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
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 D-modules 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 non-singular complex
affine algebraic variety of complex dimension n. Then U has the ho-
motopy type of a CW-complex of real dimension n. In particular,
(U, Q) = 0 for i > n and Hi