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Math-2542 · mathematics · Algebraic geometry

Kähler log abundance assuming logarithmic Iitaka

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assume that for every connected-fiber surjective morphism $f:U\to V$ of smooth connected projective complex varieties with reduced effective SNC $D_U,D_V$ and $Supp(f*D_V)\subset Supp(D_U)$, the very general fiber F with $D_F=D_U|F$ satisfies $\kappa (U,K_U+D_U)\geq \kappa (V,K_V+D_V)+\kappa (F,K_F+D_F)$, taking $-\infty $ as absorbing. For every normal irreducible compact Kähler space X with an effective rational lc boundary $\Delta $ and Q-Cartier analytically nef $K_X+\Delta $, must some positive Cartier multiple of the actual adjoint line be globally generated?

Short English statement

Assume that for every connected-fiber surjective morphism $f:U\to V$ of smooth connected projective complex varieties with reduced effective SNC $D_U,D_V$ and $Supp(f*D_V)\subset Supp(D_U)$, the very general fiber F with $D_F=D_U|F$ satisfies $\kappa (U,K_U+D_U)\geq \kappa (V,K_V+D_V)+\kappa (F,K_F+D_F)$, taking $-\infty $ as absorbing. For every normal irreducible compact Kähler space X with an effective rational lc boundary $\Delta $ and Q-Cartier analytically nef $K_X+\Delta $, must some positive Cartier multiple of the actual adjoint line be globally generated?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2542-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2542

Manuscript 1 (family 034, 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.

Kähler log abundance assuming logarithmic Iitaka

OpenAI's claim

Assuming the stated logarithmic Iitaka inequality, OpenAI claims semiampleness of nef rational log adjoints on compact Kähler log-canonical spaces in arbitrary dimension. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Assume logarithmic Iitaka subadditivity for every connected-fiber surjective morphism $f:U\to V$ of smooth connected projective complex varieties with reduced simple-normal-crossing divisors $D_U,D_V$ satisfying $\operatorname{Supp}(f^*D_V)\subseteq\operatorname{Supp}D_U$: for a very general smooth fiber $F$ and $D_F=D_U|_F$, \[ \kappa(U,K_U+D_U)\ge\kappa(F,K_F+D_F)+\kappa(V,K_V+D_V). \] Under that premise, OpenAI claims that every normal irreducible compact Kähler complex space $X$ with effective rational $\Delta$, log canonical pair $(X,\Delta)$ and analytically nef rational Cartier $D=K_X+\Delta$ has a positive Cartier multiple generated everywhere by its global sections: \[ H^0(X,\mathcal O_X(mD))\otimes\mathcal O_X\twoheadrightarrow\mathcal O_X(mD). \] The premise covers all dimensions, uses $\kappa=-\infty$ for empty positive systems, and is a hypothesis of this Kähler result.

Outline of the argument

Manuscript 1 (family 034, 2026-10-04): The induction retains both the nef line and the effective negative part through a restricted birational program. Boundary generation, signed-support root constructions, Hodge-module lifting, and compact cycle deformations produce nonvanishing; affine changes in negative multiplicities make the selected wall crossings finite.

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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