Equivariant Hodge theory and noncommutative geometry

TL;DR

This paper extends Hodge theory to equivariant settings of quotient stacks and Landau-Ginzburg models, showing degeneration of spectral sequences and linking cyclic homology with topological K-theory, thus bridging noncommutative geometry and equivariant topology.

Contribution

It develops a noncommutative Hodge theory framework for quotient stacks and Landau-Ginzburg models, demonstrating spectral sequence degeneration and canonical Hodge structures in these contexts.

Findings

Degeneration of the noncommutative Hodge-de Rham sequence for equivariant categories.

Identification of periodic cyclic homology with topological equivariant K-theory.

Establishment of Hodge-de Rham degeneration for equivariant Landau-Ginzburg models.

Abstract

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 equivariant 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.