It is not hard to see that a holomorphically symplectic form defines a complex structure uniquely.

The notion of а C-symplectic structure is used to define the holomorphically symplectic manifolds with no reliance on complex structures.

Deformations of complex structures are usually obtained using the solutions of the appropriate Maurer-Cartan equation. I would explain what equation plays its role in the C-symplectic geometry, and construct a recursive solution for the C-symplectic Maurer-Cartan, which is in fact much easier than the classical deformation theory.

This gives a new and very simple proof of Bogomolov's local Torelli theorem for hyperkahler manifolds. I would explain how this construction is used to give a generalization of Voisin’s famous theorem about deformations of holomorphic Lagrangian subvarieties. This is a joint work with Nikon Kurnosov.