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Math-2583 · mathematics · Algebraic geometry

Invariant anticanonical sections under structure-sheaf acyclicity

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a smooth connected projective complex X with smoothly semipositive $-K_X$ and $H^i(X,O_X)=0$ for all $i>0$, must every algebraic torus action admit a nonzero invariant section of some positive anticanonical power with its natural linearization?

Short English statement

For a smooth connected projective complex X with smoothly semipositive $-K_X$ and $H^i(X,O_X)=0$ for all $i>0$, must every algebraic torus action admit a nonzero invariant section of some positive anticanonical power with its natural linearization?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2583-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2583

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 under structure-sheaf acyclicity

OpenAI's claim

Asymptotically zero curvature-null valuation for a smoothly semipositive anticanonical line is claimed to be attained exactly by an effective rational anticanonical divisor. 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 $X$ be smooth connected projective complex, with a smooth semipositive anticanonical metric and $H^i(X,\mathcal O_X)=0$ for every $i>0$. Given an algebraic torus action by $T$ on $X$, use its differential linearization on $-K_X$, OpenAI claims \[ \exists m>0:\quad H^0(X,\mathcal O_X(-mK_X))^T\ne0. \] The section is invariant under the natural action, not merely an arbitrary character eigensection; smooth semipositivity and the cohomological vanishings are retained.

Outline of the argument

Manuscript 1 (family 068, 2026-09-26): In a finite bigraded ring, sections approaching a boundary ray use off-ray generators in negligible proportion. A boundary generator with suitably small normalized valuation is selected before the exceptional correction is pushed away; centeredness upgrades the resulting upper bound to exact order zero.

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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