On the Moduli space of diffeomorphic algebraic surfaces

It is proved that the number of deformation types of complex structures on a fixed oriented smooth four-manifold can be arbitrarily large. The considered examples are locally simple abelian covers of rational surfaces.Comment: Plain Tex, 41page

On some formality criteria for DG-Lie algebras

We give some formality criteria for a differential graded Lie algebra to be formal. For instance, we show that a DG-Lie algebra $L$ is formal if and only if the natural spectral sequence computing the Chevalley-Eilenberg cohomology $H^*_{CE}(L,L)$ degenerates at $E_2

On deformations of pairs (manifold, coherent sheaf)

We analyse infinitesimal deformations of pairs $(X,\mathcal{F})$ with $\mathcal{F}$ a coherent sheaf on a smooth projective manifold $X$ over an algebraic closed field of characteristic $0$. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai-Artamkin Theorem about the trace map.Comment: final version accepted for publication in Canad. J. Mat

Deformations of algebraic schemes via Reedy-Palamodov cofibrant resolutions

Let $X$ be a Noetherian separated and finite dimensional scheme over a field $\mathbb{K}$ of characteristic zero. The goal of this paper is to study deformations of $X$ over a differential graded local Artin $\mathbb{K}$-algebra by using local Tate-Quillen resolutions, i.e., the algebraic analog of the Palamodov's resolvent of a complex space. The above goal is achieved by describing the DG-Lie algebra controlling deformation theory of a diagram of differential graded commutative algebras, indexed by a direct Reedy category.Comment: Final version. To appear in Indagationes Mathematica

Extended deformation functors

We introduce a precise notion, in terms of few Schlessinger's type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessinger and obstruction theories. The inverse mapping theorem holds for natural transformations of extended deformation functors and all such functors with finite dimensional tangent space are prorepresentable in the homotopy category.Comment: Contains the previously announced part I

Uniqueness and intrinsic properties of non-commutative Koszul brackets

There exists a unique natural extension of higher Koszul brackets to every unitary associative algebras in a way that every square zero operator of degree 1 gives a curved L-infinity structure