On the Lie algebroid of a derived self-intersection

TL;DR

This paper constructs Lie algebraic structures on the shifted normal bundle of a smooth embedding, linking formal neighborhoods with Lie theory and exploring applications of classical Lie constructions in algebraic geometry.

Contribution

It introduces two Lie-type structures on the shifted normal bundle that encode the formal neighborhood of the embedding, connecting Lie theory with geometric embeddings.

Findings

Constructed two Lie-type structures on N[-1]

Linked formal neighborhoods with Lie algebraic structures

Applied classical Lie constructions to geometric embeddings

Abstract

Let $i:X\hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. Denote by $N$ the normal bundle of $X$ in $Y$. We describe the construction of two Lie-type structures on the shifted bundle $N[-1]$ which encode the information of the formal neighborhood of $X$ inside $Y$. We also present applications of classical Lie theoretic constructions (universal enveloping algebra, Chevalley-Eilenberg complex) to the understanding of the geometry of embeddings.