Shafarevich-Tate groups of holomorphic Lagrangian fibrations II

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