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In its simplest form the Decomposition Theorem asserts that the rationalintersection cohomology of a complex projective variety occurs as a summand ofthe cohomology of any resolution. This deep theorem has found importantapplications in algebraic geometry, representation theory, number theory andcombinatorics. It was originally proved in 1981 by Beilinson, Bernstein,Deligne and Gabber as a consequence of Deligne's proof of the Weil conjectures.A different proof was given by Saito in 1988, as a consequence of his theory ofmixed Hodge modules. More recently, de Cataldo and Migliorini found a much moreelementary proof which uses only classical Hodge theory and the theory ofperverse sheaves. We present the theorem and outline the main ideas involved inthe new proof.

The Hodge theory of the Decomposition Theorem (after de Cataldo and Migliorini)