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Math-2265 · mathematics · Algebraic geometry

A cyclotomic resolution-Betti obstruction for the global canonical index

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.
Fix a dimension $n$. For projective $n$-dimensional varieties with log-canonical singularities and numerically trivial canonical divisor $K_X$, is the index of $K_X$ bounded by a constant depending only on $n$? The index is the least positive integer $m$ for which $mK_X$ is Cartier and linearly trivial in the stated canonical-index setting.

Formulation references: [S1], [S2].

Short English statement

Fix a dimension $n$. For projective $n$-dimensional varieties with log-canonical singularities and numerically trivial canonical divisor $K_X$, is the index of $K_X$ bounded by a constant depending only on $n$? The index is the least positive integer $m$ for which $mK_X$ is Cartier and linearly trivial in the stated canonical-index setting.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2265-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2265

For an algebraically closed characteristic-zero base, the theorem bounds a common trivializing multiple of the adjoint for semi-log-canonical pairs of dimension at least four whose adjoint is already rationally linearly trivial. With zero boundary it covers normal log-canonical varieties in that class. This page assumes numerical triviality, without spelling out a characteristic or a dimension lower bound. Those conventions and the implication from numerical to rational linear triviality must be supplied before claiming full equivalence; the record retains partial coverage.

A cyclotomic resolution-Betti obstruction for the global canonical index

OpenAI's claim

A dimension- and coefficient-dependent multiple is claimed to trivialize the log canonical line of every connected projective semi-log-canonical Calabi--Yau pair, independently of its number of components. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $k$ have characteristic zero and be algebraically closed; over $k$, fix $d\ge4$ and a finite $\Phi\subset[0,1]\cap\mathbb Q$. OpenAI asserts an integer $a=a(d,\Phi)>0$ such that every connected projective equidimensional semi-log-canonical pair $(X,B)$ of dimension $d$, with nonzero boundary coefficients in $\Phi$ and $K_X+B\sim_{\mathbb Q}0$, satisfies \[ a(K_X+B)\text{ is Cartier},\qquad \mathcal O_X(a(K_X+B))\simeq\mathcal O_X. \] Semi-log-canonical means $X$ is reduced, $S_2$ and nodal in codimension one, $B$ avoids its nonnormal locus, and the normalization with the conductor boundary is log canonical. The hypothesis is rational linear triviality, not merely numerical triviality.

Outline of the argument

Manuscript 1 (family 034, 2026-10-05): The normal components first receive a uniform pluricanonical exponent. Residues along conductor strata turn descent into a scalar-gluing problem, whose possible characters are bounded using Hodge-norm rank estimates and moving-subvariety fibrations; this produces a common exponent for the glued pair.

Formal verification

Family 034 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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