# When is the Albanese morphism an algebraic fiber space in positive characteristic?

## TL;DR

This paper investigates conditions under which the Albanese morphism in positive characteristic is an algebraic fiber space, focusing on varieties with nef anti-canonical divisors and $F$-splitting properties, leading to new characterizations of abelian varieties.

## Contribution

It establishes that the Albanese morphism is an algebraic fiber space for varieties with nef anti-canonical divisors and $F$-pure fibers, and explores $F$-splitting properties of these morphisms.

## Key findings

- Albanese morphism is an algebraic fiber space under nef anti-canonical divisor and $F$-purity.
- $F$-split varieties have $F$-split Albanese morphisms.
- New characterization of abelian varieties in positive characteristic.

## Abstract

In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is $F$-pure. Furthermore, we consider a notion of $F$-splitting for morphisms, and investigate it of the Albanese morphisms. We show that an $F$-split variety has $F$-split Albanese morphism, and that the $F$-split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.