An overview of Hodge theory, and Relative versions of the Du Bois complex

Abstract

The cohomology of smooth projective varieties over the complex numbers posseses a rich structure given by Hodge theory.  This structure can be extended to arbitrary complex varieties by the mixed Hodge theory of Deligne.  In this talk we will review this classical theory and discuss variations of Hodge structure following work of Griffiths, Schmid, Steenbrink, and others.  Time permitting, we will also touch on the non-abelian Hodge theory of Simpson as well as analogues in characteristic p and p-adic Hodge theory.

Research talk abstract: When X is a singular complex variety, its Du Bois complex can be used in place of its de Rham complex to obtain the Hodge filtration in its mixed Hodge structure.  The Du Bois complex has found numerous applications to studying singularities and vanishing theorems, and a well-behaved relative version, generalizing the relative de Rham complex, may have such applications as well. In this talk I will describe work in progress with Kovács and Taji towards constructing and using such a complex in various settings.