Abstract

This is a series of 3 talks, where we will focus on geometry and topology of abelian fibrations—- these are maps whose general fibers are complex tori but special fibers may be highly singular and complicated. The decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber (BBDG) and the support theorem of Ngô provide powerful tools for studying these maps; Corti-Hanamura further predicted that the sheaf-theoretic BBDG decomposition is governed by algebraic cycles. In recent years, the study of Hitchin system predicts a list of surprising properties concerning the cohomological shadow of the BBDG decomposition theorem for the Hitchin system and related geometries. In my talks, I will explain a geometric tool, a theory of Fourier transform, which helps us to understand various questions and conjectures for abelian fibrations. I will start with the case of an abelian scheme (i.e. an abelian fibration without singular fiber), where the Fourier theory has been established by Beauville and Deninger-Murre more than 30 years ago. Then I will discuss the case with singular fibers. Our ultimate goal is to explain how to use the Fourier transform to construct the desired algebraic cycles for Hitchin’s integrable system as predicted by Corti-Hanamura. If time permits, I will discuss further applications of the Fourier transform and the algebraic cycles constructed from it; this includes connections to the P=W conjecture, \chi-independence phenomanon etc. Based on joint work (in progress) with Davesh Maulik and Qizheng Yin.