diff 0a159df..eab029b fano_cartier_sections_v2.txt
diff --git a/fano_cartier_sections_v2.txt b/fano_cartier_sections_v2.txt
index b301ae2..9229b4a 100644
--- a/fano_cartier_sections_v2.txt
+++ b/fano_cartier_sections_v2.txt
@@ -1,1900 +1,47 @@
-================================================================================
- UNIFORM CARTIER SECTIONS FOR FANO TYPE CONTRACTIONS
- A plain-English companion version, written for curious high school readers
-================================================================================
+Fix an algebraically closed field of characteristic zero throughout. A contraction is a projective surjective morphism f: X -> Z of normal integral quasiprojective varieties with connected fibres and f_* O_X = O_Z. The contraction is of Fano type when there exists an effective rational boundary C on X such that the pair (X, C) is klt and minus (K_X + C) is ample over Z. Projective space, smooth quadrics and the del Pezzo surfaces are Fano type; the boundary C is an auxiliary device recording a controlled slack in the negativity of K_X, and it need not agree with the boundary appearing in any theorem below. Discrepancy conditions are formulated on the total space and do not see the base.
-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.
+For a pair (X, B) with B an effective rational divisor and K_X + B rational Cartier, and a prime divisor E over X, the log discrepancy is
-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.
+ a(E; X, B) = 1 + coeff_E(K_Y - pi^*(K_X + B)),
-The document has these parts:
+where pi: Y -> X is a smooth birational model on which E is a divisor. The pair is klt when every a(E; X, B) is strictly positive, log canonical when every a(E; X, B) is nonnegative, and epsilon-lc for a real epsilon > 0 when every a(E; X, B) is at least epsilon. Log discrepancies extend linearly to positive rational multiples of divisorial valuations; a uniformity assertion quantified over divisorial valuations is always checked on primitive orders, because any divisorial order is a positive rational multiple of a primitive one.
- 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.
+For an effective divisor D on X and a closed point x of X, the log canonical threshold lct_x(X, B, D) is the supremum of real t >= 0 such that (X, B + tD) remains log canonical near x. On a smooth curve with D the divisor m times the point x, the threshold equals 1/m, so an arbitrary divisor through x gives no uniform positive lower bound for the threshold. The obstruction carried by this example is the only obstruction in the Fano type setting, in the following precise sense.
-"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.
+Theorem (birational klt complements). For each integer d > 0 and each real epsilon > 0 there is a positive integer n = n(d, epsilon) with the following property. Let X be an epsilon-lc rationally Gorenstein variety of dimension d, and let f: X -> Z be a birational Fano type contraction with minus K_X nef over Z. For every closed point z of Z there is an effective rational divisor Delta on X such that, over some open neighbourhood of z, the pair (X, Delta) is klt and n(K_X + Delta) is Cartier and linearly trivial.
-================================================================================
-PART I. WHAT THIS PAPER IS ABOUT
-================================================================================
+Theorem (uniform Cartier sections). For each integer d > 0 and each 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 quasiprojective varieties with dim Z > 0. Suppose minus (K_X + B) is nef over Z. For every closed point z of Z there are an open neighbourhood U of z and a nonzero effective Cartier divisor D on U, containing z in its support, such that the pair (f^{-1}(U), B restricted to f^{-1}(U) plus tau times the pullback of D) is log canonical.
-1. THE QUESTION IN ONE SENTENCE
--------------------------------
-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?
+Corollary (real boundaries). Over the complex numbers the conclusion of the uniform Cartier sections theorem holds for effective real boundaries B with K_X + B real Cartier, with the same uniform dependence on d and epsilon.
-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.
+The conjecture behind these statements asks for bounded complements in the sense of Shokurov's 2004 formulation. The log canonical version was proved by Birkar in 2019: for every dimension bound and every fixed rational coefficient set there is a uniform index I and an effective divisor Lambda at least B with (X, Lambda) log canonical and I(K_X + Lambda) linearly trivial. The klt version is strictly stronger, because a klt complement has strictly positive discrepancies rather than merely nonnegative ones; bigness of an index is compatible with discrepancy having collapsed to zero along some direction.
-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.
+Bingyi Chen has shown that the klt version of bounded complements is equivalent to uniform Cartier sections for Fano type contractions of bounded base dimension. That equivalence carries the theorem above for pairs into the birational statement, using only base dimension d as the input. No Cartier-index hypothesis on the nef part of any boundary enters the equivalence. A real boundary passes through the same equivalence in Chen's real-pair form; the boundary is not approximated, and real-Cartier integral Weil divisors are already rationally Cartier on the models where the argument is applied.
-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.
+./shokurov-bounded-complements-oberwolfach-2004.pdf
+./birkar-boundedness-fano-annals-2021.pdf
+./chen-boundedness-klt-complements-2403.01154v4.pdf
-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.
+The remainder is the proof of the birational klt complement theorem. It proceeds by contradiction. Suppose no n(d, epsilon) exists. Then there is a sequence of birational Fano type contractions X_i -> Z_i of dimensions d_i bounded by d, with closed points z_i of Z_i, such that the least positive integer n_i for a klt complement over a neighbourhood of z_i tends to infinity along the sequence.
-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.
+Birkar's bounded log canonical complements theorem reads as follows.
-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.
+Theorem (bounded lc complements). For each integer d >= 1, each set R of rational numbers in [0, 1] satisfying the descending chain condition, and each real epsilon > 0, there is a positive integer I = I(d, R, epsilon) such that the following holds. Let (X, B) be a projective log pair of dimension d, with coefficients of B in R, klt and epsilon-lc and of Fano type. Then there is an effective rational divisor Lambda on X with Lambda >= B, for which (X, Lambda) is log canonical and I(K_X + Lambda) is linearly trivial.
-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".
+./birkar-boundedness-fano-annals-2021.pdf
-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.
+Applied to the failing sequence of the preceding paragraph, this theorem supplies an lc complement Lambda_i on X_i of uniformly bounded index I(d), with coefficients in a fixed finite set, and with a uniform positive discrepancy gap away from the base point z_i. The support of Lambda_i is the correct support, and every coefficient lies in a bounded set. The only defect is that some coefficients may fail the strict positivity required for klt.
-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 birational statement is proved by an induction whose first half reduces each failing example to a local problem and whose second half excludes the local problem outright.
-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."
+Reduction to the local model. The first normal klt cover of the pair (X_i, Lambda_i), ramified along the support of Lambda_i, has degree bounded by the index I(d) and therefore by a constant depending only on d. Its generic fibre over z_i is a normal complex klt germ x_0 in S_0 of dimension d, and the covering group restricts to a finite group Gamma_0 acting on S_0 by a quasi-etale map with epsilon-lc quotient. A bounded-order further cover replaces Gamma_0 by a strictly cyclic group Gamma and produces a germ S_i of bounded structure. The character of the canonical sheaf is unchanged by the covering. There is a nowhere-vanishing local canonical eigen-generator omega_i of some character chi_i, and a nonzero regular eigenfunction h_i of the same character. The pair (S_i, div(h_i)) is lc, and there is a centred quasi-monomial valuation T_i with A_{S_i}(T_i) = T_i(h_i) = 1. The two cases of birational contractions (nowhere-exceptional and general) both fall under this construction: the first uses the canonical cover, and the second uses the signed anticanonical Z-graded torsor, the total space being the affine cone on the anticanonical section ring with the positive chamber of the torsor realising the germ as a Zariski open subset of codimension at least two.
-5. WHAT A LOG CANONICAL THRESHOLD IS
-------------------------------------
-Given a pair (X, B) and an effective divisor D on X, define
+Failure of a klt n-complement on Z_i pulls back to a quantitative saturation condition on S_i. For each fixed positive integer m there is an index i(m) such that, for every i >= i(m) and every regular semi-invariant s of character chi_i^m,
- lct(X, B, D) = sup{ t > 0 : (X, B + t D) is log canonical }.
+ T_i(s) >= m.
-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 valuation is constructed from the two slopes on the signed torsor: the positive slope is a rational Gauss weight of the anti-canonical grading, the negative slope is the order of the diagonal-vanishing negative section, and the canonical discrepancy evaluates to zero on the first and to the character h on the second, so the two slopes match.
-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.
+Saturation obstruction (pinning). Fix an integer n > 0 and a real epsilon > 0. There is no sequence, in dimension n, of the following shape. For each index i there is a normal complex klt germ x_i in S_i of dimension n, a finite cyclic group Gamma_i acting on S_i by a quasi-etale map that fixes x_i and has epsilon-lc quotient S_i / Gamma_i, a nowhere-vanishing local canonical eigen-generator omega_i of some character chi_i, a nonzero regular eigenfunction h_i of the same character chi_i, an lc pair (S_i, div h_i), a centred quasi-monomial valuation T_i with A_{S_i}(T_i) = T_i(h_i) = 1, and additionally the saturation condition: for every fixed integer m > 0 there is an index i(m) such that T_i(s) >= m for every regular semi-invariant s of character chi_i^m and every i >= i(m).
-================================================================================
-PART II. EXACT STATEMENTS OF THE MAIN THEOREMS
-================================================================================
-
-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.
-
-================================================================================
-PART III. THE SHAPE OF THE PROOF (NO DETAILS)
-================================================================================
-
-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.
-
-================================================================================
-PART IV. GLOSSARY
-================================================================================
-
-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.
-
-================================================================================
-PART V. THE ABSTRACT, AND THE CONVENTIONS
-================================================================================
-
- 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.
-
-================================================================================
-PART VI. WHAT A HIGH SCHOOL READER CAN ACTUALLY DO
-================================================================================
-
-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.
-
-================================================================================
-PART VII. WHY THIS MATTERS
-================================================================================
-
-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.
-
-================================================================================
-PART VIII. FULL LEMMA INDEX OF THE ORIGINAL
-================================================================================
-
-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}
-
-
-================================================================================
-END OF fano_cartier_sections_v2.txt
-================================================================================
+The remainder of this text is the proof of the saturation obstruction; once it is proved, the failing sequence is impossible, and the birational klt complement theorem follows directly, and the uniform Cartier sections theorem follows from the birational complement theorem by Chen's transfer.