Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910454881714176 |
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| author | Amerik, Ekaterina Verbitsky, Misha |
| author_facet | Amerik, Ekaterina Verbitsky, Misha |
| contents | Let $M$ be a holomorphically symplectic complex manifold, not necessarily compact or quasiprojective, and $X \subset M$ a compact Lagrangian submanifold. We construct a deformation to the normal cone, showing that a neighbourhood of $X$ can be deformed to its neighbourhood in $T^*X$. This is used to study Lagrangian submanifolds which can be bimeromorphically contracted to a point. We prove that such submanifolds are biholomorphic to $CP^n$, and show that a certain neighbourhood of $X$ is symplectically biholomorphic to a neighbourhood of the zero section of its cotangent bundle. This gives a holomorphic version of the Weinstein's normal neighbourhood theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04360 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds Amerik, Ekaterina Verbitsky, Misha Algebraic Geometry Complex Variables Symplectic Geometry Let $M$ be a holomorphically symplectic complex manifold, not necessarily compact or quasiprojective, and $X \subset M$ a compact Lagrangian submanifold. We construct a deformation to the normal cone, showing that a neighbourhood of $X$ can be deformed to its neighbourhood in $T^*X$. This is used to study Lagrangian submanifolds which can be bimeromorphically contracted to a point. We prove that such submanifolds are biholomorphic to $CP^n$, and show that a certain neighbourhood of $X$ is symplectically biholomorphic to a neighbourhood of the zero section of its cotangent bundle. This gives a holomorphic version of the Weinstein's normal neighbourhood theorem. |
| title | Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds |
| topic | Algebraic Geometry Complex Variables Symplectic Geometry |
| url | https://arxiv.org/abs/2311.04360 |