Shafarevich–Tate Groups of Holomorphic Lagrangian Fibrations

Consider a holomorphic fibration P : X → B whose general fiber is a torus. Its Shafarevich–Tate group parametrizes fibrations that are isomorphic to P locally over the base, i.e., fibers are the same but are glued in a different way. The fibrations with this property are called Shafarevich–Tate twists. I’ll describe the Shafarevich–Tate group in the case when P is a Lagrangian fibration on a compact hyperkähler manifold X. Then we’ll figure out which twists are projective, which are Kähler, and which are non-Kähler. In particular, I’ll show how to obtain the Bogomolov–Guan manifold, which is the only known example of a non-Kähler holomorphic symplectic manifold, as a Shafarevich-Tate twist of a Kähler manifold.