The Hochschild cohomology ring of a global quotient orbifold

TL;DR

This paper investigates the structure of Hochschild cohomology for quotient orbifolds, constructing a sheaf of graded algebras that captures the cup product and exploring its relation to orbifold cohomology and formality theorems.

Contribution

It introduces a G-equivariant sheaf of graded algebras on X that models Hochschild cohomology of [X/G], revealing new insights into the algebraic structure and limitations of formality in stacks.

Findings

Constructed a G-equivariant sheaf of graded algebras on X.

Showed the failure of Kontsevich's formality theorem for stacks.

Connected Hochschild cohomology with orbifold cohomology theories.

Abstract

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] 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 [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) 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 relationships with orbifold cohomology, extending Ruan's cohomological conjectures. This employs a trivialization of the determinants in the case of a symplectic group action on a symplectic variety X, which requires (for the cup product) a nontrivial normalization missing in previous literature.