The Decomposition Theorem and the topology of algebraic maps

TL;DR

This paper introduces the theory of perverse sheaves and explains the Decomposition Theorem, highlighting its development from classical topological methods and its applications in algebraic geometry.

Contribution

It provides a motivated overview of perverse sheaves and the Decomposition Theorem, connecting modern theory with classical topological constructions.

Findings

The Decomposition Theorem is a fundamental result in algebraic geometry.

Multiple approaches to the theorem are discussed.

Applications demonstrate the theorem's significance in 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.