Deformations of Lagrangian subvarieties of holomorphic symplectic manifolds

In this talk I will highlight an approach to study deformations of holomorphically symplectic manifolds which preserve Lagrangian subvarieties. In the case of Kahler manifold and smooth subvarieties the problem was solved by C. Voisin, later C. Lehn extended her approach to the snc subvarieties. We approach this question from scratch and focus on understanding the deformations of C-symplectic structures, which works for large classes of Kahler and non-Kahler manifolds, and many Lagrangian subvarieties. Joint with M.Verbitsky.