Shafarevich-Tate groups of holomorphic Lagrangian fibrations

Consider a Lagrangian fibration $\pi\colon X\to \mathbb P^n$ on a hyperk\"ahler manifold $X$. There are two ways to construct a holomorphic family of deformations of $\pi$ 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 K\"ahler. 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 $\mathbb C/\Lambda$ where $\Lambda$ 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.