The Decomposition Theorem and the topology of algebraic maps

We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the Decomposition Theorem, indicate some important applications and examples.Comment: 117 pages. New title. Major structure changes. Final version of a survey to appear in the Bulletin of the AM

The Gysin map is compatible with mixed Hodge structures

We prove that the Gysin map is compatible with mixed Hodge Structures.Comment: Published in CRM Proceedings and Lecture Series, vol. 38, 200

What is a perverse sheaf?

Three-page article on the notion of perverse sheaf to appear in the "What is?" series in the Notices of the AMS.Comment: to appear in the May 2010 issue of the Notices of the AMS http://www.ams.org/notice

The perverse filtration and the Lefschetz Hyperplane Theorem

We describe the perverse filtration in cohomology using the Lefschetz Hyperplane Theorem.Comment: Revised version with minor changes. To appear in Annals of Mathematic

The projectors of the decomposition theorem are motivic

We prove that the projectors arising from the decomposition theorem applied to a projective map of quasi projective varieties are absolute Hodge, Andr\'e motivated, Tate and Ogus classes. As a by-product, we introduce, in characteristic zero, the notions of algebraic de Rham intersection cohomology groups of a quasi projective variety and of intersection cohomology motive of a projective variety