Equivariant Hodge theory and noncommutative geometry

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.

47 pages, updated to match the published version, to avoid confusion. Following referee's suggestion, we reorganized the paper so that all matrix factorization material appears in its own separate section

References in corpus (4)

• Perfect complexes on algebraic stacks
• Functorial destackification of tame stacks with abelian stabilisers
• Geometricity for derived categories of algebraic stacks
• Derived Kn"orrer periodicity and Orlov's theorem for gauged Landau-Ginzburg models

Cited by in corpus (3)

• Categorical wall-crossing formula for Donaldson-Thomas theory on the resolved conifold
• Dévissage for periodic cyclic homology of complete intersections
• Equivariant Elliptic Cohomology and Mapping Stacks