 
Summary: THE DECOMPOSITION THEOREM AND THE
TOPOLOGY OF ALGEBRAIC MAPS
Abstract. Notes from fives lectures given by Luca Migliorini in
Freiburg in February 2010. Notes by Geordie Williamson.
1. Lecture 1: Hodge theory
This first lecture will comprise a review of a few classical facts about
the topology of complex algebraic varities. All varieties will be complex
algebraic varieties.
Everything probably can (and should) be translated into the char
acteristic p world, but there are some points that are not clear.
The important connection with Dmodules will not be treated. This
is also very important and should not be forgotten by (young) re
searchers interested in the area.
The main character in this story is the category of perverse sheaves.
The starting point is the Lefschetz hyperplane theorem:
Theorem 1.1 (S. Lefschetz 1924). Let U be a nonsingular complex
affine algebraic variety of complex dimension n. Then U has the ho
motopy type of a CWcomplex of real dimension n. In particular,
Hi
(U, Q) = 0 for i > n and Hi
