# The non-commutative Hodge theory of quotient stacks?

*Abstract:*

Non-commutative Hodge theory is the study of Hodge structures on the cyclic homology groups of dg-categories C. In this talk, we will study the case C= D^bCoh(X/G), where X is a projective-over-affine variety and G is an algebraic group. Using Halpern-Leistner’s theory of derived Kirwan surjectivity, we prove the collapse of nc Hodge-to-de Rham spectral sequence in variety of situations for example when X is smooth, G is reductive and G(X,OX)G is finite dimensional. These results on the degeneration of the spectral sequence also extend to categories of singularities. Using homotopy theoretic methods, there is a Chern character from a topological version of algebraic K-theory to this periodic cyclic homology. We show that this map is an isomorphism, thereby putting an integral structure and ultimately a weight zero Hodge structure on the periodic cyclic homology. Along the way, we identify the periodic cyclic homology with a complexified version of equivariant K-homology in the sense of Atiyah and Segal. This is joint work with Dan Halpern-Leistner.