Math-2585 · mathematics · Algebraic geometry

Anticanonical nonvanishing from smooth semipositivity

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.

For every smooth connected projective complex X, if $-K_X$ admits a smooth Hermitian metric with semipositive Chern curvature, must $H^{0}(X,-mK_X)\neq 0$ for some positive integer m?

Short English statement

For every smooth connected projective complex X, if $-K_X$ admits a smooth Hermitian metric with semipositive Chern curvature, must $H^{0}(X,-mK_X)\neq 0$ for some positive integer m?

Sources

S1 Original question source or mathematical context. https://doi.org/10.1007/s40574-020-00250-9.
S2 Original question source or mathematical context. https://doi.org/10.4171/owr/2006/40.
S3 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/README.md.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2585-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for Math-2585

Manuscript 1 (family 068, 2026-09-26): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Anticanonical nonvanishing from smooth semipositivity

OpenAI's claim

Smooth semipositivity of the anticanonical bundle is claimed to force a positive-degree anticanonical section on every smooth complex projective variety. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

For every smooth connected projective complex variety $X$ whose anticanonical line has a smooth Hermitian metric with nonnegative Chern curvature, OpenAI claims \[ \exists m\in\mathbb Z_{>0}:\quad H^0(X,\mathcal O_X(-mK_X))\ne0. \] No Euler-characteristic or rational-connectedness assumption occurs in this headline theorem. Smooth semipositivity is stronger than an assumed nef numerical class.

Outline of the argument

Manuscript 1 (family 068, 2026-09-26): Section ratios define a rational base on which the generic relative adjoint spaces stay one-dimensional. Bergman positivity transfers the specified line, while compact-monodromy structure requires an invariant section on the remaining factor; multiplication with the flat canonical frames and a finite étale norm descend the resulting section.

Formal verification

Family 068 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

OpenAI, Anticanonical nonvanishing from smooth semipositivity, 2026-09-26; citation key OAI:Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:headline.
OpenAI catalogue entry.

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Record details

Subjects: Algebraic geometry
Collection: Top Open Problems
Mathematics rank: 2585