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**Moduli spaces of surfaces and real structures.**
*(English)*
Zbl 1042.14011

Summary: 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 of 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.

(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 of 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.

### MSC:

14J29 | Surfaces of general type |

14J10 | Families, moduli, classification: algebraic theory |

14P25 | Topology of real algebraic varieties |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |

32J15 | Compact complex surfaces |