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.

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 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.

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.

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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 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.

./birkar-boundedness-fano-annals-2021.pdf

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 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 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.

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 evaluates to zero on the first and to the character h 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).

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.