Derived operadic centers in algebraic geometry and deformation quantization

Abstract

In algebraic geometry, it is well known that Hochschild cohomology and, in particular, the algebraic structure it carries, plays a central role in studying the infinitesimal noncommutative deformations of geometric spaces. This thesis provides, for the first time, an explicit interface between J. Lurie's work on higher centers and the Hochschild cohomology of an algebraic $\mathbb{k}$-scheme within the framework of formal deformation quantization. Our motivation stems from the mysterious appearance of the square root of the Todd genus in Kontsevich's formality theorem for algebraic varieties, as well as the conjectural relationships between these objects and the motivic Galois group. Our main result is a canonical solution to Deligne's conjecture for Hochschild cochains in the affine and global cases, even for singular schemes, by exhibiting the Hochschild complex as an $\infty$-operadic center. We show that this equips the Hochschild complex with a universal $\mathbb{E}_2$-algebra structure that precisely agrees with the classical Gerstenhaber bracket and cup product on cohomology in the affine and smooth cases. Finally, we interpret this universal $\mathbb{E}_2$-algebra structure in terms of operadic formal moduli problems by exhibiting a relationship between $\infty$-operadic centers and formal automorphism groups.