On the triviality of direct image of vector bundles

TL;DR

This paper characterizes when the direct image of a vector bundle under a finite morphism is trivial, linking it to the branching divisor, and shows all ramified abelian Galois covers of projective space support Ulrich bundles.

Contribution

It provides necessary and sufficient conditions for the triviality of direct images of vector bundles under finite morphisms, especially for ramified abelian Galois covers.

Findings

Criteria based on branching divisor properties for trivial direct images

Complete characterization for ramified abelian Galois coverings

Existence of Ulrich bundles on all such covers of projective space

Abstract

Let $\pi\,:\, X \,\longrightarrow\, Y$ be a finite morphism of smooth projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for $\pi$ such that there exists a vector bundle $E$ on $X$ whose direct image $\pi_*E$ is trivial. We show that the existence of $E$ is guided by the properties of the branching divisor of $\pi$. When the covering $\pi\,:\, X \,\longrightarrow\, Y$ is ramified abelian Galois, we give a complete answer. As an application, we prove every smooth ramified abelian Galois covering of $\mathbb{P}^n$ supports an Ulrich bundle.