Moduli spaces of surfaces and real structures

Abstract

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 counterexamples 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 triangle curve, meaning that $C_2 / G \equiv \mathbb{P}^1$ and the covering is branched in exactly three points.