Pushforwards of lc pairs under morphisms to abelian varieties

Abstract

Let $f$ be a morphism from a log canonical pair $(X,Δ)$ to an abelian variety $A$ and $D\sim_{\mathbb Q}m(K_X+Δ)$ a Cartier divisor on $X$ with $m>1$ a rational number. We prove that $f_*\mathcal{O}_X(D)$ is globally generated after pullback by an isogeny. Our methods involve birational geometry, mixed Hodge modules and singular Hermitian metrics with the minimal extension property. Moreover, we provide simple counterexamples where the above result fails for plt pairs when $m=1$ and where the existence of Chen--Jiang decompositions fails for plt pairs for any $m\ge1$.