Hermitian-symplectic and Kahler structures on degenerate twistor deformations

Let (M, ) be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration: M B, and a closed (1, 1) -form on B. Then + ^* is a holomorphically symplectic form on a complex manifold which is called the degenerate twistor deformation of M. We prove that degenerate twistor deformations of compact holomorphically symplectic K\"ahler manifolds are also K\"ahler. First, we prove that degenerate twistor deformations are Hermitian symplectic, that is, tamed by a symplectic form; this is shown using positive currents and an argument based on the Hahn--Banach theorem, originally due to Sullivan. Then we apply a version of Huybrechts's theorem showing that two non-separated points in the Teichm\"uller space of holomorphically symplectic manifolds correspond to bimeromorphic manifolds if they are Hermitian symplectic.