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Math-2558 · mathematics · Algebraic geometry

B-semiampleness for compact Kähler log-smooth fibrations

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let $f:Y\to X$ be surjective with connected fibers between smooth connected compact Kähler manifolds. Let $\Delta $ be effective rational SNC with coefficients in [0,1] and $K_Y+\Delta $ Q-linearly equal to f*L for $L\in Pic(X)\otimes Q$. Is the canonical-bundle-formula moduli rational line b-semiample: on some smooth Kähler modification $S\to X$, does it become a globally generated line after a positive integral multiple, and pull back as the same rational line on every subsequent smooth modification?

Short English statement

Let $f:Y\to X$ be surjective with connected fibers between smooth connected compact Kähler manifolds. Let $\Delta $ be effective rational SNC with coefficients in [0,1] and $K_Y+\Delta $ Q-linearly equal to f*L for $L\in Pic(X)\otimes Q$. Is the canonical-bundle-formula moduli rational line b-semiample: on some smooth Kähler modification $S\to X$, does it become a globally generated line after a positive integral multiple, and pull back as the same rational line on every subsequent smooth modification?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2558-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2558

Manuscript 1 (family 033, 2026-09-10): 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.

B-semiampleness for compact Kähler log-smooth fibrations

OpenAI's claim

The threshold-moduli part of a compact log-smooth Kähler fibration is claimed to become globally generated on a suitable modification, even with horizontal coefficient-one divisors. 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 $f:Y\to X$ be a connected-fiber surjective holomorphic map of smooth compact connected Kähler manifolds, with effective rational simple-normal-crossing $\Delta$ of coefficients in $[0,1]$, and $K_Y+\Delta\sim_{\mathbb Q}f^*L$ for $L\in\operatorname{Pic}(X)_{\mathbb Q}$. On a smooth compact Kähler modification $\mu:X'\to X$, resolve the main fiber product to $f':Y'\to X'$ and $h:Y'\to Y$, and set $\Delta'=h^*\Delta-K_{Y'/Y}$. Define \[ t_P=\inf_{E\mapsto P}\frac{a(E;Y',\Delta')}{\operatorname{ord}_E((f')^*P)},\quad B_{X'}=\sum_P(1-t_P)P,\quad M_{X'}=\mu^*L-K_{X'}-B_{X'}. \] The infimum is over divisors dominating $P$ with positive denominator. OpenAI claims a smooth compact Kähler modification $S\to X$ for which every further smooth compact Kähler $\nu:S_1\to S$ has $M_{S_1}=\nu^*M_S$ in rational Picard groups and some actual positive integral multiple of $M_S$ is globally generated. Point bases and relative dimension zero are included, and no projectivity assumption on the spaces or morphism is made.

Outline of the argument

Manuscript 1 (family 033, 2026-09-10): Cyclic roots and iterated residues identify the relevant pure Hodge line with a deepest boundary stratum. Compact Douady spaces parameterize product covers of the fibers, and norm estimates extend the precise line identity across the boundary; finite analytic norms then descend one generating exponent.

Formal verification

For family 033, the scope page documents the following selections. Only the case of a negative-infinite logarithmic fiber dimension is selected; it forces negative-infinite dimension upstairs and vanishing of positive pluriforms. Finite-dimensional fibers and the negative-base case remain outside the linked statement. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion.

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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