COHOMOLOGY OF COMPACT HYPERKAHLER MANIFOLDS AND ITS APPLICATIONS

We announce the structure theorem for the H-2(M)-generated part of coh omology of a compact hyperkahler manifold. This computation uses an ac tion of the Lie algebra so(4, n-2) where n=dim H-2(M) on the total coh omology space of M. We also prove that every two points of the connect ed component of the moduli space of holomorphically symplectic manifol ds can be connected with so-called ''twistor lines'' - projective line s holomorphically embedded in the moduli space and corresponding to th e hyperkahler structures. This has interesting implications for the ge ometry of compact hyperkahler manifolds and of holomorphic vector bund les over such manifolds.