Pushforwards of klt pairs under morphisms to abelian varieties

TL;DR

The paper proves that for morphisms from klt pairs to abelian varieties, certain pushforward sheaves become globally generated after an isogeny and admit Chen-Jiang decompositions, leading to effective results on irregular varieties.

Contribution

It establishes the global generation and Chen-Jiang decomposition of pushforward sheaves under morphisms from klt pairs to abelian varieties, with applications to irregular varieties.

Findings

Sheaves become globally generated after pullback by an isogeny.

Pushforward sheaves admit Chen-Jiang decomposition.

Results lead to effective criteria for line bundles on irregular varieties.

Abstract

Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{\mathbb Q}m(K_X+\Delta)$. We prove that the sheaf $f_*\mathcal{O}_X(D)$ becomes globally generated after pullback by an isogeny and has the Chen-Jiang decomposition, along with some related results. These are applied to some effective results for $\mathcal{O}_X(D)$ when $X$ is irregular.