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Math-2568 · mathematics · Algebraic geometry

Zero Lelong numbers for minimal metrics on nef adjoints

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a normal connected projective complex klt pair $(H,\Theta )$ with effective rational $\Theta $ and nef Q-Cartier $D=K_H+\Theta $, does every semipositive singular Hermitian metric with minimal singularities on $\pi *D$ have zero Lelong number at every point, for every projective log resolution $\pi :V\to H$? Existence of such a minimal metric is included; $K_H$ and $\Theta $ need not separately be Q-Cartier.

Short English statement

For a normal connected projective complex klt pair $(H,\Theta )$ with effective rational $\Theta $ and nef Q-Cartier $D=K_H+\Theta $, does every semipositive singular Hermitian metric with minimal singularities on $\pi *D$ have zero Lelong number at every point, for every projective log resolution $\pi :V\to H$? Existence of such a minimal metric is included; $K_H$ and $\Theta $ need not separately be Q-Cartier.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2568-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2568

Manuscript 1 (family 034, 2026-09-27): 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.

Zero Lelong numbers for minimal metrics on nef adjoints

OpenAI's claim

Minimal semipositive metrics of nef klt adjoints are claimed to have no positive Lelong numbers, and zero-Lelong endpoint metrics are claimed to give the stated interior-boundary $H^1$ injection. 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 every normal connected projective complex klt pair $(H,\Theta)$ with effective rational $\Theta$ and nef rational Cartier $D_H=K_H+\Theta$, and every projective log resolution $\pi:V\to H$, OpenAI claims that $N=\pi^*D_H$ admits a semipositive singular Hermitian metric of minimal singularities and \[ \nu(h,x)=0\quad\text{for every }x\in V \] for every such metric $h$. Here $\nu$ is the Lelong number; minimal singularities means its weights are no more singular, up to bounded additive error, than the weights of any other semipositive metric on $N$. The components $K_H$ and $\Theta$ need not be separately rational Cartier.

Outline of the argument

Manuscript 1 (family 034, 2026-09-27): An extremal normalized-section argument contradicts a positive Lelong number by improving the minimal weight locally. For injectivity, harmonic forms and local primitives are compared with both endpoint weights; weak convergence in a fixed Hilbert space retains the ordinary coherent-cohomology class.

Formal verification

Family 034 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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