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Math-2534 · mathematics · Algebraic geometry

Bounded-metric anticanonical nonvanishing for klt pairs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let (W,D) be projective Q-factorial klt with effective rational boundary and nef rational Cartier $P=-(K_W+D)$. If on a smooth projective resolution $\rho $ the pullback $\rho *P$ has a semipositive singular metric with locally bounded weights and $\chi (O_{Wtilde})\neq 0$, must some positive Cartier multiple mP have a nonzero global section?

Short English statement

Let (W,D) be projective Q-factorial klt with effective rational boundary and nef rational Cartier $P=-(K_W+D)$. If on a smooth projective resolution $\rho $ the pullback $\rho *P$ has a semipositive singular metric with locally bounded weights and $\chi (O_{Wtilde})\neq 0$, must some positive Cartier multiple mP have a nonzero global section?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2534-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2534

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.

Bounded-metric anticanonical nonvanishing for klt pairs

OpenAI's claim

A nef anti-log-canonical divisor with bounded semipositive metric is claimed to acquire a section when a resolution has nonzero structure-sheaf Euler characteristic; the torus-quotient application gives naturally invariant sections. 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 a projective rational-factorial klt complex pair $(W,D)$ with effective rational boundary, set $P=-(K_W+D)$ and assume $P$ nef. Require a smooth projective resolution $\rho:\widetilde W\to W$ on which $\rho^*P$ has a semipositive singular Hermitian metric with locally bounded weights, and \[ \chi(\widetilde W,\mathcal O_{\widetilde W})\ne0. \] OpenAI claims a positive Cartier multiple with $H^0(W,\mathcal O_W(mP))\ne0$. Locally bounded weights need not be continuous. Nefness alone is not the stated hypothesis; the bounded metric and Euler-characteristic condition are retained.

Outline of the argument

Manuscript 1 (family 068, 2026-09-26): The nonzero Euler characteristic first supplies twisted differentials, and cotangent-subsheaf and birational arguments convert them into the specified line. A stable projective torus quotient is built with the same bounded-metric hypotheses, so its section pulls back as an invariant anticanonical plurisection.

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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