On the triviality of direct image of coherent sheaves

Let $\pi\,:\, X \,\longrightarrow\, Y$ be a finite morphism of 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 coherent sheaf $E$ on $X$ whose direct image $\pi_*E$ 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 $\pi$.

When the covering $\pi\,:\, X \,\longrightarrow\, Y$ is admissible abelian Galois, we give a complete answer.

As an application, it is shown that every smooth admissible abelian Galois covering of $\mathbb{P}^n$ supports an Ulrich bundle.