Julien Grivaux: Closed subschemes as a geometrization of Lie algebra pairs

Let X be a closed smooth subscheme of an ambient scheme Y. Due to a standard construction in algebraic geometry (deformation to the normal cone), the closed immersion from X to Y can be deformed to the injection of X into the total space of its normal bundle in Y. However, these two immersions are not isomorphic in general. In the case where Y=X x X and X is the diagonal in Y, the situation is understood thanks to the work of Kapranov and Markarian: the failure of the existence of such an isomorphism is encoded in a specific algebraic structure: a Lie structure on the shifted tangent bundle TX[-1].

In the first part of this talk, we will recall basic things about derived categories and Lie objects in symmetric monoidal categories. We will explain how the classical Hochschild–Kostant–Rosenberg theorem is related with Lie theory. Then we will introduce quantized analytic cycles, and explain why this setting is a geometrization of standard problems in Lie theory.

In the second part of the talk, we will introduce tame analytic cycles (the tameness condition was discovered by Shilin Yu), and compute their enveloping algebras. Then we will explain how general considerations on Lie algebras allows to compute intrinsic invariants attached to quantized analytic cycles. 

This is joint work with Damien Calaque.