  Given a projective morphism of compact, complex, algebraic varieties and a
relatively ample line bundle on the domain we prove that a suitable choice,
dictated by the line bundle, of the decomposition isomorphism of the
Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields
isomorphisms of pure Hodge structures. The proof is based on a new
cohomological characterization of the decomposition isomorphism associated with
the line bundle. We prove some corollaries concerning the intersection form in
intersection cohomology, the natural map from cohomology to intersection
cohomology, projectors and Hodge cycles, and induced morphisms in intersection
cohomology.
