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Equivariant Hodge theory and noncommutative geometry

Daniel Halpern-Leistner and Daniel Pomerleano

Geometry & Topology 24 (2020) 2361–2433
Abstract

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks XG 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.

Keywords
$K$–theory, Hochschild homology, Hodge structures, derived categories, equivariant geometry
Mathematical Subject Classification 2010
Primary: 14A22, 14C30, 19D55, 19L47
References
Publication
Received: 9 November 2018
Revised: 20 October 2019
Accepted: 1 January 2020
Published: 29 December 2020
Proposed: Richard P Thomas
Seconded: Dan Abramovich, Frances Kirwan
Authors
Daniel Halpern-Leistner
Mathematics Department
Cornell University
Ithaca, NY
United States
Daniel Pomerleano
Mathematics Department
University of Massachusetts Boston
Boston, MA
United States