Shafarevich-Tate groups of holomorphic Lagrangian fibrations
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912736573652992 |
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| author | Abasheva, Anna Rogov, Vasily |
| author_facet | Abasheva, Anna Rogov, Vasily |
| contents | Consider a Lagrangian fibration $π\colon X\to \mathbb P^n$ on a hyperkähler manifold $X$. There are two ways to construct a holomorphic family of deformations of $π$ over $\mathbb C$. The first one is known under the name Shafarevich-Tate family while the second one is the degenerate twistor family constructed by Verbitsky. We show that both families coincide. We prove that for a very general $X$ all members of the Shafarevich-Tate family are Kähler. There is a related notion of the Shafarevich-Tate group associated to a Lagrangian fibration. Its connected component of unity can be shown to be isomorphic to $\mathbb C/Λ$ where $Λ$ is a finitely generated subgroup of $\mathbb C$ and $\mathbb C$ is thought of as the base of the Shafarevich-Tate family. We show that for a very general $X$, projective deformations in the Shafarevich-Tate family correspond to the torsion points in the connected component of unity of the Shafarevich-Tate group. A sufficient condition for a Lagrangian fibration $X$ to be projective is existence of a holomorphic section. We find sufficient cohomological conditions for existence of a deformation in the Shafarevich-Tate family that admits a section. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_10921 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Shafarevich-Tate groups of holomorphic Lagrangian fibrations Abasheva, Anna Rogov, Vasily Algebraic Geometry Complex Variables 14J42, 14F25, 32Q15, 32J25 Consider a Lagrangian fibration $π\colon X\to \mathbb P^n$ on a hyperkähler manifold $X$. There are two ways to construct a holomorphic family of deformations of $π$ over $\mathbb C$. The first one is known under the name Shafarevich-Tate family while the second one is the degenerate twistor family constructed by Verbitsky. We show that both families coincide. We prove that for a very general $X$ all members of the Shafarevich-Tate family are Kähler. There is a related notion of the Shafarevich-Tate group associated to a Lagrangian fibration. Its connected component of unity can be shown to be isomorphic to $\mathbb C/Λ$ where $Λ$ is a finitely generated subgroup of $\mathbb C$ and $\mathbb C$ is thought of as the base of the Shafarevich-Tate family. We show that for a very general $X$, projective deformations in the Shafarevich-Tate family correspond to the torsion points in the connected component of unity of the Shafarevich-Tate group. A sufficient condition for a Lagrangian fibration $X$ to be projective is existence of a holomorphic section. We find sufficient cohomological conditions for existence of a deformation in the Shafarevich-Tate family that admits a section. |
| title | Shafarevich-Tate groups of holomorphic Lagrangian fibrations |
| topic | Algebraic Geometry Complex Variables 14J42, 14F25, 32Q15, 32J25 |
| url | https://arxiv.org/abs/2112.10921 |