Configurations of Lagrangian spheres in $K3$ surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915401266364416 |
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| author | Muñoz-Echániz, Juan |
| author_facet | Muñoz-Echániz, Juan |
| contents | We study Dehn--Seidel twists on configurations of Lagrangian spheres in symplectic $K3$ surfaces, using tools from Seiberg--Witten theory. In the case of $ADE$ configurations of Lagrangian spheres, we prove that a naturally associated representation of the generalised Braid group in the symplectic mapping class group is always faithful after abelianising, in a suitable sense. More generally, we prove that squared Dehn--Seidel twists on homologically-distinct Lagrangian spheres are algebraically independent in the abelianisation of the smoothly-trivial symplectic mapping class group, and deduce from this new infinite-generation results. Beyond symplectic $K3$ surfaces, we also establish analogues of these results at the level of the fundamental group of the space of symplectic forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Configurations of Lagrangian spheres in $K3$ surfaces Muñoz-Echániz, Juan Geometric Topology Symplectic Geometry We study Dehn--Seidel twists on configurations of Lagrangian spheres in symplectic $K3$ surfaces, using tools from Seiberg--Witten theory. In the case of $ADE$ configurations of Lagrangian spheres, we prove that a naturally associated representation of the generalised Braid group in the symplectic mapping class group is always faithful after abelianising, in a suitable sense. More generally, we prove that squared Dehn--Seidel twists on homologically-distinct Lagrangian spheres are algebraically independent in the abelianisation of the smoothly-trivial symplectic mapping class group, and deduce from this new infinite-generation results. Beyond symplectic $K3$ surfaces, we also establish analogues of these results at the level of the fundamental group of the space of symplectic forms. |
| title | Configurations of Lagrangian spheres in $K3$ surfaces |
| topic | Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2507.15039 |