Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

TL;DR

This paper demonstrates that infinitesimal deformations of coherent sheaves are governed by a specific differential graded Lie algebra, linking classical Ext groups with modern homotopical algebra techniques.

Contribution

It introduces a Thom-Whitney construction approach to control deformations via DGLAs, generalizing the understanding of deformation theory for coherent sheaves.

Findings

Deformations controlled by global sections of acyclic resolutions.

Tangent space identified with Ext^1(F,F).

Obstructions lie in Ext^2(F,F).

Abstract

We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees with a noncommutative Cech cohomology-type functor.