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arXiv · 2601.20460

On the triviality of direct image of coherent sheaves

Indranil Biswas·Jagadish Pine
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Abstract

Let $π\,:\, X \,\longrightarrow\, Y$ be a finite morphism of projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for $π$ such that there exists a coherent sheaf $E$ on $X$ whose direct image $π_*E$ 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 $π$. When the covering $π\,:\, X \,\longrightarrow\, Y$ 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.

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Indranil Biswas, Jagadish Pine. 2026-07-20. On the triviality of direct image of coherent sheaves. https://arxiv.org/abs/2601.20460

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