Shafarevich-Tate groups of holomorphic Lagrangian fibrations

Abstract

Consider a Lagrangian fibration $\ddot{I}€:X\to {\mathbb{P}}^n$ on a hyperkähler manifold $X$. There are two ways to construct a holomorphic family of deformations of $\ddot{I}€$ 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 $\mathbb{C}/\hat{I}›$ where $\hat{I}›$ 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.

27 pages. v4: published version

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• Shafarevich-Tate groups of holomorphic Lagrangian fibrations

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• Shafarevich-Tate groups of holomorphic Lagrangian fibrations