Modul: MAT770 Oberseminar: Algebraische Geometrie

## Derived intersections and dg-Lie algebroids

Dr. Julien Grivaux talk

Speaker invited by: Prof. Dr. Joseph Ayoub

**Date:** 25.03.19 **Time:** 13.15 - 14.45 **Room:** Y27H25

Let Y be a smooth algebraic scheme and X be a closed subscheme of Y. The derived intersection of X in Y is a derived scheme supported on X, whose algebro-geometric properties partially encode some of the formal geometry of X in Y. This object has been studied in many cases: in the case of the diagonal embedding of a smooth scheme, the celebrated Hochschild-Kostant-Rosenberg isomorphism gives a full description of this derived scheme; and for arbitrary pairs, the problem was first studied by Arinkin and Cāldāraru, and then in full generality by Calaque, Cāldāraru, and Tu. The main achievement of the latter article is the construction of a dg-Lie algebroid structure up to homotopy on the shifted normal bundle N_{X/Y} [-1] (which generalizes an earlier construction of Kapranov and Markarian) that governs the situation. In this talk, I will explain how to provide N_{X/Y} [-1] a derived dg-Lie structure under a specific geometric condition called moderation. This is a joint work with Damien Calaque.