tailieunhanh - Đề tài " Moduli spaces of surfaces and real structures "

We give infinite series of groups Γ and of compact complex surfaces of general type S with fundamental group Γ such that 1) Any surface S with the same Euler number as S, and fundamental group Γ, 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). | Annals of Mathematics Moduli spaces of surfaces and real structures By Fabrizio Catanese Annals of Mathematics 158 2003 577 592 Moduli spaces of surfaces and real structures By Fabrizio Catanese This article is dedicated to the memory of Boris Moisezon Abstract We give infinite series of groups r and of compact complex surfaces of general type S with fundamental group r such that 1 Any surface S with the same Euler number as S and fundamental group r 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 r cannot be the fundamental group of a real surface. Our surfaces are surfaces isogenous to a product . they are quotients Cl X C2 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 C1 while the second curve is a triangle curve meaning that C2 G P1 and the covering is branched in exactly three points. The research of the author was performed in the realm of the SCHWERPUNKT Globale Methode in der komplexen Geometrie and of the EAGER EEC Project. 578 FABRIZIO CATANESE 1. Introduction Let S be a minimal surface of general type then to S we attach two positive integers x x OS y KS which are invariants of the oriented topological type of S. The moduli space of the surfaces with invariants x y is a quasi-projective variety defined over the integers in particular it is a real variety similarly for the Hilbert scheme of 5-canonical embedded canonical models of which the moduli space is a quotient cf. Bo Gie . For fixed x y we have several possible topological types but by the .

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