Deformation types of real and complex manifolds

Abstract

The Leit-Faden of the article (which is partially a survey) is a negative answer to the question whether, for a compact complex manifold which is a $K(π, 1)$ the diffeomorphism type determines the deformation type. We show that a deformation in the large of complex tori is again a complex torus, and then that the same holds for products of a torus with a curve of genus $g\geq 2$. Together with old results of Blanchard, Calabi, and Sommese, who showed the existence of non Kähler complex structures on the product of a curve with a two dimensional complex torus, this gives the first counterexamples. We generalize these constructions and we study the small deformations of what we call Blanchard-Calabi 3-folds. These give infinitely many deformation components. We give also a criterion for a complex manifold to be a complex torus, namely, to have the same integral cohomology algebra of a complex torus, and to possess $n$ independent holomorphic and d-closed 1-forms. As a corollary, we give a classification of deformation types of real tori. We illustrate then simple new examples of the author of surfaces of general type which are $K(π, 1)$'s, and for which there are, for a fixed differentiable structure, two different deformation types. These surfaces are quotients of products of curves, and the moduli space for the fixed topological type has two connected components, exchanged by complex conjugation. We also report, after an elementary introduction to the problem, on recent results of the author and P. Frediani on the Enriques classification of real surfaces (namely, the classification of real hyperelliptic surfaces).