The smooth Mordell-Weil group and mapping class groups of elliptic surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909713079205888 |
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| author | Farb, Benson Looijenga, Eduard |
| author_facet | Farb, Benson Looijenga, Eduard |
| contents | This is a paper in smooth $4$-manifold topology, inspired by the Néron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(π)$ of Mordell-Weil group of an elliptic fibration $π:M\to\Pb^1$ is finitely generated. We compute $\MW(π_d)$ explicitly for elliptic fibrations $π_d:M_d\to\Pb^1$, where $M_d$ is a simply-connected complex surfaces $M_d$ of arithmetic genus $d\geq 1$ and all fibers of $π_d$ are nodal.
We prove in this case that the fibered structure is unique up topological isotopy. By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of $M_d$ with $d\geq 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15960 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The smooth Mordell-Weil group and mapping class groups of elliptic surfaces Farb, Benson Looijenga, Eduard Geometric Topology Algebraic Geometry Number Theory This is a paper in smooth $4$-manifold topology, inspired by the Néron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(π)$ of Mordell-Weil group of an elliptic fibration $π:M\to\Pb^1$ is finitely generated. We compute $\MW(π_d)$ explicitly for elliptic fibrations $π_d:M_d\to\Pb^1$, where $M_d$ is a simply-connected complex surfaces $M_d$ of arithmetic genus $d\geq 1$ and all fibers of $π_d$ are nodal. We prove in this case that the fibered structure is unique up topological isotopy. By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of $M_d$ with $d\geq 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers. |
| title | The smooth Mordell-Weil group and mapping class groups of elliptic surfaces |
| topic | Geometric Topology Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2403.15960 |