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*script O signX ≅ Ω Si 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) = h p,q(ℙn), where n = dim S. If S ≅ ℙ n and X is an irreducible symplectic manifold, there exists a hypersurface Μf of the Kuranishi space of X such that every member of the Kuranishi family over Μf admits a Lagrangian fibration over ℙn.