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

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

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

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

A relative version of the ordinary perturbation lemma

The perturbation lemma and the homotopy transfer for L-infinity algebras is proved in a elementary way by using a relative version of the ordinary perturbation lemma for chain complexes and the coalgebra perturbation lemma

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