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Math-2314 · mathematics · Algebraic geometry

Bounded Complements for $\varepsilon-Log-Canonical$ Varieties

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every dimension bound $d$ and $\varepsilon\geq 0$, does there exist a finite set of complementary indices $N_d(\varepsilon)$ such that every $\varepsilon$-log-canonical variety $X$ of dimension at most $d$, with $-K_X$ nef and big over $Z$, has a general bounded complement $(X,D/n)$ that is $\varepsilon$-log-canonical?

Formulation references: [S1], [S2].

Short English statement

For every dimension bound $d$ and $\varepsilon\geq 0$, does there exist a finite set of complementary indices $N_d(\varepsilon)$ such that every $\varepsilon$-log-canonical variety $X$ of dimension at most $d$, with $-K_X$ nef and big over $Z$, has a general bounded complement $(X,D/n)$ that is $\varepsilon$-log-canonical?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2314-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2314

Manuscript 1 (family 066, 2026-09-25): The manuscript assumes a positive epsilon-lc Fano contraction and gives a bounded klt complement locally over the base. The problem includes epsilon zero, a nef-and-big rather than ample setting, and an epsilon-lc complement. Those scope and retained-singularity conditions are not all covered.

Bounded Complements for \$
\textbackslash{}
varepsilon-Log-Canonical\$ Varieties

OpenAI's claim

Fano contractions with a fixed positive log-discrepancy gap are claimed to admit locally bounded-index klt complements, with the stated monotone finite-coefficient extension over the complex numbers. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix $d\ge1$ and rational $\varepsilon>0$. OpenAI claims an integer $N(d,\varepsilon)>0$ for all projective surjective connected-fiber contractions $f:X\to Z$ of normal integral varieties over a field that has characteristic zero and is algebraically closed, where $\dim X=d$, $K_X$ is rational Cartier, $X$ is $\varepsilon$-log canonical and $-K_X$ is relatively ample. For every closed $z\in Z$, it claims effective rational $B$ and a Zariski neighborhood $U$ of $z$ such that $(X,B)$ is klt over $U$ and \[ N(K_X+B)|_{f^{-1}(U)}\text{ is integral Cartier and linearly equivalent to }0. \] The bound is uniform in the contraction and point, and the complement is klt rather than merely log canonical.

Outline of the argument

Manuscript 1 (family 066, 2026-09-25): The problem is reduced to a birational threshold contradiction while a bounded lc complement supplies an auxiliary graded section system. Constrained cone valuations retain exact divisor normalization, and two-stage Cartier-trace lifting provides controlled sections whose rank growth would otherwise force a prime with discrepancy below the prescribed gap.

Formal verification

Family 066 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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