Higher direct images of dualizing sheaves of Lagrangian fibrations

Let f : X → S be a Lagrangian fibration between projective varieties. We prove that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] ≅ Ωⁱ_S 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 h^p,q(S) = h^p,q[inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], where n = dim S. If S ≅ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] and X is an irreducible symplectic manifold, there exists a hypersurface M_f of the Kuranishi space of X such that every member of the Kuranishi family over M_f admits a Lagrangian fibration over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /].