On the triviality of direct image of vector bundles
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915758793031680 |
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| author | Biswas, Indranil Pine, Jagadish |
| author_facet | Biswas, Indranil Pine, Jagadish |
| contents | Let $π\,:\, X \,\longrightarrow\, Y$ 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 $π$ such that there exists a vector bundle $E$ on $X$ whose direct image $π_*E$ is trivial. We show that the existence of $E$ is guided by the properties of the branching divisor of $π$. When the covering $π\,:\, X \,\longrightarrow\, Y$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20460 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the triviality of direct image of vector bundles Biswas, Indranil Pine, Jagadish Algebraic Geometry 14H30, 14H60, 14J60, 14M10 Let $π\,:\, X \,\longrightarrow\, Y$ 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 $π$ such that there exists a vector bundle $E$ on $X$ whose direct image $π_*E$ is trivial. We show that the existence of $E$ is guided by the properties of the branching divisor of $π$. When the covering $π\,:\, X \,\longrightarrow\, Y$ 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. |
| title | On the triviality of direct image of vector bundles |
| topic | Algebraic Geometry 14H30, 14H60, 14J60, 14M10 |
| url | https://arxiv.org/abs/2601.20460 |