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Math-2325 · mathematics · Algebraic geometry

Nonvanishing or Uniruledness for Smooth Projective Varieties

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Does every smooth projective variety $X$ either have a nonzero pluricanonical section in $H^0(X,\mathcal O_X(mK_X))$ for some $m>0$, or is it covered by rational curves?

Formulation references: [S1], [S2].

Short English statement

Does every smooth projective variety $X$ either have a nonzero pluricanonical section in $H^0(X,\mathcal O_X(mK_X))$ for some $m>0$, or is it covered by rational curves?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2325-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2325

Manuscript 1 (family 034, 2026-09-27): For smooth projective complex fourfolds with pseudo-effective canonical class the primary theorem supplies a pluricanonical section. The complementary uniruled fourfold branch is the standard pseudo-effectivity/uniruledness alternative, but the problem asks every dimension; this paper only provides the missing dimension-four nonvanishing branch.

Nonvanishing or Uniruledness for Smooth Projective Varieties

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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