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Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 1, Pages 133–152 (Mi izv374)  

This article is cited in 34 scientific papers (total in 34 papers)

On real structures on rigid surfaces

Vik. S. Kulikova, V. M. Kharlamovb

a Steklov Mathematical Institute, Russian Academy of Sciences
b University Louis Pasteur

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 $p_g=q=0$ and $K^2=9$.
These surfaces also provide new counterexamples to the “Dif = Def” problem.


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English version:
Izvestiya: Mathematics, 2002, 66:1, 133–150

Bibliographic databases:

UDC: 512.7+515.1
MSC: 14P25, 14J29
Received: 09.01.2001

Citation: Vik. S. Kulikov, V. M. Kharlamov, “On real structures on rigid surfaces”, Izv. RAN. Ser. Mat., 66:1 (2002), 133–152; Izv. Math., 66:1 (2002), 133–150

Citation in format AMSBIB
\by Vik.~S.~Kulikov, V.~M.~Kharlamov
\paper On real structures on rigid surfaces
\jour Izv. RAN. Ser. Mat.
\yr 2002
\vol 66
\issue 1
\pages 133--152
\jour Izv. Math.
\yr 2002
\vol 66
\issue 1
\pages 133--150

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    This publication is cited in the following articles:
    1. Kharlamov V., Kulikov V., “Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curves”, C. R. Acad. Sci. Paris Sér. I Math., 333:9 (2001), 855–859  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Degtyarev A., Kharlamov V., “Real rational surfaces are quasi-simple”, J. Reine Angew. Math., 551 (2002), 87–99  crossref  mathscinet  zmath  isi  elib
    3. Catanese F., “Moduli spaces of surfaces and real structures”, Ann. of Math. (2), 158:2 (2003), 577–592  crossref  mathscinet  zmath  isi  scopus
    4. Catanese F., Frediani P., “Real Hyperelliptic Surfaces and the Orbifold Fundamental Group”, J. Inst. Math. Jussieu, 2:2 (2003), 169–233  crossref  mathscinet  isi  scopus
    5. Degtyarev A., Itenberg I., Kharlamov V., “Finiteness and quasi-simplicity for symmetric K3-surfaces”, Duke Math. J., 122:1 (2004), 1–49  crossref  mathscinet  zmath  isi  scopus
    6. Bauer I., Catanese F., Grunewald F., “Beauville surfaces without real structures”, Geometric Methods in Algebra and Number Theory, Progress in Mathematics, 235, 2005, 1–42  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Vik. S. Kulikov, V. M. Kharlamov, “Surfaces with DIF$\ne$DEF real structures”, Izv. Math., 70:4 (2006), 769–807  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Bauer I., Catanese F., Grunewald F., “Chebycheff and Belyi polynomials, Dessins d'Enfants, Beauville surfaces and group theory”, Mediterr. J. Math., 3:2 (2006), 121–146  crossref  mathscinet  zmath  isi  scopus
    9. van Opstall M., “Stable degenerations of surfaces isogenous to a product of curves”, Proc. Amer. Math. Soc., 134:10 (2006), 2801–2806  crossref  mathscinet  zmath  isi  scopus
    10. Kharlamov V., “Overview of topological properties of real algebraic surfaces”, Algebraic Geometry and Geometric Modeling, Mathematics and Visualization, 2006, 103–117  crossref  mathscinet  zmath  isi  scopus
    11. Vik. S. Kulikov, “Hurwitz curves”, Russian Math. Surveys, 62:6 (2007), 1043–1119  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Catanese F., “Q.E.D. for algebraic varieties”, J. Differential Geom., 77:1 (2007), 43–75  crossref  mathscinet  zmath  isi  scopus
    13. Catanese F., Wajnryb B., “Diffeomorphism of simply connected algebraic surfaces”, J. Differential Geom., 76:2 (2007), 177–213  crossref  mathscinet  zmath  isi  scopus
    14. Catanese F., “Differentiable and deformation type of algebraic surfaces, real and symplectic structures”, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., 1938, Springer, Berlin, 2008, 55–167  crossref  mathscinet  zmath  adsnasa  isi
    15. Manetti M., “Smoothings of singularities and deformation types of surfaces”, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., 1938, Springer, Berlin, 2008, 169–230  crossref  mathscinet  zmath  isi
    16. Vik. S. Kulikov, V. M. Kharlamov, “Automorphisms of Galois coverings of generic $m$-canonical projections”, Izv. Math., 73:1 (2009), 121–150  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. Catanese F., “Canonical symplectic structures and deformations of algebraic surfaces”, Commun. Contemp. Math., 11:3 (2009), 481–493  crossref  mathscinet  zmath  isi  scopus
    18. Nurislamov O.R., Shagapov V.Sh., “Gas injection into a moist porous medium with the formation of a gas hydrate”, J. Appl. Math. Mech., 73:5 (2009), 581–591  crossref  mathscinet  zmath  isi  elib  scopus
    19. Cartwright D.I., Steger T., “Enumeration of the 50 fake projective planes”, C. R. Math. Acad. Sci. Paris, 348:1-2 (2010), 11–13  crossref  mathscinet  zmath  isi  scopus
    20. Prasad G., Yeung Sai-Kee, “Addendum to “Fake projective planes” Invent. Math. 168, 321–370 (2007)”, Invent Math., 2010  crossref  mathscinet  isi  scopus
    21. Bauer I., Catanese F., Pignatelli R., “Surfaces of General Type with Geometric Genus Zero: a Survey”, Complex and Differential Geometry, Springer Proceedings in Mathematics, 8, eds. Ebeling W., Hulek K., Smoczyk K., Springer-Verlag Berlin, 2011, 1–48  crossref  mathscinet  zmath  isi  scopus
    22. NERMİN SALEPCİ, “CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS VIA NECKLACE DIAGRAMS”, J. Knot Theory Ramifications, 21:09 (2012), 1250089  crossref  mathscinet  zmath  isi  scopus
    23. Sai-Kee Yeung, “Exotic structures arising from fake projective planes”, Sci. China Math, 2012  crossref  mathscinet  isi  scopus
    24. Prasad G., Yeung S.-K., “Nonexistence of Arithmetic Fake Compact Hermitian Symmetric Spaces of Type Other Than a(N) (N <= 4)”, J. Math. Soc. Jpn., 64:3 (2012), 683–731  crossref  mathscinet  zmath  isi  scopus
    25. Keum J., “Toward a Geometric Construction of Fake Projective Planes”, Rend. Lincei-Mat. Appl., 23:2 (2012), 137–155  crossref  mathscinet  zmath  isi  scopus
    26. Yeung S.-K., “Classification of Surfaces of General Type with Euler Number 3”, J. Reine Angew. Math., 679 (2013), 1–22  crossref  mathscinet  zmath  isi  scopus
    27. Vik. S. Kulikov, V. M. Kharlamov, “On numerically pluricanonical cyclic coverings”, Izv. Math., 78:5 (2014), 986–1005  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    28. Sergey Galkin, Ludmil Katzarkov, Anton Mellit, Evgeny Shinder, “Derived categories of Keum's fake projective planes”, Advances in Mathematics, 278 (2015), 238  crossref  mathscinet  zmath  scopus
    29. V.S.. Kulikov, Eugenii Shustin, “On rigid plane curves”, European Journal of Mathematics, 2015  crossref  mathscinet  scopus
    30. F. Catanese, “Topological methods in moduli theory”, Bull. Math. Sci, 2015  crossref  mathscinet  scopus
    31. Keum J., “Q-Homology Projective Planes With Nodes Or Cusps”, Algebraic Geometry in East Asia - Taipei 2011, Advanced Studies in Pure Mathematics, 65, eds. Chen J., Chen M., Kawamata Y., Keum J., Math Soc Japan, 2015, 143–158  mathscinet  zmath  isi
    32. Yeung S.-K., “Foliations Associated to Harmonic Maps on Some Complex Two Ball Quotients”, Sci. China-Math., 60:6, SI (2017), 1137–1148  crossref  mathscinet  zmath  isi  scopus
    33. Dubouloz A. Mangolte F., “Fake Real Planes: Exotic Affine Algebraic Models of R-2”, Sel. Math.-New Ser., 23:3 (2017), 1619–1668  crossref  mathscinet  zmath  isi  scopus
    34. Vik. S. Kulikov, “On divisors of small canonical degree on Godeaux surfaces”, Sb. Math., 209:8 (2018), 1155–1163  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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