Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Formal deformation quantization of symplectic singularities

Given a complex smooth affine algebraic variety X with an algebraic symplectic form and a finite group G acting by symplectomorphisms, the product in the structure algebra $\mathcal{O}(X)^G$ of the categorical quotient variety X/G can be formally deformed into a G-invariant star product * via a G-equivariant Fedosov formal deformation quantization of X. In the early 2000's Vasily Dolgushev and Pavel Etingof conjectured that the Fedosov formal quantization $A^{\hbar}_{X/G}:=(\mathcal{O}(X)((\hbar))^G,*)$ possesses a universal formal deformation, parametrized by the second Chen-Ruan cohomology group of the quotient stack [X/G] modeled on X/G. In this talk, I discuss a joint work with Pavel Etingof which settles Dolgushev-Etingof's conjecture relating the theory of factorial and terminal symplectic singularities with Bezrukavnikov-Kaledin's theory of formal deformation quantization of admissible schemes.