Marco Manetti

Extended deformation functors

We introduce a precise notion, in terms of few Schlessingers 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.

On the moduli space of diffeomorphic algebraic surfaces

Abstract.In this paper we show that the number of deformation types of complex structures on a fixed smooth oriented four-manifold can be arbitrarily large. The examples that we consider in this paper are locally simple abelian covers of rational surfaces. The proof involves the algebraic description of rational blowdowns, classical Brill-Noether theory and deformation theory of normal flat abelian covers.

Lie description of higher obstructions to deforming submanifolds

To every morphism

Normal degenerations of the complex projective plane.

\chi\colon L\to M

Formality of Koszul brackets and deformations of holomorphic Poisson manifolds

of differential graded Lie algebras we associate a functors of artin rings

Semiregularity and obstructions of complete intersections

\Def_\chi

A period map for generalized deformations

whose tangent and obstruction spaces are respectively the first and second cohomology group of the cylinder of

Cosimplicial DGLAs in Deformation Theory

\chi

An algebraic proof of Bogomolov-Tian-Todorov theorem

. Such construction applies to Hilbert and Brill-Noether functors and allow to prove with ease that every higher obstruction to deforming a smooth submanifold of a Kaehler manifold is annihilated by the semiregularity map.

Enumerative Invariants in Algebraic Geometry and String Theory

Albeit this investigation is interesting for its own sake, there are also applications to other problems in algebraic geometry. For example, Catanese [Ca] uses normal degenerations of P x P in order to study the Zariski closure of some open subsets of moduli spaces of certain surfaces S of general type. We also believe that the study of normal degenerations can be applied to the classification of normal surfaces. We actually give here a result in this direction.