Speaker: Fabrizio Catanese
Title: Algebraic Surfaces: real structures, topological and differentiable types.
Affiliation: Göttingen -Florida State University.
Date: Friday, February 16, 2001.
Place and Time: Room 101 - Love Building, 3:35-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. In the talk, after explaining two different ways to look at real varieties, one leading to the famous example of the Klein bottle, we shall concentrate on the important role of the orbifold fundamental group of real varieties. After recalling the Enriques' classification of complex algebraic surfaces, we shall see one class of surfaces, the so called hyperelliptic surfaces, where this notion allows a fine classification. We shall illustrate the following theorem by showing its analogy to the classification of real elliptic curves ( a much easier task).

Theorem ( -, Frediani). The orbifold fundamental group of a real hyperelliptic surface determines the differentiable type of the pair (S,\sigma). There are exactly 78 such types, and for each one the corresponding moduli space is connected and irreducible. Moreover, this notion applied to a general type analogue of such surfaces, namely, the surfaces isogenous to a product ( they are the quotients C_1 × C_2 / G of a product of curves by the free action of a finite group G ), yield interesting examples.

Theorem. There exist infinitely many examples of moduli spaces of complex surfaces having exactly two connected components, which are exchanged by real conjugation.

Corollaries.
(1) There are moduli spaces without real points.
(2) There are surfaces which are differentiably equivalent but not deformation equivalent.

    We will end the talk by mentioning several questions and open problems.