Abstract:

The perverse (Leray) filtration captures key topological information of algebraic maps. Recent studies of integrable systems (e.g. Hitchin system, Beauville-Mukai system) suggests two common features of the perverse filtration of abelian fibrations:

1) the perverse filtration is multiplicative with respect to the cup product;

2) the perversity of tautological classes is governed by the Chern degree.

In this talk, I will explain a unified approach to the two statements above for a large class of abelian fibrations, namely fibrations in compactified Jacobians. It uses a combination of Arinkin’s Fourier-Mukai theory, Ngô’s support theorem, and Chow-theoretic/motivic techniques. I will also discuss some applications to earlier work of Maulik-Yun and to the P = W conjecture of de Cataldo-Hausel-Migliorini. Joint work in progress with Davesh Maulik and Junliang Shen.