Poisson structures and generalized Kähler submanifolds

Abstract

Let $X$ be a compact Kähler manifolds with a non-trivial holomorphic Poisson structure $\beta$ . Then there exist deformations $\{(J_{\beta t},{\psi }_t)\}$ of non-trivial generalized Kähler structures with one pure spinor on $X$ . We prove that every Poisson submanifold of $X$ is a generalized Kähler submanifold with respect to $(J_{\beta t},{\psi }_t)$ and provide non-trivial examples of generalized Kähler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bihermitian structures constructed from Poisson structures.