Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

Abstract

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 E nd * (E · ), where E

Deformations and Obstructions of Pairs (X,D)

“…[FIM09,FMM12] If each g i is concentrated in degree zero, i.e., g ∆ is a semicosimplicial Lie algebra, then the functor Def T W (g ∆ ) has another explicit description; namely, it is isomorphic to the following functor:…”

On Deformations of Pairs (Manifold, Coherent Sheaf)

“…Proof. The proof follows the general lines already used in [6]. According to Definition 6.6, we have to prove that for any affine open cover U = {U i } of X, there exists an isomorphism of functors of Artin rings Def Tot(U,D * K (X,E * )) → Def (X,F) .…”

“…The main references for this section are [6,8,17,18,19]. From this section, and throughout the rest of the paper, we work over a fixed algebraically closed field K of characteristic zero.…”

“…Indeed, an element (l, m) ∈ Z 1 sc (exp(D * K (X, E * )(U)))(A) gives a deformation of the pair (X, F) as gluing of deformations on each U i . We only stress the fact that the gluing condition on the isomorphisms involves an element in i,j,k Hom −1 OX (E * , E * )(U ijk ) ⊗ m A ; this is due to the fact that we do not have to glue the deformed complexes but rather their cohomology to get a sheaf (see [6,Section 2] for more details about this). As regards the equivalence relation ∼, the first condition is the isomorphism of the induced deformation on each open, the second condition gives the gluing of the local isomorphism to have a global isomorphism of the induced deformations.…”

“…For a coherent sheaf F on a projective smooth variety X, it is known that the deformations of F are controlled by the sheaf of DG-Lie algebra of endomorphisms of any finite locally free resolution of F [6].…”