Abstract:

We present in this lecture series some recent progress in understanding the topology of abelian fibrations. We first introduce the “perverse filtration” which encodes key cohomological information of the fibrations. Then we explain how various structures (such as Fourier-Mukai transforms, intersection theory, BBDG decomposition theorem, and Ngô support theorem) can be combined to give a new characterization of the perverse filtration for certain abelian fibrations. Applications include the P=W conjecture, the motivic decomposition conjecture for the Hitchin fibration, and the construction of an intrinsic cohomology ring of the universal compactified Jacobian. Finally we specialize to Lagrangian fibrations and survey a few known results and open questions in this direction. Based on joint work with Davesh Maulik, Junliang Shen, and others.