The Hochschild cohomology ring of a global quotient orbifold (Q2302230)

The authors 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, they provide 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. Using Hochschild cohomology, the autors provide a proof that Kontesvich's formality theorem does not hold for Deligne-Mumford stacks for the cup product. In the case of a symplectic group action on a symplectic variety \(X\), the authors discuss relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, the authors employ compatible trivializations of the determinants of the normal bundles of the fixed loci in \(X\), which requires (for the cup product) a nontrivial normalization missing in previous literature. The structure of the paper is: in Section \(2\) we find a discussion on some projective examples. Section \(3\) is dedicated to background material. In Section \(4\) the authors introduce the smash product and establish the necessary relationships between the local Hochschild cohomology \(HH_\star(X)\) and the cohomology of the stack quotient \(X\). Section \(5\) is devoted to some generic relations between normal bundles and tangent bundles for the fixed spaces, which are used in Sections \(6\) and \(7\) to give a geometric description the Hochschild cohomology \(HH_\star(X)\) as a sheaf of algebras on \(X\). The authors discuss formality and Calabi-Yau orbifolds in Section \(8\) and in Section \(9\) they describe the Hochschild cohomology of the quotient orbifold of a symplectic variety by symplectic automorphisms. Finally, in Section \(10\) the authors establish some linear identifications between Hochschild cohomology and orbifold cohomology, and provide a number of conjectures regarding the relationship between the cup product on Hochschild cohomology and the orbifold product on orbifold cohomology.