Symplectic realizations of holomorphic Poisson manifolds

Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X,\pi )$, we prove that there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of $T^{\ast }X$ such that the projection map is a symplectic realization of the given Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic symplectic structure on $Y\subseteq T^{\ast }X$.