Algebraic and Symplectic Geometry Seminar

The classical Hochschild-Kostant-Rosenberg (HKR) theorem supplies an identification of Hochschild homology groups in terms of differential forms. In the setting of derived geometry, Ben Zvi and Nadler and Toën and Vezzosi supplied a geometric interpretation of the HKR theorem in terms of an equivalence between the derived loop space and the shifted tangent bundle of a scheme. There is however a nontrivial problem in globalising this equivalence from schemes to stacks, and it is well known that derived loops and shifted tangent bundle do not agree already on Deligne-Mumford stacks. Ben Zvi and Nadler provided an infinitesimal version that holds for Artin stacks ; on the other hand, Arinkin Caldararu and Hablicsek [ACH] provided a global formula for smooth global quotient Deligne-Mumford stacks. In joint work with Fu, Scherotzke and Sibilla we extended this global formula to general derived Deligne-Mumford stacks, establishing a coordinate-free version of the previous result of [ACH]. In this talk I will explain the main ideas as well as some more recent developments.