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Math-2534-A1 · revision r1 · on Math-2534

OpenAI claim summary for Math-2534

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Bounded-metric anticanonical nonvanishing for klt pairs
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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