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On real structures on rigid surfaces. (English. Russian original) Zbl 1055.14060

Izv. Math. 66, No. 1, 133-150 (2002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 1, 133-152 (2002).
Summary: We construct examples of rigid surfaces with a particular behaviour with respect to real structures. In one example the surface has no real structure. In another it has a unique real structure, which is not a maximal with respect to the Smith-Thom inequality. These examples give negative answers to the following problems: the existence of real surfaces in each deformation class of complex surfaces, and the existence of a maximal real surface in every complex deformation class that contains real surfaces. Moreover, we prove that there are no real surfaces among surfaces of general type with \(p_g=q=0\) and \(K^2=9\). These surfaces also provide new counterexamples to the “Dif=Def” problem.

MSC:

14P25 Topology of real algebraic varieties
14J29 Surfaces of general type
14N20 Configurations and arrangements of linear subspaces
14J10 Families, moduli, classification: algebraic theory
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