On algebraic vector bundles of rank $2$ over smooth affine fourfolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916867409444864 |
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| author | Brazelton, Thomas Opie, Morgan Syed, Tariq |
| author_facet | Brazelton, Thomas Opie, Morgan Syed, Tariq |
| contents | The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds is a notoriously difficult problem. Isomorphism classes of such vector bundles are not uniquely determined by their Chern classes, in contrast to the situation in lower dimensions. Given a smooth affine fourfold over an algebraically closed field of characteristic not equal to $2$ or $3$, we study cohomological criteria for finiteness of the fibers of the Chern class map for rank $2$ bundles. As a consequence, we give a cohomological classification of such bundles in a number of cases. For example, if $d\leq 4$, there are precisely $d^2$ non-isomorphic algebraic vector bundles over the complement of a smooth hypersurface of degree $d$ in $\mathbb P^4_{\mathbb C}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_21029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On algebraic vector bundles of rank $2$ over smooth affine fourfolds Brazelton, Thomas Opie, Morgan Syed, Tariq Algebraic Geometry K-Theory and Homology 14F42, 14F45, 14C17, 14C35, 55S35, 13C10, 19A13 The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds is a notoriously difficult problem. Isomorphism classes of such vector bundles are not uniquely determined by their Chern classes, in contrast to the situation in lower dimensions. Given a smooth affine fourfold over an algebraically closed field of characteristic not equal to $2$ or $3$, we study cohomological criteria for finiteness of the fibers of the Chern class map for rank $2$ bundles. As a consequence, we give a cohomological classification of such bundles in a number of cases. For example, if $d\leq 4$, there are precisely $d^2$ non-isomorphic algebraic vector bundles over the complement of a smooth hypersurface of degree $d$ in $\mathbb P^4_{\mathbb C}$. |
| title | On algebraic vector bundles of rank $2$ over smooth affine fourfolds |
| topic | Algebraic Geometry K-Theory and Homology 14F42, 14F45, 14C17, 14C35, 55S35, 13C10, 19A13 |
| url | https://arxiv.org/abs/2507.21029 |