Consider a Lagrangian fibration \(\pi :X\rightarrow {\mathbb {P}}^n\) on a hyperkähler 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 Kähler. There is a related notion of the Shafarevich–Tate group associated to a Lagrangian fibration. The connected component of unity of 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 . A sufficient condition for a Lagrangian fibration X to be projective is the existence of a holomorphic section. We find sufficient cohomological conditions for the existence of a deformation in the Shafarevich–Tate family that admits a section.
