Geodesic
On real structures on rigid surfaces
Izvestiya. Mathematics , Tome 66 (2002) no. 1, pp. 133-150.

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We construct examples of rigid surfaces (that is, surfaces whose deformation class consists of a unique surface) 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 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 maximal real surfaces 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. 
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Vik. S. Kulikov; V. M. Kharlamov. On real structures on rigid surfaces. Izvestiya. Mathematics , Tome 66 (2002) no. 1, pp. 133-150. http://geodesic.mathdoc.fr/item/IM2_2002_66_1_a5/

[1] Barth W., Peters C., Van de Ven A., Compact Complex Surfaces, Springer-Verlag, N. Y., 1984 | MR | Zbl

[2] Barthel G., Hirzebruch F., Höfer Th., Geradenkonfigurationen und Algebraische Flöhen, Friedr. Vieweg Sohn, Braunschweig, 1987 | Zbl

[3] Degtyarev A., Kharlamov V., “Topologicheskie svoistva veschestvennykh algebraicheskikh mnogoobrazii”, UMN, 55:4 (2000), 129–212 | MR | Zbl

[4] Grauert H., Remmert R., “Komplexe Raüme”, Math. Ann., 136 (1958), 245–318 | DOI | MR | Zbl

[5] Hirzebruch F., “Arrangements of lines and algebraic surfaces”, Arithmetics and Geometry, V. II, Prog. Math., 36, Birkháuser, 1983, 113–140 | MR

[6] Ishida M.-N., “The Irregularities of Hirzebruch's Examples of Surfaces of General Type with $c_1^2=3c_2$”, Math. Ann., 262 (1983), 407–420 | DOI | MR | Zbl

[7] Itenberg I., “Contre-exemples à la conjecture de Ragsdale”, C. R. Acad. Sci. Paris, 317 (1993), 277–282 | MR | Zbl

[8] Kharlamov V., “Variétés de Fano réelles”, Sém. Bourbaki, no. 872, 2000

[9] Manetti M., “On the moduli space of diffeomorphic algebraic surfaces”, Invent. Math., 143 (2001), 29–76 | DOI | MR | Zbl

[10] Miyaoka Y., “On algebraic surfaces with positive index”, Classification of algebraic and analytic manifolds, Prog. Math., 39, Birkháuser, 1983, 281–301 | MR

[11] Mumford D., “An algebraic surface with $K$ ample, $K^2=9$, $p_g=q=0$”, Amer. J. Math., 101 (1979), 233–244 | DOI | MR | Zbl

[12] Namba M., Branched coverings and algebraic functions, Longman Scientific Technical, N.Y., 1987 | MR | Zbl

[13] Preissman A., “Quelques propriétés globales des espaces de Riemann”, Comment. Math. Helv., 15 (1943), 175–216 | DOI | MR

[14] Yau S-T., “Calaby's conjecture and some new results in algebraic geometry”, Proc. Nat. Acad. Sci. USA, 74 (1977), 1798–1799 | DOI | MR | Zbl