Perverse Sheaves and Hodge Theory

Luca Migliorini (Università di Bologna)

The aim of the course is to illustrate some applications of the theory of perverse sheaves to Hodge theory and in particular to the study of algebraic cycles. I will recall some fundamental results about the category of perverse sheaves, emphasizing their connection to classical results of Hodge theory such as the local invariant cycle theorem and its generalization. Then I will discuss some recent results and conjectures in this circle of ideas, namely:

1. The Hodge theoretic nature of the Leray filtration (D. Arapura) and of the perverse Leray filtration (M. de Cataldo-L. M.).

2. Green-Griffiths theory of normal functions in several variables (M. Green-P. Griffiths, P. Brosnan-G. Pearlstein, C. Schnell).

3. The projectors associated to the decomposition theorem for a projective map are motivated cycles in the sense of Y. Andreé (M. de Cataldo-L. M.).