On varieties of maximal Albanese dimension

TL;DR

This paper investigates the structure of morphisms between varieties with maximal Albanese dimension, showing they are closely related to quotients by finite abelian groups and analyzing pluricanonical maps.

Contribution

It extends previous results by characterizing such morphisms as quotients by finite abelian groups and studies the pluricanonical map's role in the Iitaka model.

Findings

Morphisms are birationally equivalent to quotients by finite abelian groups.

The linear series |5K_X| induces the Iitaka model of X.

Provides new insights into the structure of varieties with maximal Albanese dimension.

Abstract

Let $f: X\to Y$ be a surjective morphism of smooth $n$-dimensional projective varieties, with $Y$ of maximal Albanese dimension. Hacon and Pardini studied the structure of $f$ assuming $P_m(X)=P_m(Y)$ for some $m\geq 2$. We extend their result by showing that, under the above assumtions, $f$ is birationally equivalent to a quotient by a finite abelian group. We also study the pluricanonical map of varieties of maximal Albanese dimesnion. The main result is that the linear series $|5K_X|$ incuces the Iitaka model of $X$.