# Pushforwards of pluricanonical bundles under morphisms to abelian varieties

## TL;DR

The paper proves that pluricanonical sheaves on smooth projective varieties mapping to abelian varieties become globally generated after certain isogenies, leading to a decomposition theorem and applications to pluricanonical linear series on irregular varieties.

## Contribution

It establishes new global generation and decomposition results for pluricanonical sheaves under morphisms to abelian varieties, extending previous work for the case m=1.

## Key findings

- Sheaves become globally generated after pullback by an isogeny.
- Decomposition theorem for pluricanonical sheaves when m ≥ 2.
- Effective results for pluricanonical linear series on irregular varieties.

## Abstract

Let $f \colon X \to A$ be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves $f_* \omega_X^{\otimes m}$ become globally generated after pullback by an isogeny. We use this to deduce a decomposition theorem for these sheaves when $m \ge 2$, analogous to that obtained by Chen-Jiang when $m = 1$. This is in turn applied to effective results for pluricanonical linear series on irregular varieties with canonical singularities.