Algebraic Geometry Seminar
Abstract: The decomposition theorem of Beilinson, Bernstein, Deligne and Gabber is an important result in Hodge theory, related to the hard Lefschetz theorem. It states that the derived pushforward of a perverse sheaf of rational vector spaces of geometric origin along a proper morphism of complex algebraic varieties is isomorphic to the direct sum of its perverse cohomology sheaves. The original proof relies on Deligne's proof of the Weil conjectures and positive-characteristic methods. M. Saito gave a purely Hodge-theoretic proof by painstaking analysis of degenerations of variations of mixed Hodge structures. In this talk, we will recall the notions of perverse sheaves and Hodge structures, describe a dg-enhanced theory of mixed Hodge modules, and use it to give a proof of the decomposition theorem based on formal properties of Grothendieck's six functors.