Equivariant Hodge theory and noncommutative geometry

We develop a version of Hodge theory for a large class of smooth cohomologically 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 equivariant $K$-theory of $X$ with respect to a maximal compact subgroup $M subset G$. The result is a natural pure Hodge structure of weight $n$ on $K^n_M(X^{an})$. We also treat categories of matrix factorizations for equivariant Landau-Ginzburg models.