We identify Čech cocycles in nonabelian (formal) group cohomology with Maurer-Cartan elements in a suitable L∞-algebra. Applications to deformation theory are described.

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

Abstract. We propose a generalisation of Mori dream spaces to stacks. We show that this notion is preserved under root constructions and taking abelian gerbes. Unlike the case of Mori dream spaces, such a stack is not always given as a quotient of the spectrum of its Cox ring by the Picard group. We give a criterion when this is true in terms of Mori dream spaces and root constructions. Finally, we compare this notion with that of smooth toric Deligne-Mumford stacks.

We identify dglas that control infinitesimal deformations of the pairs (manifold, Higgs bundle) and of Hitchin pairs. As a consequence, we recover known descriptions of first order deformations and we refine known results on obstructions. Secondly we prove that the Hitchin map is induced by a natural L∞-morphism and, by standard facts about L∞-algebras, we obtain new conditions on obstructions to deform Hitchin pairs.

We give a proper definition of the multiplicative structure of the following rings: Cox ring of invertible sheaves on a general algebraic stack; Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish its (non-)uniqueness in terms of an Ext-group. Moreover, we compare this definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.