The Hochschild cohomology ring of a global quotient orbifold
File(s)
Accepted version
Author(s)
Negron, Cris
Schedler, Travis
Type
Journal Article
Abstract
We study the cup product on the Hochschild cohomology of the stack quotient of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of , under a natural filtration. This sheaf is an algebra over the polyvector fields on X, and is generated as a -algebra by the sum of the determinants of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne–Mumford stacks in general. We discuss, in the case of a symplectic group action on a symplectic variety X, relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, we employ compatible trivializations of the determinants , which requires (for the cup product) a nontrivial normalization missing in previous literature
Date Issued
2020-04-15
Date Acceptance
2019-12-20
Citation
Advances in Mathematics, 2020, 364, pp.1-49
ISSN
0001-8708
Publisher
Elsevier BV
Start Page
1
End Page
49
Journal / Book Title
Advances in Mathematics
Volume
364
Copyright Statement
© 2020 Elsevier Ltd. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence http://creativecommons.org/licenses/by-nc-nd/4.0/
Sponsor
National Science Foundation
Identifier
https://www.sciencedirect.com/science/article/pii/S0001870820300037?via%3Dihub
Grant Number
DMS-1406553
Subjects
0101 Pure Mathematics
General Mathematics
Publication Status
Published online
Article Number
106978
Date Publish Online
2020-01-29