Splitting vector bundles over real algebraic varieties
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| Format: | Preprint |
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2025
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| author | Asok, Aravind Fasel, Jean Lerbet, Samuel |
| author_facet | Asok, Aravind Fasel, Jean Lerbet, Samuel |
| contents | Suppose $X$ is a smooth affine real variety and $\mathscr{E}$ is a vector bundle over $X$. We analyze the problem of splitting off a free rank one summand from $\mathscr{E}$ in corank $0$ and $1$. The problem in corank $0$ can be viewed as the search for a real analog of Murthy's celebrating splitting theorem in the algebraically closed case: to wit, beyond the vanishing of the top Chern class in Chow theory, are the obstructions to splitting ``purely topological''? In a sense, the answer in this case is yes, and we give a proof, using motivic techniques, of a mild extension of the results of Bhatwadekar-Sridharan and Bhatwadekar-Das-Mandal.
In corank $1$, in the algebraically closed situation, Murthy's splitting conjecture (now a theorem in characteristic $0$) predicts that the vanishing of the top Chern class in Chow theory is the only obstruction to splitting off a free rank $1$ summand, and we can search for a suitable ``real'' analog of this assertion. We observe that several natural guesses for a ``real'' analog of Murthy's splitting conjecture cannot be true, i.e., that the situation over the real numbers is rather complicated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_15616 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splitting vector bundles over real algebraic varieties Asok, Aravind Fasel, Jean Lerbet, Samuel Algebraic Geometry Commutative Algebra Algebraic Topology K-Theory and Homology 14F42 (Primary) 14F25, 13C10, 55R25 (Secondary) Suppose $X$ is a smooth affine real variety and $\mathscr{E}$ is a vector bundle over $X$. We analyze the problem of splitting off a free rank one summand from $\mathscr{E}$ in corank $0$ and $1$. The problem in corank $0$ can be viewed as the search for a real analog of Murthy's celebrating splitting theorem in the algebraically closed case: to wit, beyond the vanishing of the top Chern class in Chow theory, are the obstructions to splitting ``purely topological''? In a sense, the answer in this case is yes, and we give a proof, using motivic techniques, of a mild extension of the results of Bhatwadekar-Sridharan and Bhatwadekar-Das-Mandal. In corank $1$, in the algebraically closed situation, Murthy's splitting conjecture (now a theorem in characteristic $0$) predicts that the vanishing of the top Chern class in Chow theory is the only obstruction to splitting off a free rank $1$ summand, and we can search for a suitable ``real'' analog of this assertion. We observe that several natural guesses for a ``real'' analog of Murthy's splitting conjecture cannot be true, i.e., that the situation over the real numbers is rather complicated. |
| title | Splitting vector bundles over real algebraic varieties |
| topic | Algebraic Geometry Commutative Algebra Algebraic Topology K-Theory and Homology 14F42 (Primary) 14F25, 13C10, 55R25 (Secondary) |
| url | https://arxiv.org/abs/2511.15616 |