Higher direct images of dualizing sheaves of Lagrangian fibrations

Abstract. Let f: X → S be a Lagrangian fibration between projective varieties. We prove that Rif∗OX ∼ = ΩiS if S is smooth. Suppose that X is an irreducible symplectic manifold or a certain moduli space of semistable torsion free sheaves on a K3 surface, the Hodge numbers satisfy hp,q(S) = hp,q(Pn), where n = dim S. If S ∼ = Pn and X is an irreducible symplectic manifold, there exists a hypersurface Mf of the Kuranishi space of X such that every member of the Kuranishi family over Mf admits a Lagrangian fibration over Pn. 1. Introduction. W