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Math-2565 · mathematics · Algebraic geometry

Rational effectivity modulo a controlled boundary

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let Y be smooth projective complex of positive dimension and C an effective integral divisor. If $-K_Y+C$ has a singular Hermitian metric whose curvature dominates a Kähler form and whose multiplier ideal contains $O_Y(-C)$, must every pseudo-effective rational divisor M admit an effective rational correction $C_{0}$ supported on C such that $M+C_{0}$ is Q-linearly equal to an effective rational divisor?

Short English statement

Let Y be smooth projective complex of positive dimension and C an effective integral divisor. If $-K_Y+C$ has a singular Hermitian metric whose curvature dominates a Kähler form and whose multiplier ideal contains $O_Y(-C)$, must every pseudo-effective rational divisor M admit an effective rational correction $C_{0}$ supported on C such that $M+C_{0}$ is Q-linearly equal to an effective rational divisor?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2565-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2565

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.

Rational effectivity modulo a controlled boundary

OpenAI's claim

A globally strict anticanonical metric with controlled boundary integrability is claimed to make every pseudo-effective rational divisor effective after a correction on that boundary, and to remove fixed errors from unbounded anticanonical twists. 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 $Y$ be a complex projective variety that is smooth and has positive dimension and $C\ge0$ integral. Assume $-K_Y+C$ carries a singular Hermitian metric $h$ with curvature at least a Kähler form and whose multiplier ideal satisfies $\mathcal J(h)\supset\mathcal O_Y(-C)$. OpenAI claims \[ \forall\text{ pseudo-effective rational }M\ \exists C_0\ge0\text{ supported on }C:\quad M+C_0\sim_{\mathbb Q}E\ge0. \] The multiplier condition means local squared integrability of the canonical section of $C$ in the metric. A separate conversion theorem says that if $X$ is smooth connected projective complex, $L=-K_X$ smoothly semipositive and $P$ pseudo-effective Cartier, then effective integral $N_j\sim m_jL-P$ for unbounded positive $m_j$ imply $H^0(X,mL)\ne0$ for some $m>0$.

Outline of the argument

Manuscript 1 (family 068, 2026-09-26): Singular adjoint direct-image positivity descends a strict metric with an exceptional integral boundary and trivial total-space multiplier ideal. Controlled section rings and a single-Iitaka conversion return the corrected sections upstairs; a supported Fano-type base model supplies the required effectivity without losing the disappearing-prime condition.

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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