Equivariant Hodge theory and noncommutative geometry

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks X = G analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge–de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equi-variant K–theory of X with respect to a maximal compact subgroup of G, equipping the latter with a canonical pure Hodge structure. We also establish Hodge–de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau–Ginzburg models.