Poisson structures and generalized Kähler submanifolds

Abstract

Let X be a compact Kähler manifolds with a non-trivial holomorphic Poisson structure β. Then there exist deformations {(\mathscr{J}_βt, ψ_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 (\mathscr{J}_βt, ψ_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.