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Math-2559 · mathematics · Algebraic geometry

Numerical semiampleness of Kähler adjoints

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let X be smooth connected compact Kähler, B effective rational SNC with coefficients below one, and M a nef rational holomorphic line. If $K_X+B$ is pseudo-effective and $D=K_X+B+M$ is nef, must some semiample rational holomorphic line L have $c_{1}(L)=c_{1}(D)$ in real Bott–Chern cohomology? Actual semiampleness of D is not required.

Short English statement

Let X be smooth connected compact Kähler, B effective rational SNC with coefficients below one, and M a nef rational holomorphic line. If $K_X+B$ is pseudo-effective and $D=K_X+B+M$ is nef, must some semiample rational holomorphic line L have $c_{1}(L)=c_{1}(D)$ in real Bott–Chern cohomology? Actual semiampleness of D is not required.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2559-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2559

Manuscript 1 (family 036, 2026-10-04): 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.

Numerical semiampleness of Kähler adjoints

OpenAI's claim

On the stated compact Kähler pairs, a nef adjoint with an additional nef rational line is claimed to have a semiample representative of its Bott--Chern class, allowing a flat change of bundle. 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 smooth connected compact Kähler, $B\ge0$ rational simple-normal-crossing with coefficients below one, and $M\in\operatorname{Pic}(X)\otimes\mathbb Q$. Assume pseudo-effectivity of $K_X+B$, and nefness of both $M$ and $D=K_X+B+M$, OpenAI claims \[ \exists\text{ semiample rational line bundle }L:\quad c_1(L)=c_1(D)\text{ in }H^{1,1}_{\mathrm{BC}}(X,\mathbb R). \] Nef means the Chern class lies in the closed Kähler cone; pseudo-effective means it contains a positive current. Semiampleness means an actual positive integral multiple is globally generated. Equality is of Bott--Chern classes, not of line bundles.

Outline of the argument

Manuscript 1 (family 036, 2026-10-04): The induction uses projective positive-part results when algebraic reduction has positive-dimensional image. In algebraic dimension zero, decomposition into torus and generically symplectic pieces, extension estimates, and compact cycle spaces provide the numerical representative while preserving the distinction from an actual line identity.

Formal verification

Family 036 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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