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Math-2579 · mathematics · Algebraic geometry

Schnell’s fiber-space assertion in Kodaira dimension zero

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a connected-fiber surjective morphism $f:X\to Y$ of smooth connected projective complex varieties with smooth geometric generic fiber F of $\kappa (F)=0$, suppose $m_{0}K_X-f*H$ is pseudo-effective for some $m_{0}>0$ and ample Cartier H on Y. Must $\kappa (X)=dim$ Y?

Short English statement

For a connected-fiber surjective morphism $f:X\to Y$ of smooth connected projective complex varieties with smooth geometric generic fiber F of $\kappa (F)=0$, suppose $m_{0}K_X-f*H$ is pseudo-effective for some $m_{0}>0$ and ample Cartier H on Y. Must $\kappa (X)=dim$ Y?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2579-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2579

Manuscript 1 (family 034, 2026-09-24): 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.

Schnell’s fiber-space assertion in Kodaira dimension zero

OpenAI's claim

The Campana--Peternell Kodaira inequality and the specified Schnell fiber-space conclusion are claimed from a pseudo-effective difference between a canonical multiple and the chosen effective or pulled-back divisor. 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 $f:X\to Y$ be a connected-fiber surjective morphism of smooth connected projective complex varieties, with smooth geometric generic fiber $F$ satisfying $\kappa(F)=0$. If some ample Cartier divisor $H$ on $Y$ and integer $m_0>0$ have $m_0K_X-f^*H$ pseudo-effective, OpenAI claims \[ \kappa(X)=\dim Y. \] The manuscript also invokes its companion's canonical good-model theorem: every smooth connected projective complex $T$ with pseudo-effective $K_T$ has a $K_T$-negative birational contraction to a normal projective rational-factorial klt $V$ with $K_V$ semiample, and on a common smooth resolution $p:W\to T$, $q:W\to V$, \[ p^*K_T=q^*K_V+E,\qquad E\ge0,\quad q_*E=0. \] The good-model statement is explicitly credited to the companion theorem, not presented as an independent proof here.

Outline of the argument

Manuscript 1 (family 034, 2026-09-24): The total-space canonical model, with its good-model property, supplies the geometric reduction. Cartier intersection theory and proper pushforward compare the base with the canonical image without requiring either map to factor through the other; transfer of sections then yields the Kodaira and fiber formulas.

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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