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Math-2557 · mathematics · Algebraic geometry

Kähler fourfold abundance after nonvanishing

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a normal connected compact Kähler fourfold with effective rational klt boundary $\Delta $, let $D=K_X+\Delta $ be Q-Cartier and analytically nef. If $\kappa (X,D)\geq 0$, must the actual adjoint have a globally generated positive Cartier multiple? When $\kappa =0$, must some positive multiple be the trivial line?

Short English statement

For a normal connected compact Kähler fourfold with effective rational klt boundary $\Delta $, let $D=K_X+\Delta $ be Q-Cartier and analytically nef. If $\kappa (X,D)\geq 0$, must the actual adjoint have a globally generated positive Cartier multiple? When $\kappa =0$, must some positive multiple be the trivial line?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2557-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2557

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.

Kähler fourfold abundance after nonvanishing

OpenAI's claim

A nef rational adjoint on a compact Kähler klt fourfold is claimed to be semiample once a nonzero section exists, including triviality of a multiple in Iitaka dimension zero. 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 $X$ be a compact complex space, required to be Kähler, normal and connected, of dimension four and let $\Delta\ge0$ be rational, with $(X,\Delta)$ Kawamata log terminal and actual adjoint $D=K_X+\Delta$ rational Cartier. Assume $D$ is analytically nef: after fixing a Cartier multiple and a Kähler form, its curvature has smooth representatives bounded below by $-\varepsilon$ times that form for every $\varepsilon>0$. Assume also $\kappa(X,D)\ge0$, meaning a nontrivial section exists for at least one positive Cartier multiple. OpenAI claims \[ \exists m>0:\ mD\text{ Cartier},\qquad H^0(X,\mathcal O_X(mD))\otimes\mathcal O_X\twoheadrightarrow\mathcal O_X(mD). \] If $\kappa(X,D)=0$, the multiple may be chosen with $\mathcal O_X(mD)\simeq\mathcal O_X$. This theorem requires nonvanishing as a hypothesis.

Outline of the argument

Manuscript 1 (family 034, 2026-09-27): Contractions of threefold null loci and analytic thickenings prepare the boundary components for a fourfold birational reduction. The comparison on floors and lower strata preserves the actual holomorphic line identity, allowing boundary-generated sections to descend to the nef adjoint.

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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