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 epsilon nd*(epsilon(.)), where epsilon(.) is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to Def(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 g(Delta), whose cohomology is concentrated in nonnegative degrees, with a noncommutative Cech cohomology-type functor H-sc(1)(exp g(Delta)).
Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves / Fiorenza, Domenico; Donatella, Iacono; Elena, Martinengo. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - STAMPA. - 14:2(2012), pp. 521-540. [10.4171/jems/310]