Abstract:
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 pg=q=0 and K2=9.
These surfaces also provide new counterexamples to the “Dif = Def” problem.
\Bibitem{KulKha02}
\by Vik.~S.~Kulikov, V.~M.~Kharlamov
\paper On real structures on rigid surfaces
\jour Izv. Math.
\yr 2002
\vol 66
\issue 1
\pages 133--150
\mathnet{http://mi.mathnet.ru/eng/im374}
\crossref{https://doi.org/10.1070/IM2002v066n01ABEH000374}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1917540}
\zmath{https://zbmath.org/?q=an:1055.14060}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0042343154}
Linking options:
https://www.mathnet.ru/eng/im374
https://doi.org/10.1070/IM2002v066n01ABEH000374
https://www.mathnet.ru/eng/im/v66/i1/p133
This publication is cited in the following 40 articles:
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Vik. S. Kulikov, “On divisors of small canonical degree on Godeaux surfaces”, Sb. Math., 209:8 (2018), 1155–1163
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Cartwright D.I., Steger T., “Enumeration of the 50 fake projective planes”, C. R. Math. Acad. Sci. Paris, 348:1-2 (2010), 11–13
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