Let M be a simple hyperkähler manifold. Kuga-Satake construction gives an embedding of H 2 (M, C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H • (M, C) → H •+l (T, C) for some l, which is compatible with the Hodge structures and the Poincaré pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M .

In this paper we study the Lagrangian fibrations for projective irreducible symplectic fourfolds and exclude the case of non-smooth base. Our method could be extended to the higherdimensional cases.

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We study groups of biholomorphic and bimeromorphic automorphisms of projective hyperkähler manifolds. Using an action of these groups on some non-positively curved space, we immediately deduce many of their properties, including finite presentation, strong form of Tits' alternative, and some structural results about groups consisting of transformations with infinite order.