On canonical bundle formula for fibrations of curves with arithmetic genus one

TL;DR

This paper establishes canonical bundle formulas for fibrations of curves with arithmetic genus one in characteristic p>0, addressing both separable and inseparable cases, and applies results to classify certain fiber spaces.

Contribution

It develops canonical bundle formulas for genus one fibrations in positive characteristic, including inseparable cases, extending previous formulas and applications.

Findings

Canonical bundle formulas are obtained for separable fibrations.

Inseparable fibrations with maximal Albanese dimension are treated.

Applications include classification of fiber spaces over abelian varieties.

Abstract

In this paper, we develop canonical bundle formulas for fibrations of relative dimension one in characteristic $p>0$. For such a fibration from a log pair $f\colon (X, \Delta) \to S$, if $f$ is separable, we can obtain a formula similar to the one due to Witaszek \cite{Wit21}; if $f$ is inseparable, we treat the case when $S$ is of maximal Albanese dimension. As an application, we prove that for a klt pair $(X,\Delta)$ with $-(K_X+\Delta)$ nef, if the Albanese morphism $a_X\colon X \to A$ is of relative dimension one, then $X$ is a fiber space over $A$.