On coisotropic deformations of holomorphic submanifolds

We describe the differential graded Lie algebras governing Poisson deformations of a holomorphic Poisson manifold and coisotropic embedded deformations of a coisotropic holomorphic submanifold. In both cases, under some mild additional assumption, we show that the infinitesimal first order deformations induced by the anchor map are unobstructed. Applications include the analog of Kodaira stability theorem for coisotropic deformation and a generalization of McLean-Voisin’s theorem about the local moduli space of Lagrangian submanifold. Finally it is shown that our construction is homotopy equivalent to the homotopy Lie algebroid of Oh, Park, Cattaneo and Felder, in the cases where this is defined.