MATHEMATICS COLLOQUIUM
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, 3:354:30 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.
