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Math-2584 · mathematics · Algebraic geometry

Invariant anticanonical sections with nonzero Euler characteristic

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let X be smooth connected projective complex with smoothly semipositive $-K_X$ and $\chi (X,O_X)\neq 0$. For any algebraic torus T acting on X with the natural anticanonical linearization, must some positive anticanonical multiple have a nonzero T-invariant global section?

Short English statement

Let X be smooth connected projective complex with smoothly semipositive $-K_X$ and $\chi (X,O_X)\neq 0$. For any algebraic torus T acting on X with the natural anticanonical linearization, must some positive anticanonical multiple have a nonzero T-invariant global section?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2584-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2584

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.

Invariant anticanonical sections with nonzero Euler characteristic

OpenAI's claim

Invariant sections in unbounded anticanonical degrees with a fixed negative pseudo-effective error are claimed to produce a positive-degree untwisted invariant section, yielding smooth semipositive anticanonical nonvanishing. 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 smooth connected projective complex $X$ with smoothly semipositive $L=-K_X$, an algebraic torus $T$ acting on $X$, and $\chi(X,\mathcal O_X)\ne0$, OpenAI claims \[ \exists m>0:\quad H^0(X,mL)^T\ne0 \] for the natural anticanonical linearization. The torus may be trivial. The Euler-characteristic condition replaces the stronger higher-cohomology vanishing used in a related manuscript.

Outline of the argument

Manuscript 1 (family 068, 2026-09-26): Character-sensitive transfer chooses a base from invariant section systems and tracks exact direct-image lines. In the separate harmonic construction, approximation of the pulled-back base metric forces base canonical covectors into a weak-limit representative, making restriction to a general fiber detect its class; compact-monodromy descent then returns sections to the original variety.

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

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