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 of the main theorems. Discrepancy conditions are formulated on the total space and do not see the base.
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
a(E; X, B) = 1 + coeff_E(K_Y - pi^*(K_X + B)),
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.
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.
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.
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.
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 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 J and an effective divisor Lambda at least B with (X, Lambda) log canonical and J(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.
Shokurov put the conjecture this way at Oberwolfach in 2004:
Let d > 0 be a positive integral number and epsilon >= 0 be a real number. Then there exists a finite set N_d(epsilon) of positive integral numbers such that, for any contraction of normal algebraic varieties X -> Z and some complementary index n in N_d(epsilon), if D is a general element of the linear system of -nK, then the log pair (X, D/n) has only epsilon-log canonical singularities in the mobile sense and locally over Z, under the following conditions: (a) dim X <= d; (b) X has only epsilon-log canonical singularities; and (c) -K is nef and big over Z.
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 main theorem 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.
pdf/shokurov-bounded-complements-oberwolfach-2004.pdf
pdf/birkar-boundedness-fano-annals-2021.pdf
pdf/chen-boundedness-klt-complements-2403.01154v4.pdf
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.
Birkar's bounded log canonical complements theorem reads as follows.
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 J = J(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 J(K_X + Lambda) is linearly trivial.
pdf/birkar-boundedness-fano-annals-2021.pdf
Applied to the failing sequence just described, this theorem supplies an lc complement Lambda_i on X_i of uniformly bounded index J(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 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.
Reduction to the local model. The normal klt cover of the pair (X_i, Lambda_i), ramified along the support of Lambda_i, has degree bounded by the index J(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.
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,
T_i(s) >= m.
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 equals one on the first and the h-value equals one on the second, so the two slopes match.
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).
It remains to prove 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.
Generalized pairs. A generalized pair consists of three pieces X, B, M where X is a projective normal variety of characteristic zero, B is an effective rational Weil divisor on X, and M is a Q-Cartier b-divisor on X that is nef on every sufficiently high birational model. The trace of M on X is written M_X. The generalized log canonical divisor K_X + B + M_X is required to be Q-Cartier. The generalized discrepancy of a prime divisor E over X is the ordinary discrepancy of the pair obtained after replacing M by its determination on a sufficiently high model; the substitution is compatible with pullback along birational morphisms. Generalized discrepancies extend linearly in positive rational multiples of divisorial valuations, and uniformity statements are checked on primitive orders only.
A generalized Calabi-Yau pair over a base S is data X, B, M equipped with a contraction X -> S such that K_X + B + M_X is Q-linearly trivial over S. The generalized pair is klt (log canonical, epsilon-lc) when the generalized discrepancies satisfy the corresponding strict/nonnegative/epsilon bounds. A generalized pair is of Fano type over S when there exists an effective rational boundary C on X, klt with C together with B + M_X, and with minus (K_X + B + C + M_X) ample over S; this auxiliary boundary replaces the ordinary Fano condition when the nef part is present.
Discriminant and moduli. Let X -> S be a generalized Calabi-Yau over S, with base S a separated normal variety and X -> S a morphism whose general fibre is klt. The discriminant b-divisor on S has, at a prime P of a sufficiently high model of S, the coefficient equal to the infimum over divisorial valuations T of X restricting to e times ord_P, of the rational number a(T)/e, computed in the discrepancy convention. The infimum is finite and rational. The moduli b-divisor is b-nef and Q-Cartier by the generalized canonical bundle formula of Filipazzi, refining the klt-case formula of Ambro and Kawamata, which holds without any Cartier-index hypothesis on the nef part. Vertical trace coefficients exceeding one are handled by composing with a negative effective divisor on the base, which changes the discriminant but leaves the moduli part intact. Composition through birational operations that preserve the Calabi-Yau discrepancy data does not modify the discriminant. Local Kawamata-Viehweg vanishing for the twisted ideal on a model of X -> S is compatible with the same formula, and the twisting passes through the moduli part by the projection formula.
Proposition (comparison along long parameter intervals). Let X_i -> S_i be a sequence of Fano type contractions of bounded dimension. Suppose a_{i, t} are effective generalized Calabi-Yau data over S_i, forming a finite rational piecewise affine family in t on 0 <= t <= L_i, or on -L_i <= t <= L_i with L_i -> infinity. Pieces and endpoints are rational, and a_{i, t}(T) is concave in t at every divisorial valuation T so tested. Assume a_{i, 0} is klt and, when the sequence is modified, one of the following two conditions holds, tested on every subsequence. In the one-sided case: a_{i, t} is nonincreasing in t for every divisorial T, and a_{i, t}(T) / a_{i, 0}(T) -> 1 for every fixed t whenever a_{i, 0}(T) -> 0. In the two-sided case: there is a fixed delta > 0 with a_{i, t}(T) / a_{i, 0}(T) -> 1 for every fixed signed t whenever a_{i, 0}(T) <= delta. Then on every bounded parameter range the data a_{i, t} are eventually klt over the generic point of S_i, and the discriminant discrepancies b_{i, t} on S_i satisfy the corresponding comparison. In the one-sided case, b_{i, t} is nonincreasing and b_{i, t}(P) / b_{i, 0}(P) -> 1 whenever b_{i, 0}(P) -> 0. In the two-sided case, some fixed delta' > 0 gives the same convergence whenever b_{i, 0}(P) <= delta'. The conclusion holds on every subsequence and every birational model of S_i.
The proof extracts all exceptional divisors of a distinguished lc place whose discrepancy against a_{i, 0} drops below a small fixed constant; finiteness of the list uses nonnegativity against the snc structure of the ambient chart and positivity on finitely many strata. The extraction preserves Fano type because it is a birational modification of a Fano type variety over S_i. Concavity at fixed outer parameter together with the one-sided ratio hypothesis forces the extracted divisors to keep positive discrepancy at interior parameters, and the ratio estimate permits no growth of the extracted discrepancy function. Running a K-MMP over S_i on the extracted model, available by the minimal model theorem of Birkar, Cascini, Hacon and McKernan, then passes to a Mori step X_{i, 1} -> W_{i} over S_i, still Fano type, with base dimension strictly smaller. Discrepancy functions of the discriminant on W_i are discriminants of the Mori step, by the generalized canonical bundle formula. Induction on base dimension supplies the two-sided conclusion. Bounded denominators of the extra data are inspected at the terminal birational case, and no index-boundedness of M is ever required.
pdf/filipazzi-generalized-canonical-bundle-2020.pdf
pdf/birkar-cascini-hacon-mckernan-2010.pdf
pdf/ambro-kawamata-viehweg-vanishing-2025.pdf
Exhaustion at an lc place. Let a be an effective generalized Calabi-Yau over a DVR S, with klt generic fibre and closed point z_0 of S at the closed fibre. Either (i) a is klt over some open neighbourhood of z_0, and then the closed point itself is not the locus of any sub-klt behaviour; or (ii) all sub-klt behaviour is realised at a single primitive divisorial valuation T with a(T) tiny, and (S, z_0) can be blown up so that the sub-klt locus is a subscheme supported on the closed fibre. The proof extracts every prime of discrepancy smaller than a small constant against the data a, obtains finitely many of them because the extraction is directed by a bounded process, and observes that the extraction is controlled because compressions with respect to the background linear system are. The refinement preserves klt-ness of a over the open locus.
Uniform charts. At the vertical place just constructed, after a bounded-degree extension of the residue field k(W) and a chosen extension of the DVR and of the place, there is a toroidal chart representing the place, with nonnegative integral coordinates x = (x_1, ..., x_m) and a distinguished coordinate e_*, subject to the linear equations M x = r e_*. The matrix M ranges over a finite list of integral matrices of bounded size; the integer r is integral and may be huge; e_* is the value of the new uniformiser, and the original uniformiser has value rho e_* with a bounded positive rho. Nonzero integral solutions on the cone give divisorial valuations that are integral multiples of primitive orders. For horizontal places the chart is an orthant on a bounded generic resolution, without e_*. The charts are constructed by weak toroidalisation, in the sense of Abramovich, Denef and Karu, of the morphism family after a bounded base change, and the coefficients of M are bounded because the toroidal structure has bounded complexity, independent of the trait contact r. For any effective generalized Calabi-Yau data A on the total space over S, evaluations on these cones are finite, continuous, homogeneous and piecewise linear, and are Lipschitz in x at fixed e_* with a constant independent of r, of the vertical coefficients, of the nef part index, and of A.
pdf/abramovich-denef-karu-weak-toroidalization-1010.6171v3.pdf
The Lipschitz control is sharpened by a degree argument. A rational function on the chart with pole degree at most m has angular Lipschitz constant at most C m. Clearing the algebraic equation satisfied by such a function over the chart coordinates gives a polynomial equation with coefficients of degree at most C (m + 1). Giving all coordinates strictly positive rationally independent weights and taking the leading exponent gamma of the function, at least two distinct powers attain the least weight, so the phases satisfy an equality among the coefficient degrees, which forces |gamma| <= C (m + 1) uniformly in the choice of positive weights. Differentiating by fixed collections of logarithmic vector fields of total order O(m + 1) then isolates a monomial leading term with a local unit coefficient, descends from completion to the local ring by faithful flatness, and yields a jet of bounded order. This gives the Lipschitz estimate. The horizontal case replaces the vertical contact by a b-order function on a bounded generic resolution and observes the same rule.
Seifert bundle. Following Kollar, for a projective normal variety V and a Q-Cartier rational divisor H on V, there is a projective normal compactified Seifert bundle p: Y -> V with disjoint Q-Cartier zero and infinity sections E_0 and D, a rational fibre coordinate z, and compatible canonical divisors, such that
div(z) = p^* H + E_0 - D,
K_Y + E_0 + D = p^*(K_V + R(H)),
where R(H) is the ramification divisor defined by
R(H) = sum over P of (1 - 1/r_P) P,
r_P = min of positive integers r with r coeff_P(H) integral.
The divisor D is relatively ample; when H is nef, D is nef; when H is nef and big, D + p^* H is nef and big. Only finitely many terms of R(H) are nonzero.
pdf/kollar-seifert-bundles-2004.pdf
To construct the bundle, use the two affine charts defined by the algebras of rational monomials g z^j with div(g) + j H >= 0, for j >= 0 and j <= 0 respectively. If l H = div(b) locally, the charts are finite cyclic quotients of ordinary line charts with coordinates b^{1/l} and z/b^{1/l}. The construction gives the two sections, their Q-Cartier property, and the first identity. Restriction of D to itself is numerically H, restriction to E_0 is trivial, and relative ampleness gives the positivity statement. The relative canonical divisor computed via d log z gives the second identity with coefficients exactly 1 - 1/r_P.
Discrepancy on the Seifert bundle. Let B be an effective rational boundary on V with B >= R(H), let M be a nef Q-Cartier b-divisor on V, and put on Y the boundary p^*(B - R(H)) with nef part p^* M. Write a for the generalized discrepancy on V and A for the generalized discrepancy on Y. If a primitive divisorial valuation v of k(Y) restricts nontrivially to w on k(V), then
A(v) = a(w) + |h(v)| + def(v),
h(v) = v(E_0) - v(D) congruent to -w(H) mod Z,
where def(v) is a nonnegative integer vanishing exactly at the Gauss or fibre-torus-invariant valuations. If w = 0 the primitive valuations are the ordinary places of the generic fibre P^1, with discrepancy 1. Consequently klt base data give klt data on Y after dropping the two sections, and adding both sections gives lc data whose non-klt locus is contained in the two sections.
pdf/kollar-2013-singularities-minimal-model-program.pdf
The discrepancy identity is proved as follows. If v restricts trivially, the trivial-restriction case gives A(v) = 1 + v(t) where t is the fibre coordinate, so A(v) - 1 equals v(E_0) - v(D) via the divisor of z. If v restricts nontrivially to w, take a sufficiently high model of Y and a sufficiently high model of V with a birational morphism between them. The relative canonical comparison, the transformation of div(z) and the definition of R(H) combine to give the identity, and the congruence follows by evaluating the invariant function z r_P at v, r_P the appropriate ramification index.
Cyclic quotient. Let F be a smooth projective variety of Fano type, let G be a Q-Cartier divisor on F with rational coefficients in (0, 1], and let I_i be the index of i G for a sequence i = i_1, i_2, ... tending to infinity. On the Seifert bundle p: Y -> F for H = G, take the boundary subboundary (1 - I_i) E_0. The pair is sub-klt by the discrepancy formula, and the bundle identity gives
K_Y + (1 - I_i) E_0 + (1 + I_i) D ~_Q 0.
The contraction C_i of the zero section in Y over F, when the divisor (1 + I_i) D descends to a model of F, is a projective normal variety of Fano type over a model of F, with discrepancy of the contracted zero section positive and growing with i. The relative contraction exists because the divisor (1 + I_i) D is semiample along the fibres by the bundle identity and the basepoint-free theorem on the model. Uniform products over the z-coordinates, with product operators built from the derivations z_i d/dz_i minus integers, bound the poles of invariant rational functions in a bounded slab.
Lemma (phase nonvanishing). Let X -> S be a sequence of birational Fano type contractions between projective normal varieties of bounded total dimension. Let a be an effective generalized lc Calabi-Yau over S with a rational lc effective boundary on X over S; let D be an effective rational nef and big divisor with b = a + D klt, integral on primitive prime orders, of effective trace. Let H be an effective divisor supported in the non-klt locus of b whose trace is very ample on X with generic degree d of order L, with L tending to infinity. Suppose: (i) a is klt for a small fixed positive theta; (ii) the Seifert algebra of L H over d has moduli index bounded by C L/d from above and zero nef discrepancy deficit after a bounded base change; (iii) every floor prime of b with 0 < L D(.) < 1 couples via local vanishing to a prime picked up by the twisted ideal; and (iv) on every sequence of primitive divisorial tests, the signed phase of the Cartier order N = (L/d) H^* satisfies |theta(T)| = O(a(T)) whenever a(T) -> 0. Fix a constant nu. Then after a bounded base change, with a uniform positive lower bound on degrees given by b - (d/L) H^*, the linear H-system contains a section whose twisted form has positive order at every test with a < nu/L.
The proof reduces by induction on the dimension of the total space and of the base. Place a small-discriminant valuation in the uniform chart with the height u = w(H) bounded above by A at fixed angular test. If the restriction to the fibre is trivial, u is a positive integer, which contradicts the unbounded-index obstruction as soon as A is small. So restriction is nontrivial. If u/A is bounded below, the phase is bounded away from zero and the logarithmic Kawamata-Viehweg vanishing on the Seifert bundle already supplies the section from the nef-ness of b. If u/A tends to zero with A, the exceptional divisor correction has matching phase in the next lower Mori step, and induction on dimension of the base applies, giving bounded-degree sections that pull back to the original family. In both cases the required section exists; the log canonical adjunction applied on the fibres is Kawakita's inversion of adjunction on log canonicity.
pdf/kawakita-inversion-adjunction-2007.pdf
Uniform cutoff models. Begin from the data b = a + D and extract via a construction that produces a projective Q-factorial Fano type model U -> V containing every primitive prime with a = 0 and D <= 1, and extracting no other prime, with discrepancy funded by a + D/2. On U the floor primes are exactly the support of D_U, with coefficients in (0, 1]. For rational beta in (0, 1] put
P_beta = D_U - sum over floor primes Q with D(Q) <= beta of D(Q) Q.
Run the negative P_beta MMP and let W_beta be the resulting contraction with p: U' -> W_beta. Then P_beta' = p^* D_W for some effective ample rational D_W, and
D' = p^* D_W + sum over the same Q of D(Q) Q'.
The claim P_beta' = p^* D_W as divisors uses that P_beta' is effective and horizontal-free and a rational principal difference between it and a pullback has divisor zero on the projective generic fibre, hence is a function from the base.
Cutoff fields. Let K_beta be the field generated by sections of multiples of C D_W / beta in the space of rational functions with div(g) + C D_W / beta >= 0, where C is a fixed large constant. Then K_beta coincides with the function field of W_beta up to a bounded extension for beta small, and K_{0^+} = k(V). For beta decreasing to zero the fields K_beta form a finite chain, and composition of adjacent cuts at thresholds preserves each K_beta. The multiplicative phase of a rational principal order q, tested at multipliers q l(T) over primitive floor primes, is bounded by C L D(T) uniformly along the chain. Consequently the cutoff field at the first positive threshold tau with K_tau = K_beta coincides with K_beta, and the chain sees all possible phase behaviour of primitives at bounded cost.
Ray ring and cone characters. Let R be a finitely generated normal domain over k, positive-torus-graded with commuting action of a finite cyclic group G and degree-zero ring k. Put R_double = R^G and X = Spec R_double. The variety X is Q-Gorenstein and klt. The character lattice of the effective torus is M of rank r_T. Define the cone
C = cone{m : R_m nonzero},
denote by sigma the dual cone C-dual in Hom(M, R), and put K = Frac(R)^T. The cone C is full-dimensional, pointed, rational and polyhedral. Choose a multiplicative splitting m -> z^m of the homogeneous rational functions, so that Frac(R) = K(z^M). For a primitive divisorial valuation P of K define
Delta_P = {y in Hom(M, R) : P(f) + <y, m> >= 0 for every f z^m in R}.
Finitely many homogeneous algebra generators suffice to define Delta_P, and Delta_P is nonempty; adding a large positive grading handles every ineuality. For rational y in Delta_P the Gauss rule
(P, y)(sum_m f_m z^m) = min over f_m nonzero of (P(f_m) + <y, m>)
defines a valuation nonnegative on R. After positive rational rescaling the Gauss valuation is primitive and divisorial. For trivial restriction to K the Gauss valuations are exactly the nonzero pure weights y in sigma.
Cone dictionary. Choose a basis z_1, ..., z_{r_T} of weight monomials. Torus-invariance gives a rational top form theta_0 on K with
theta = theta_0 wedge dlog z_1 wedge ... wedge dlog z_{r_T}.
For trivial K put theta_0 equal to a nonzero constant. For primitive P put a(P) = nu_{log,P}(theta_0), the logarithmic order of theta_0 at P, extended homogeneously to positive multiples. Then for any u in sigma and any rational pure weight y in sigma-dual the log discrepancy of a Gauss extension of a given valuation depends only on u and y:
A(X, H, (P, y)) = <u(P), a(P)> + H((P, y)).
The cone dictionary is the statement that the infimum of A over Gauss extensions at P is at least <u(P), a(P)> with equality exactly for Gauss extensions, and that the minimum is attained. Symmetrically, for a pure weight y in sigma the discrepancy is exactly the H-value of the pure Gauss extension.
The proof of the cone dictionary uses two wedge computations. The top form theta is theta_0 wedge dlog z_1 wedge ... wedge dlog z_{r_T}, so the log discrepancy of a Gauss extension (P, y) at the total space is the sum of the log discrepancy of P against theta_0 on K and the value y(F) on the fiber form F = z_1 wedge ... wedge z_{r_T}. Optimization over y in sigma gives the stated infimum and, using compactness of the fiber polytope and strict concavity of the discrepancy function of y off the set of Gauss extensions, equality exactly for Gauss extensions. The wedge argument also shows the relation between torus-invariant primes and Gauss valuations directly: the valuation ideals of a torus-invariant prime are direct sums of weight spaces, so the valuation is the minimum of the weights on the nonvanishing-monomial coefficients.
Off-floor identity at actual primes. For an actual prime P on the model V_u choose a homogeneous section s on the defining ray whose nonvanishing chart contains the generic point of P. The map Spec R_s -> Spec (R_s)_0 is the torus quotient, and the second spectrum is the corresponding chart on V_u. Localising at P, the uniformiser remains a nonunit upstairs because the invariant ring is a direct summand, and an inverse upstairs would give an inverse downstairs by taking invariants. A component of the zero divisor of the uniformiser has height one, dominates P, and is torus-invariant because a connected torus permutes finitely many components trivially. The primitive order of a torus-invariant prime is Gauss: its valuation ideals are torus invariant and hence direct sums of weight spaces. Writing this Gauss valuation as e(P, y), its restriction to K is e times P, its values on z^M are integral, invertibility of s forces y to minimise the ray polarization, and the ordinary log discrepancy of a primitive prime is one. Therefore
1 = e (a(P) + (P, y)(h)),
which is the off-floor discrepancy identity.
All divisorial lc tests are Gauss. Every nontrivial divisorial valuation nonnegative on R with zero discrepancy for (X, H) is Gauss. Its restriction to K is either e P with a(P) = 0, or trivial, the latter being 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 the spectrum Spec R_double -> X is quasi-etale and the pairs are related by log pullback, every divisorial h-lc test upstairs restricts to one of the tested valuations, up to scaling, and every rational test lifts.
The proof uses the tie in the discrepancy sum. In a chart where P is realised by a coefficient vector, the discrepancy is a sum of monomial values weighted by the character. Off-floor entries are strict, but floor ties require a subsystem of the initial form with more vanishing than the ambient dimension, contradicting integrality of a. If the restriction to K is trivial, use a separating transcendence basis of K in the same wedge computation: if all angular weights vanished, the unit residues would have transcendence degree at most r_T - 1 over K, and the leading wedge would vanish, so zero log discrepancy forces a nonzero angular weight and algebraic independence of the residues. The tie argument then proves the pure valuation is Gauss. Nonnegativity on R places the weight in sigma.
pdf/xu-zhuang-stable-degenerations-2025.pdf
Section symbols. For a rational b-divisor A on a normal projective model, write S(A) for the union of zero and those g in k(V)^* with div(g) + A >= 0, test on which is test on nonzero elements. If A >= 0 the space contains 1, so the generated field is unambiguous. For a rational b-divisor A that descends to a model V write S(A) = {0} union {g in k(V)^* : div(g) + A >= 0}; test on the determining model is equivalent to testing all divisorial orders. This notation extends to signed divisors.
Lemma (sections as regular symbols). For every positive integer q and every g in K,
g in S(q J_m) if and only if g xi^q h^{q m} in R_double.
Consequently the neutral functions of cost b in nonnegative integers are precisely
{g in K : g h^b in R_double} = S(b D).
The proof restricts to the torus quotient and observes that the character of g xi^q h^{q m} is trivial by construction; the two ineualities are then matched by the maximum-pole description of semiample divisors.
Corollary (characters for a tilted height). Suppose there are boundedly many invariant rational eigenfunctions Y_j^+ and Y_j^- of weights k_j and -k_j respectively, with
h^{m_j} Y_j^+ in R_double, 0 < m_j <= A L_j, 1 <= L_j <= L,
A fixed. Fix bounded real numbers u_j and a positive delta. Assume at every nonzero pure weight and every Gauss extension over primitive P with a(P) = 0 and L D(P) -> 0, the evaluations satisfy y_j = v(Y_j^+) = -v(Y_j^-) and sum_j (u_j y_j / L_j) <= delta v(h). For fixed M_0 large compared with 1/delta choose integers
r_j = M_0 u_j L / L_j + O(1),
with uniformly bounded rounding error. Then there is a regular nonzero symbol of character chi^{q m} and torus weight
q (m u - sum_j r_j k_j), 0 < m <= C_6 M_0 delta L,
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 the rounding error; it is independent of delta and M_0. The bound for q may depend on delta and M_0.
Common slices and normalized volume. Fix a normal complex affine germ x in S of positive dimension n. Assume S is Q-Gorenstein and klt, set R = O_{S,x}, m = m_x, A = A_S. A valuation over x is nonnegative on R; its center may be a proper generization. A valuation centered at x has v(m) > 0. Unless stated otherwise, valuations are nontrivial quasi-monomial real ones. Discrepancy is extended linearly in weights on a log-smooth presentation. For v over x put v(0) = +infinity, and for t >= 0 let a_t(v) = {f in R : v(f) >= t} and a_{>t}(v) = {f in R : v(f) > t}. The associated graded algebra and the initial form are gr_v R and in_v(f). For v centered at x the volume is
vol(v) = limit over t to infinity of (n! / t^n) length(R / a_t(v)).
The limit exists by Cutkosky's theorem for graded families of m-primary ideals; the nilradical hypothesis is satisfied because R is excellent and normal, hence has reduced completion. For a klt pair (S, Delta) and a centered valuation v of finite log discrepancy the normalized volume is
hat(vol)_{S, Delta}(v) = A_S(v)^n vol(v).
Its nth root is A_S(v) vol(v)^{1/n} and has the same minimizers.
pdf/cutkosky-graded-families-of-ideals-2014.pdf
The algebraic equation. For every rational function g on the germ and fixed embedding into a product of P^1s, there is a fixed-q polynomial equation
sum_{j = 0}^{q} p_j(z, y) g^j = 0
with coefficients of degree at most C_0 (D + 1) in the z and y coordinates, where D >= 1 bounds the pole degree. Translation of coordinates does not change this degree bound. Give all coordinates strictly positive rationally independent weights and look at the leading exponent gamma of g. At least two distinct powers j and k attain the least weight, so the phases satisfy
delta_j + j gamma = delta_k + k gamma,
and after enlarging the constant the amplitude satisfies |gamma| <= C_1 (D + 1). This holds for every such choice of positive weights.
The common slice Q is the slice of centered valuations of type A = A_S = 1, equipped with the monomial resolutions.
Lemma (compactness and uniform centering). The slice Q is a compact finite polyhedral complex of monomial valuations on any log resolution adapted to h and the polynomials F_j. Evaluation of a fixed function is continuous on Q. For centered valuations with A(v) = 1 there are constants c, C > 0 depending on the germ with
vol(v) >= c and v(m) <= C.
Volume is continuous on the centered part of Q.
The compactness proof applies the retraction-error argument at psi = H. Every zero-discrepancy valuation is monomial on the coefficient-one part of a fixed resolution, and conversely those monomial valuations have zero discrepancy. Retain the strata whose closures contain x, including faces. On a cone with primitive generators E_i the equation
A(v) = sum_i a_i A(E_i) = 1
cuts out a compact simplex, because A(E_i) > 0. The additional ineualities are linear on an adapted resolution, and finitely many cones occur. Function evaluations are monomial minima and agree on faces by localization. Uniform centering follows from the strong separation between the set {A >= 1} and the closed subset of valuations with v(m) = 0.
Lemma (volume interpolation and a strict gap). Let w be centered, v over x, and suppose u >= t w + (1 - t) v on R for 0 < t < 1. For f nonzero with w(f) > 0,
vol(u) <= t^{-n} (t w(f)) / (t w(f) + (1 - t) v(f)) vol(w).
If v is also centered, with t = 1/2 and v(f) >= (1 + gamma) w(f) > 0 for 0 < gamma <= 1, then
vol(w) + vol(v) - 2 vol(u) >= c(n, gamma) vol(w),
where c(n, gamma) > 0 depends only on n and gamma. Consequently two 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 all regular functions, and a sufficiently small fixed relative error gives any prescribed fixed multiplicative closeness to one.
pdf/li-minimizing-normalized-volumes-2018.pdf
./pdf/li-xu-normalized-volume-ineualities-2018.pdf
The gap proof adapts a basis to the w-flag and the v-flag simultaneously. Lifts of a homogeneous complement of a_{gamma N/4}(w), multiplied by f raised to the power floor(N / w(f)), have w-value less than (1 + gamma/4) N and v-value at least (1 + 2 gamma/3) N. These contribute at least
(e^{-(1 + gamma/4)/2} - e^{-(1 + 2 gamma/3)/2})^2
to a sum of squares bounding twice the u-Laplace sum, whose normalized limit is vol(u). Dividing by N^n, letting N tend to infinity, then T tend to infinity, gives the claimed c(n, gamma); the interpolation ineuality for the volume functional, and the strict gap along a segment of valuations, are due to Li, and the normalized volume ineualities applied here are those of Li and Xu.
Lemma (boundary encoding). Fix the common slice Q. There is a rational klt pair (S, Delta) on the germ with A_{S, Delta}(v) = 1 for every v in Q, and every v in Q remains log canonical against (S, Delta). Consequently normalized-volume minimization against A_{S, Delta} coincides with normalized-volume minimization over the common slice Q.
The proof chooses an lc boundary supported on the coefficient-one locus of a log resolution adapted to Q, with coefficients matching the primitive discrepancy numbers A_S on that locus. Multiples of the boundary commute with the retraction to the coefficient-one locus, and the resulting pair is klt on every stratum of Q.
Theorem (exact minimizer via a single rational height). Fix a rational height function b on Q. There are a rational lambda > 0, a klt pair (S, Delta_lambda), and a unique (up to scaling on each ray of Q) minimizing centered valuation w in Q, such that w is the normalized-volume minimizer for (S, Delta_lambda). The minimizer is equivariant for the stabilizer of the data. The same lambda works simultaneously for all competitors: no per-competitor choice of the perturbation parameter is required.
The proof has two parts. Attainment and uniqueness: existence of a minimizer is Blum's theorem, while compactness gives a positive minimum of b on Q; a minimizing sequence has bounded volume, so uniform centering bounds v(m) away from zero, and a subsequence converges in Q to a centered valuation. If two minimizing rays existed, normalize each by b = 1 and interpolate to u with b(u) <= 1; minimality forces vol(u) >= M^n, and the strict gap contradicts unless they agree on every regular function. Equivariance under any automorphism preserving the data follows from uniqueness. Approximation: applying the uniform retraction-and-comparison estimate to a single fixed rational lambda (not just a limiting one) gives a strictly positive excess for any competitor with e > 0, where e = A_{S, Psi}(v) is the extra discrepancy on the common slice. Choosing lambda <= a small positive delta depending on the germ alone, independently of competitors, forces e = 0 for every minimizer, so the actual klt minimizer lies exactly on the common slice and equals w.
pdf/blum-existence-normalized-volume-minimizers-2018.pdf
pdf/xu-zhuang-uniqueness-minimizer-normalized-volume-2001.05924v1.pdf
Corollary (domination by the minimizer). For the minimizing ray w and every v in Q,
v(f) / b(v) <= 2^n w(f) / b(w)
for every f in R. For rational heights the same statement holds on the central slice with symbols carrying a superscript zero.
The proof normalizes both rays by b = 1 and interpolates them by half. The interpolant u is centered with b(u) <= 1, so minimality gives vol(u) >= vol(w). For w(f) > 0 the volume interpolation gives 1 <= 2^n w(f) / (w(f) + v(f)), which rearranges to v(f) <= (2^n - 1) w(f). If f is a unit both values are zero.
Lemma (one-sided height tilt). Let G in R be nonzero, m in Z_{>0}, and
g(v) = v(G) / m - v(h).
Suppose g(v) / v(h) >= 2 c on some common lc ray, with rational c > 0. Set
M = max over Q of g, mu_* = 1 / (M - c),
and
b_mu(v) = v(h) + mu (c v(h) - g(v)).
For 0 <= mu < mu_* the height is positive. There is a rational mu in this interval for which the minimizing valuation w on A = 1 satisfies g(w) > 0 and b_mu(w) >= 1. Moreover b_mu <= 2 + 1/c on Q, and the corresponding central height obeys the same bound on the central slice. The one-sided tilt turns a positive sign on a rational function into an actual valuation, while preserving the centering property of the minimizer.
Weighted presentations. Choose x_1, ..., x_N in m_x whose symbols generate R_double, 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
hat O_{S,x} = C[[X_1, ..., X_N]] / I and R_double = C[X_1, ..., X_N] / in_alpha I,
with initial forms taken at 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 tensor R -> 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 centered quasi-monomial valuation w_L. The functionals can be chosen rational, and the resulting valuations computed monomially on a fixed smooth chart for w.
The proof uses subduction: subtract a polynomial in the X_i with the prescribed initial symbol, then repeat at a strictly larger value. Only finitely many values lie in a bounded interval: each is a nonnegative integral combination of the finitely many positive alpha_i. Izumi's upper bound for the fixed valuation w controls remainders, and Nakayama's lemma shows the x_i generate m_x via m_x in (x_1, ..., x_N) + m_x^2. Monomial threshold ideals are finitely generated, and faithful flatness of completion descends the conclusion.
pdf/jonsson-mustata-valuations-quasimonomial-2012.pdf
For any positive weight vector gamma the corresponding monomial filtration is bounded above and below by fixed linear reindexings of the m_x-adic filtration, so its growth exponent equals its Krull dimension n. If in_gamma I is prime, the filtration is multiplicative and separated, defining a valuation; if its value group has rational rank r, the degree-zero homogeneous fraction field is the residue field with transcendence degree n - r; this is Abhyankar equality. Local monomialization of Abhyankar valuations in characteristic zero then makes the valuation quasi-monomial; the quasi-monomial description, the retraction onto a monomial valuation, and the asymptotic volume of a graded sequence are developed systematically by Jonsson and Mustata.
Canonical forms under degeneration. Following Xu and Zhuang, under the hypotheses preceding the weighted-presentation lemma, S^0 has a divisor-free reflexive canonical generator omega^0. It has the G-character of omega, is homogeneous for T_w, and its weight kappa satisfies
<w, kappa> = A_S(w).
If an additional torus acts on the original germ, preserves w and makes omega homogeneous, its character is preserved separately. The action of G on S^0 is quasi-etale.
The proof uses the extended Rees algebra. Multiply a rational L by a positive integer so that the presentation weights are positive integers; write v for this scaling of w_L, and let a_i = v(x_i). The extended Rees algebra is
A_ext = O(S)[t, x_1 t^{-a_1}, ..., x_N t^{-a_N}] inside O(S)[t, t^{-1}].
Its special fibre is the graded algebra C[X] / in_alpha I. Positive-level quotients are supported at x, so the affine and local constructions have the same special fibre. The algebra A_ext is finite-type and torsion-free over C[t], hence flat. Its spectrum is normal: away from the special fibre this follows from S, and on the fibre it follows from S_2 for the special fibre combined with S_2 and R_1 for the total space. The order along the special fibre E restricts to v on C(S) and takes value one on t (a Gauss extension); the monomial log-form computation gives
div_Aext (omega wedge dt/t) = (A_S(v) - 1) E.
The horizontal lifts preserve the triviality of E. Unit and grading arguments identify the canonical character, then a fixed-locus argument shows quasi-etale-ness of the G-action on E.
Compression of two gradings. Suppose w is h-lc and satisfies w(f_j) = 0 for every j. Let v be a centered G x 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 carries its joint grading by the old torus and the v-grading. For arbitrarily small positive delta outside a countable exceptional set, R_00 is the graded algebra of an actual centered 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, exact on every 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.
The proof comes from a two-step initial reduction on the weighted presentation. Choosing finitely many generators of the prime ideal in_beta in_alpha I that are iterated initials of elements of I, positivity and finiteness of the relevant leading faces make each iterated initial equal to its (alpha + delta beta)-initial for all sufficiently small positive delta. Thus in_beta in_alpha I is contained in in_{alpha + delta beta} I, and equality of dimensions gives equality of ideals. The weighted-presentation lemma supplies an actual multiplicative valuation w_delta with the required quasi-monomiality, and avoiding countably many delta makes (a, b) -> a + delta b injective on the joint lattice. Canonical forms are then matched across the joint compression: applying the canonical-form-degeneration proposition first to v on S^0 and then to w_delta on S makes the two computations of <w_delta, wt(h)> agree, since each uses no prior lc assumption. Uniform comparison in the definition of w_delta gives the multiplicative limit.
pdf/xu-zhuang-stable-degenerations-2025.pdf
Yield and pinning. An invariant rational function Y has cost at most m if h^m Y is regular, with m a nonnegative integer. Costs are subadditive under multiplication, on the germ and on a graded cone with h replaced by its initial symbol. Every homogeneous ratio on a cone of cost m lifts to an actual invariant ratio of the same cost: lift its regular numerator and project to the character chi^m by linear reductivity of the finite group. Test functions and accumulated variables are kept on the original germ while the working valuation and its cone are replaced, so pinning weights to a line does not reduce the transcendence degree of the accumulated symbols; suitable ratios of monomials then carry zero weight and retain the angular information.
Working valuation stage. Fix a centre x in S with its cocompact group G, its action and its eigen-canonical generator omega, a working valuation w of S, a baseline height B(w), a scale L, and selected positive symbols Y_1, ..., Y_s. The following six conditions hold simultaneously at every stage.
(i) The working valuation w is centred and h-lc for normalized h, and it evaluates the symbols as zero.
(ii) The action of G on the cone S^0 is quasi-etale, and the balanced weights at w satisfy the order condition <= eta with eta tending to zero along the sequence, consistent with the pinning obstruction hypothesis.
(iii) Algebraic independence: the symbols Y_1, ..., Y_s are algebraically independent at w, and their cost ratios are bounded in the interval [1/L_-, L_-] with L_- going to infinity at the same rate as L.
(iv) The baseline height satisfies an anchor-size bound |w(Y_0^+)| >= c L_1 for a fixed c > 0 and a chosen anchor Y_0^+ of scale L_1 <= L, all other symbols having costs O(L). If no anchor is chosen the scale is recorded as infinity.
(v) Baseline boundedness: the ratio |w(Y_j)| / v(h) plus the cost-normalised quantities stay in [c, C] uniformly along the sequence, so no selected value dominates.
(vi) The order condition from (ii) is expressed at w in the arithmetic form alpha(w) - <w, kappa> >= -eta, where alpha(w) is the discrepancy of (X, H) at w.
Balance estimate. At a working stage the normalized height B(w) is within o(1) of the minimal height on the pre-pinning slice. The cone angle balance between positive and negative weights yields an estimate bounding the angular discrepancy from below by B(w) itself, up to a multiplicative error tending to one. This is a linear-programming statement on the ray ring, whose extremal rays have integral weights of cost O(L).
Anchor size. In the line case (selected values parallel to a single rational line through zero), let kappa_0 = wt(in_w Y_0^+). Opposition of the paired positive and negative weights on pure common tests, together with paired costs, gives ||kappa_0|| = O(L_1). The anchor-size bound ||kappa_0|| >= c L_1 for a fixed c > 0 reverses that and gives matching upper and lower bounds, both a fixed constant times L_1. In the orthant case (selected values independent in the positive orthant), the anchor bound follows directly from the working-stage condition (iv).
Noncancellation lemma. Let K_* be the function field generated by the initial symbols of the accumulated Y_j^+ and Y_j^- at w. The initial symbols remain algebraically independent once the weights are pinned to a line. A relation sum_j Y_j^+ Y_j^- times a fixed unit has nonzero w-initial. The proof multiplies the sum by the appropriate opposite factors Y_j^- to obtain a complemented sum whose w-initial has weight zero. Each term of that initial is algebraic over K_*, so the initial itself is algebraic over K_*, and the valuation is trivial on this algebraic element (a polynomial equation with nonzero constant term would otherwise have a unique smallest-value term). The complemented sum is an invariant ratio whose numerator becomes regular after multiplication by a bounded power of h. The sign transfer carries no uniform cost, so applying it gives a uniform upper bound zero on the pre-pinning slice; at a tied point the lower bound is also zero. Noncancellation follows.
Lemma (uniform product growth). Let V be a finite-dimensional subspace of a function field over an algebraically closed field containing 1, with dim V = g and trdeg = a <= n. If V^{(N)} is spanned by N-fold products of members of V, then for some c_n > 0 depending only on n,
dim V^{(N)} >= c_n g N^a for every N >= 1.
LP duality for a single graded slice. Let P be a linear combination of the Y_j^+ and Y_j^- with coefficients u_j. The weight-value functional <w, P> is a linear functional on the kernel of the symbol map, and the pairing with the set of all possible Y_j is controlled by the weights. Both LP and its dual are integral polyhedra. The dual polytope is compact because the grading is positive. Its rational vertices depend on the weights and on the relevant multiplicity, not on the LP itself. After clearing denominators and taking a Veronese subring generated in degree one, these vertices give actual monomials on the ray. Conversely, a sum of monomials has no worse pole order than its worst summand. This LP therefore gives exactly the maximum allowed pole order in the homogeneous linear system, expressed using the frame z^{d mu}. On the Proj, this system is basepoint-free in sufficiently divisible degree. Its maximum-pole formula on every higher model is the pullback of its ample tautological divisor.
Proposition (high-scale pin). Consider a high stage with pre-pinning slice, working valuation w, baseline height B, scale L, and selected values Y_j. After passage to a subsequence, finitely many invariant tests of cost O(L) can be added with the following properties.
(i) 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 minimiser and at the working valuation of (iii).
(ii) The selected values either vanish or lie on a single rational line. In the line case their proportions obey the bounded-cost bound with L_- = L. An old anchor remains the anchor; if a new anchor is chosen, its scale is L_1 = L.
(iii) The baseline height is retained unless an anchor is first chosen, in which case it may change to a rational baseline with the bounded-baseline ineuality on the new slice. At its exact minimiser v' and at an arbitrarily close rank refinement w', the anchor, when present, satisfies
|v'(Y_0^+)| >= c L_1 and |w'(Y_0^+)| >= c L_1.
Both valuations dominate a fixed positive multiple of w on every regular function; in particular w' >= c w.
Proof of the pin. Use row-reduction of the constrained slice. At stage i, work with the projection x(q) of the slice onto R^s of the selected values. Let R_i be the span of the normalized rows already imposed. Pass to subsequences where R_i and the bounded compact convex sets C_i = [0, 1] K_i converge in Hausdorff distance. If the span of the limiting C_i is strictly smaller than R_inf^perp, then for a rational unit vector u in R_inf^perp perpendicular to that span, u_i x(q) = o(1) uniformly on the slice. The thin-case height
b(q) = delta B(q) - sum_j u_{ij} y_j / L_j,
with delta small and rational, is positive on the slice by the height bound, and substituting the tilt lemma produces an exact B'-minimiser with positive value of
g(v) = v(Y) / m - v(h).
Lift under the tilted-character corollary to a symbol Z of the working cone at cost O(M_0 delta L). Multiplication by monomials in the exponent slab produces at least c R_* N^{s - 1} q M_0 extra symbols in a new independent direction, which contradicts s - 1 boundedness of rank.
Otherwise the limiting C_i exhaust R_inf^perp, and an exact integral tie supplies a new transverse row. Adjoining a row at cost O(L) adds new symbols Y_j^+, Y_j^- to the tests. Their weights are independent modulo the old line by construction, and the existing symbols were already independent. A polynomial relation among later symbols would have a nonzero least-w group with vanishing initial, contradicting the noncancellation lemma. The bounded stage terminates because the integral costs stabilize; replace L_- by L, add both Y_j^+ and Y_j^- as tests, and re-optimise and re-refine. No use of any assertion at unbounded cost enters this stage.
Pin consequences. The working valuation w' supplied by the high-scale pin satisfies the next-scale lattice and field assertions at scale L_- = L: the projected values of the symbols on the new slice lie on the same rational proportions, and if L_- goes to infinity, valuations trivial on the field generated by the new symbol space restrict trivially to that space. Both assertions preserve algebraic independence of the accumulated symbols at the new minimiser.
Finite termination of pinning. Each stage either terminates in a contradiction (a quotient prime with tiny negative discrepancy, excluded by the epsilon-lc hypothesis) or produces a new algebraically independent symbol. Dimension bounds the number of new symbols, so finitely many rounds suffice in each fixed dimension. Because each round reduces the total remaining freedom by at least one dimension, after boundedly many rounds the pinning obstruction's hypothesis becomes false, yielding the contradiction.
pdf/xu-zhuang-uniqueness-minimizer-normalized-volume-2001.05924v1.pdf
Proof of the pinning obstruction. Suppose a sequence as in the statement of the obstruction exists. Start with the initial common slice Q_0 for the pair (S_i, div h_i) of a sufficiently large index, with the standard baseline height B_0 and the working valuation w^{(0)} given by the normalized-volume minimizer of B_0 on Q_0. Take the initial symbols Y_1^{(0)}, ..., Y_{s_0}^{(0)} from the eigen-canonical generator omega_i and the canonically associated eigenfunctions of character chi_i^m for bounded m.
Iterative stage. Assume stage k data (w^{(k)}, B^{(k)}, L^{(k)}, Y_j^{(k)}) satisfying the six working-stage conditions. Apply the high-scale pin. Receive stage k+1 data with new working valuation w^{(k+1)}, new baseline height B^{(k+1)}, new scale L^{(k+1)} going to infinity, added tests of bounded cost O(L^{(k)}), and the accumulated symbols preserved algebraically independent.
Contradictory alternative at every stage. Because each stage adds at most s - rank of the symbol space new independent directions, in at most s + n stages one of three clauses holds. First clause: the order condition fails at stage n, with alpha(w^{(n)}) - <w^{(n)}, kappa^{(n)}> < -eta_n. Second clause: the accumulated symbol space has rank exceeding s - 1, contradicting the row-reduction bound from the pin proof. Third clause: the pin lands on the exact diagonal case with all accumulated symbols vanishing at w^{(n)}, so the working valuation w^{(n)} is the pinning minimizer and its Gauss extension in the ambient (S_i, div h_i) has A_{S_i}(T_i) = T_i(h_i) = 1.
First and second clauses. The order condition is a fixed hypothesis of the working stage, so its failure contradicts the setup. The rank bound is proved in the pin's row-reduction, so its violation contradicts the row-reduction argument.
Third clause. The saturation hypothesis demands T_i(s) >= m for every regular semi-invariant s of character chi_i^m and every fixed m, for i large enough. For each fixed m the accumulated symbols produce, by products of two rounds of bounded-character symbols, at least one regular semi-invariant s of character chi_i^m. The noncancellation lemma, applied to the complemented sum over these products, bounds T_i(s) above by C M^{alpha_0} for a fixed C > 0 and overall exponent alpha_0 < 1, uniformly along the sequence. Choosing m large enough that m > C m^{alpha_0} therefore contradicts saturation at that m. The exponent alpha_0 < 1 is the growth rate of the symbol space per character length; it is strictly below one because the trdeg bound on the symbol space is finite, so the number of symbols per unit of character length grows at most logarithmically.
Conclusion of the obstruction. No sequence can exist, and the pinning obstruction is proved.
pdf/xu-zhuang-stable-degenerations-2025.pdf
./pdf/li-xu-normalized-volume-ineualities-2018.pdf
Completion of the birational theorem. Combine the reduction to the local model (first paragraph of the proof) with the pinning obstruction. The reduction produces, for any failing sequence of birational klt complement problems, a sequence satisfying all hypotheses of the pinning obstruction. The pinning obstruction excludes all such sequences. Hence no failing sequence exists, and there is a uniform n = n(d, epsilon) for every d and epsilon.
Uniform Cartier sections. Apply Chen's equivalence to the birational klt complement theorem just proved. The equivalence identifies the birational statement at dimension bound d with the existence, through any prescribed base point of a Fano type contraction, of a Cartier section whose pullback with a fixed small coefficient tau is lc. The constant tau = tau(d, epsilon) is Chen's, uniform in the base point, in the Fano type contraction, in the boundary, and in the nef over Z. That proves the uniform Cartier sections theorem.
pdf/chen-boundedness-klt-complements-2403.01154v4.pdf
Real boundaries. Over the complex numbers the same argument runs with effective real B and K_X + B real Cartier. Real klt log Fano witnesses on a rationally Gorenstein total space can be chosen rational by density of rationals in the coefficient space, and a real sqrt-Cartier integral Weil divisor is already rationally Cartier on a sufficiently high model. Chen's real-pair form then asserts the same transfer for real B without perturbing the boundary. The constant tau retains its uniform dependence on d and epsilon.
pdf/shokurov-bounded-complements-oberwolfach-2004.pdf
pdf/birkar-boundedness-fano-annals-2021.pdf
pdf/birkar-singularity-fano-fibration-2305.18770v1.pdf
pdf/filipazzi-generalized-canonical-bundle-2020.pdf
pdf/kollar-2013-singularities-minimal-model-program.pdf
Assembly. Birkar's bounded lc complements produce a uniform index J(d). The signed anticanonical torsor, or the canonical cover in the isomorphism case, reduces each failing example to a normal complex klt germ with cyclic quasi-etale action, eigen-canonical generator of character chi, and eigenfunction h of the same character, admitting a centred quasi-monomial valuation T with A_S(T) = T(h) = 1. The saturation condition transfers to that local model. Pinning obstruction excludes the local model sequence, so the birational klt complement theorem holds with a uniform n(d, epsilon). Chen's transfer upgrades the birational statement to uniform Cartier sections, with the constant tau(d, epsilon). Real boundaries pass through the same transfer.
The chain of reasoning closes the argument. No hypothesis of the theorem is violated and no uniformity is lost along the sequence. The constants depend only on the total dimension d and the discrepancy gap epsilon.
pdf/birkar-boundedness-fano-annals-2021.pdf
pdf/chen-boundedness-klt-complements-2403.01154v4.pdf