Descent of Deligne groupoids

TL;DR

This paper constructs a functor linking dg Lie algebras to simplicial sets, showing it aligns homotopically with the Deligne groupoid and preserves homotopy limits, advancing understanding of deformation problems in algebraic geometry.

Contribution

It proves that the functor from dg Lie algebras to simplicial sets commutes up to homotopy with total space functors, confirming a conjecture and linking local and global deformation theories.

Findings

The functor $\Sigma_g$ is homotopy equivalent to the Deligne groupoid.

$\Sigma_g$ commutes up to homotopy with total space functors.

Global deformation problems are described by the homotopy Lie algebra $R\Gamma(X,g)$.

Abstract

To any non-negatively graded dg Lie algebra $g$ over a field $k$ of characteristic zero we assign a functor $\Sigma_g: art/k \to Kan$ from the category of commutative local artinian $k$-algebras with the residue field $k$ to the category of Kan simplicial sets. There is a natural homotopy equivalence between $\Sigma_g$ and the Deligne groupoid corresponding to $g$. The main result of the paper claims that the functor $\Sigma$ commutes up to homotopy with the "total space" functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman which implies that if a deformation problem is described ``locally'' by a sheaf of dg Lie algebras $g$ on a topological space $X$ then the global deformation problem is described by the homotopy Lie algebra $R\Gamma(X,g)$.