Semicosimplicial Dglas in Deformation Theory

We describe a canonical L∞ structure on the total complex of a semicosimplicial differential graded Lie algebra and give an explicit descriprion of the Maurer-Cartan elements and of the associated deformation functor in the particular case of semicosimplicial Lie algebras. We use these results to investigate the deformation functor associated to a sheaf of Lie algebras L and to show that it is naturally isomorphic to the deformation functor associated to the DGLA of global sections of a resolution of L by an acyclic sheaf of DGLAs. As classical examples we recover the DGLA description of infinitesimal deformations of a complex manifold X, of a locally free sheaf E of OX -modules, and of the pair (E ,X).