Shafarevich-Tate groups of holomorphic Lagrangian fibrations
Anna Abasheva, Vasily Rogov
TL;DR
This work analyzes Shafarevich–Tate groups for holomorphic Lagrangian fibrations on hyperkähler manifolds, revealing a deep link between complex-analytic twists and Verbitsky’s degenerate twistor deformations. It shows that Sha twists carry holomorphic symplectic structures and, under non–M-special hypotheses, all twists are Kähler and projectivity is governed by torsion in Sha^0; there is a precise description of the connected component Sha^0 as a complex torus, often non-Hausdorff, with a dense lattice when the Picard rank is not maximal. A central result ties the Sha–Tate deformation theory to period maps, Leray-type cohomology, and Hard Lefschetz-type theorems, enabling obstruction-theory analysis for the existence of sections via H^2(B, Γ). The paper also establishes countability of algebraic points in Sha families and provides cohomological criteria ensuring deformation to a section-admitting fibration, highlighting rich interactions between Hodge theory, deformation theory, and hyperkähler geometry.
Abstract
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.