The decomposition theorem, perverse sheaves and the topology of algebraic maps

Abstract

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.

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

“…Definition of the perverse filtration P on H * (M ). We employ freely the language of derived categories and perverse sheaves (see the seminal paper [7], the survey [13], or for example the paper [12]). Standard textbooks on the subject are [21], [37], [38].…”

On the length of perverse sheaves on hyperplane arrangements

“…The following facts about the intermediate extension functor can be found in [DM09]. The intermediate extension functor behaves well only when it comes to the simple perverse sheaves, that is, it takes a simple perverse sheaf into a simple perverse sheaf.…”

Support Theorems for Algebraic Maps

“…For a much more extensive survey, including a discussion of some significant applications of this theory, see [11] and its (huge) list of references. A shorter, extremely stimulating introduction is given in [31], while I strongly recommend S. Kleiman's paper [29] for a beautiful history of this subject.…”

“…We will also state theorems in a slightly abusive way since we didn't properly discuss the constructible derived category, in particular we did not give the (rather subtle) definition of morphism in the derived category. The survey [11] contains a quick summary of the main ideas underlying the construction of the derived category, and there are by now many texts discussing this, such as [20]. A very nice introduction is given in the already…”