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Math-2539 · mathematics · Algebraic geometry

Canonical nonvanishing for smooth fourfolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If X is a smooth connected projective complex fourfold and $K_X$ is pseudo-effective, must $H^{0}(X,mK_X)$ be nonzero for at least one positive integer m?

Short English statement

If X is a smooth connected projective complex fourfold and $K_X$ is pseudo-effective, must $H^{0}(X,mK_X)$ be nonzero for at least one positive integer m?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2539-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2539

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.

Canonical nonvanishing for smooth fourfolds

OpenAI's claim

A smooth projective complex fourfold with pseudo-effective canonical divisor is claimed to possess a nonzero pluricanonical section. 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 smooth connected projective complex fourfold $X$ with pseudo-effective canonical class, OpenAI claims \[ \exists m\in\mathbb Z_{>0}:\quad H^0(X,\mathcal O_X(mK_X))\ne0. \] Pseudo-effective means lying in the closure of the cone of effective divisor classes. This is nonvanishing in dimension four; it does not by itself assert semiampleness or nonvanishing in every dimension. The cited companion metric input says that for a projective complex pair that is normal, connected and klt $(H,\Theta)$ with effective rational boundary and nef rational Cartier $D_H=K_H+\Theta$, every projective log resolution $\pi:V\to H$ gives a rational line bundle $\pi^*D_H$ with a semipositive singular metric having minimal singularities; every metric satisfying this condition has vanishing Lelong numbers at all points and trivial multiplier ideals at every positive exponent.

Outline of the argument

Manuscript 1 (family 034, 2026-09-27): Minimal metrics and moving centers reduce the problem to compatible jet-order and rank estimates on two factors. Frobenius diagonal filtrations convert ordinary jets to truncated monomials, while instability control keeps the numerical separation needed to force nonvanishing.

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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