Equivariant Hodge theory and noncommutative geometry

Halpern-Leistner, Daniel; Pomerleano, Daniel

doi:10.2140/gt.2020.24.2361

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Equivariant Hodge theory and noncommutative geometry

Daniel Halpern-Leistner and Daniel Pomerleano

Geometry & Topology 24 (2020) 2361–2433

DOI: 10.2140/gt.2020.24.2361

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.

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

Volume 24, issue 5 (2020)