On the characterization of abelian varieties for log pairs in zero and positive characteristic

TL;DR

This paper investigates how the condition ppa(K_X + \u03b4)=0 influences the properties of the Albanese map for log pairs in both characteristic 0 and positive characteristic, extending previous results and providing new surjectivity criteria.

Contribution

It generalizes Kawamata's and Hacon-Patakfalvi's results to log pairs in different characteristics and establishes surjectivity of the Albanese morphism for certain threefolds in positive characteristic.

Findings

Generalized known results to log pairs in characteristic 0 and p>0.

Proved surjectivity of the Albanese morphism for specific threefolds in positive characteristic.

Extended the understanding of the Albanese map's behavior under ppa(K_X + \u03b4)=0.

Abstract

Let $(X,\Delta)$ be a pair. We study how the condition $\kappa(K_X + \Delta)=0$ causes surjectivity or birationality of the Albanese map and the Albanese morphism of $X$ in both characteristic $0$ and characteristic $p > 0$. In particular in characteristic $0$ we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic $p>0$ we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if $X$ is a normal projective threefold in characteristic $p>0$, the coefficients of the components of $\Delta$ are $\le 1$ and $-(K_X+\Delta)$ is semiample, then the Albanese morphism of $X$ is surjective under reasonable assumptions on $p$ and the singularities of the general fibers of the Albanese morphism. This is a positive characteristic analog in dimension 3 of a result of Zhang on a conjecture of Demailly-Peternell-Schneider.