Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

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 ε.*( ε where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to DefF 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 gδ, whose cohomology is concentrated in nonnegative degrees, with a noncommutative Č ech cohomology-type functor H 1sc.exp gδ/. © European Mathematical Society 2012.