row}
After a subsequence there is an additional row of the form
\eqref{eq:G-row}, exact at $v$, whose transverse part is independent of
the previously imposed transverse rows.
\end{claim}
\begin{lemma}\label{lem:G-pin-consequences}
The working valuation $w'$ furnished by Proposition~\ref{prop:G-high-pin}
satisfies \ref{it:G-lattice}--\ref{it:G-field} at the next scale $L_-=L$.
\end{lemma}
Section H
------------------------------------------------------------
\begin{proposition}[Factorial saturation]\label{prop:H-saturation}
For each example there is a primitive divisorial order $w$ over $X$ whose
image centre on $Z$ contains the specified point and for which
\begin{equation}\label{eq:H-saturation}
w(\Delta_s)\ge k A_X(w)\quad
(k>0,\ k\mid M_i),\qquad
w(\Delta_{s_b})=bA_X(w).
\end{equation}
After replacing the example by a geometric generic slice, one may assume
that the image centre is a closed point $z$, that
$1\le\dim X=\dim Z\le d$, and that all the inequalities hold for the
complete local systems at $z$.
\end{proposition}
\begin{lemma}[The isomorphic case]\label{lem:H-canonical-cover}
Suppose $f$ is an isomorphism near $z$. There is a normal integral
cyclic cover $S_0\to X$, with a unique point $o$ over $z$, a
quasi-\'{e}tale cyclic action fixing $o$, and a divisor-free canonical
generator of character one. It has a regular eigenfunction
$f_b=s_bt^b$ of character $b$ and a divisorial valuation $w_0$ centred
at $o$ such that, with $A_0=A_{S_0}(w_0)>0$,
\begin{equation}\label{eq:H-upstairs-saturation}
(S_0,\divisor(f_b)/b)\text{ is lc},\qquad
w_0(f_b)=bA_0,\qquad
w_0(a)\ge kA_0
\end{equation}
for every regular eigenfunction $a$ of character $k$ when
$k>0$ and $k\mid M_i$. The variety $S_0$ is klt.
\end{lemma}
\begin{lemma}[The signed anti-canonical torsor]\label{lem:H-signed-torsor}
Suppose $f$ is not an isomorphism near $z$. The signed algebra
\begin{equation}\label{eq:H-signed-algebra}
R=\bigoplus_{k\in\Z}
H^0(X,\mathcal O_X(-kK_X))t^k,\qquad S_0=\Spec R,
\end{equation}
over the affine base is a finitely generated normal domain of dimension
$\dim X+1$. Write $R_k=H^0(X,\mathcal O_X(-kK_X))t^k$ for its
homogeneous part of degree $k$. Its \emph{positive chamber} is the open
\[
S_0^+=\bigcup_{k>0}\ \bigcup_{0\ne a\in R_k}D_{S_0}(a),
\]
where $D_{S_0}(a)$ is the principal open on which $a$ does not vanish.
This chamber has complement of codimension at least two. The form
\begin{equation}\label{eq:H-torsor-form}
\Omega_0=t\omega_X\wedge d\log t
\end{equation}
is a divisor-free canonical generator of grading weight one, and every
finite scalar grading subgroup acts quasi-\'{e}talely.
The variety $S_0$ is klt, and
$(S_0,\divisor(s_bt^b)/b)$ is lc. There is a grading-fixed closed point
$o$ over $z$ and a rational Gauss valuation $w_0$ centred there which
satisfies \eqref{eq:H-upstairs-saturation} for every actual positive
degree $k\mid M_i$. Moreover there are numbers $\alpha>0$ and
$F_w>\alpha$, depending on the example, for which every nonzero
homogeneous section satisfies
\begin{equation}\label{eq:H-two-slopes}
w_0(s t^k)\ge
\begin{cases}
k\alpha,&k>0,\\
|k|(F_w-\alpha),&k<0.
\end{cases}
\end{equation}
\end{lemma}
\begin{lemma}[Padding and the actual quotient discrepancy]\label{lem:H-padding}
In either construction, set
\[
S=S_0\times\mathbf A^{b-1},
\qquad (o,0)\in S,
\]
and give every added coordinate $u_j$ character one. Then $S$ is klt,
the finite quotient $Y=S/\mu_r$ is $\epsilon$-lc, and the canonical
generator on $S$ has character $b$. There is a regular function $h$ of
character $b$ such that $(S,\divisor(h))$ is lc and a centred
quasi-monomial valuation $T_0$ such that
\begin{equation}\label{eq:H-padding-equality}
A_S(T_0)=T_0(h)=bA_0.
\end{equation}
\end{lemma}
\begin{proposition}[The pinning input]\label{prop:H-g-input}
The failure sequence produces, after a subsequence, a fixed-dimensional
sequence satisfying every hypothesis of Theorem~\ref{thm:G-pinning}.
\end{proposition}
\begin{lemma}[Point ideals and change of field]\label{lem:H-field-reduction}
It suffices to prove Theorem~\ref{thm:main} over $\C$. More precisely, a
uniform positive lower bound for the log canonical threshold of
$\mathfrak m_z\mathcal O_X$ implies the Cartier-section assertion after
replacing the bound by a smaller number less than one. Both the hypotheses
and this ideal threshold are preserved by algebraically closed field
extensions of characteristic zero.
\end{lemma}
Section Introduction
------------------------------------------------------------------------
\begin{maintheorem}[Uniform Cartier sections]\label{thm:main}
For every integer $d>0$ and every real number $\epsilon>0$, there is a real number $\tau=\tau(d,\epsilon)>0$ with the following property. Let $(X,B)$ be an $\epsilon$-lc rational pair of dimension $d$, and let $f:X\to Z$ be a Fano type contraction of normal integral quasi-projective varieties with $\dim Z>0$. Suppose that $-(K_X+B)$ is nef over $Z$. For every closed point $z\in Z$, there are an open neighbourhood $U$ of $z$ and a nonzero effective Cartier divisor $D$ on $U$, with $z\in\Supp D$, such that
\[
\bigl(f^{-1}(U),B|_{f^{-1}(U)}+\tau f^*D\bigr)
\quad\text{is log canonical.}
\]
\end{maintheorem}
\begin{maintheorem}[Birational klt complements]\label{thm:birational}
For each $d>0$ and $\epsilon>0$, there is a positive integer $n=n(d,\epsilon)$ such that the following holds. Let $X$ be an $\epsilon$-lc rationally Gorenstein variety of dimension $d$, and let $f:X\to Z$ be a birational Fano type contraction with $-K_X$ nef over $Z$. For every closed $z\in Z$, there is an effective rational divisor $\Delta$ over a neighbourhood of $z$ such that $(X,\Delta)$ is klt there and $n(K_X+\Delta)$ is Cartier and linearly trivial there, after shrinking the neighbourhood.
\end{maintheorem}
\begin{maincorollary}[Real boundaries]\label{cor:real-boundaries}
Over $\C$, the conclusion of Theorem~\ref{thm:main} holds for effective real boundaries $B$ with $K_X+B$ real Cartier, with the same uniform dependence on $d$ and $\epsilon$.
\end{maincorollary}
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