Moduli spaces of surfaces and real structures

Fabrizio Catanese

We give infinite series of groups $\Gamma$ and of compact complex surfaces of general type $S$ with fundamental group $\Gamma$ such that

1) any surface $S'$ with the same Euler number as $S$, and fundamental group $\Gamma$, is diffeomorphic to $S$

2) the moduli space of $S$ consists of exactly two connected components, exchanged by complex conjugation.

Whence,

i) on the one hand we give simple counter-examples to the DEF = DIFF question whether deformation type and diffeomorphism type coincide for algebraic surfaces

ii) on the other hand we get examples of moduli spaces without real points.

iii) Another interesting corollary is the existence of complex surfaces $S$ whose fundamental group $\Gamma$ cannot be the fundamental group of a real surface.

Our surfaces are surfaces isogenous to a product, i.e., they are quotients $(C_1 \times C_2)/ G $ of a product of curves by the free action of a finite group $G$.

They resemble the classical hyperelliptic surfaces, in that $G$ operates freely on $C_1$, while the second curve is a $\Bbb triangle \ curve$, meaning that $C_2 / G \equiv \PP ^1$ and the covering is branched in exactly three points.