Hermitian-symplectic and Kahler structures on degenerate twistor deformations

Let $(M,\mathrm{\Omega })$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $\pi :M\to B$, and $\eta$ a closed $(1,1)$-form on $B$. Then $\mathrm{\Omega }+{\pi }^{\ast }\eta$ 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 Kähler manifolds are also Kähler. 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 Teichmüller space of holomorphically symplectic manifolds correspond to bimeromorphic manifolds if they are Hermitian symplectic.