  We prove that the higher direct images of the dualizing sheaf of a Lagrangian
fibration between smooth projective manifolds are isomorphic to the cotangent
bundles of base space.
  As a corollary, we obtain that every Hodge number of the base space of a
fibre space of an irreducible symplectic manifold is the same to that of a
projective space if the base space is smooth.
