Shafarevich-Tate groups of holomorphic Lagrangian fibrations II

Abstract

Let $X$ be a compact hyperkähler manifold with a Lagrangian fibration $\pi :X\to B$. A Shafarevich-Tate twist of $X$ is a holomorphic symplectic manifold with a Lagrangian fibration ${\pi }^{\phi }:X^{\phi }\to B$ which is isomorphic to $\pi$ locally over the base. In particular, ${\pi }^{\phi }$ has the same fibers as $\pi$. A twist $X^{\phi }$ corresponds to an element $\phi$ in the Shafarevich-Tate group of $X$. We show that $X^{\phi }$ is Kähler when a multiple of $\phi$ lies in the connected component of unity of the Shafarevich-Tate group and give a necessary condition for $X^{\phi }$ to be bimeromorphic to a Kähler manifold.

31 pages, new proof of Theorem A(1) (Kählerness of twists), added Theorem B. To appear in Epiga