On the triviality of the direct image of coherent sheaves

Abstract

Let be a finite morphism of projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for π such that there exists a coherent sheaf E on X whose direct image ⁎ is a trivial vector bundle on Y of positive rank. When X is smooth and Y is Cohen-Macaulay, such a coherent sheaf is necessarily locally free. We show that the existence of such a coherent sheaf E is guided by the properties of the branching divisor of π. When the covering is admissible abelian Galois, we give a complete answer. As an application, it is shown that every smooth admissible abelian Galois covering of supports an Ulrich bundle.