Pushforwards of pluricanonical bundles under morphisms to abelian varieties

Abstract

Let $f:X\to A$ be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves $f_{\ast }{\omega }_X^{\otimes m}$ 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.