Shafarevich-Tate groups of holomorphic Lagrangian fibrations II

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