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Israel Journal of Mathematics

, Volume 133, Issue 1, pp 189–238 | Cite as

Viro theorem and topology of real and complex combinatorial hypersurfaces

  • Ilia Itenberg
  • Eugenii Shustin
Article

Abstract

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP n and are topologically “glued” out of algebraic hypersurfaces in (ℂ*) n . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces.

Keywords

Euler Characteristic Toric Variety Double Covering Real Algebraic Variety Coordinate Hyperplane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    M. F. Atiyah,Convexity and commuting Hamiltonians, The Bulletin of the London Mathematical Society14 (1982), 1–15.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. F. Atiyah,Angular momentum, convex polyhedra and algebraic geometry, Proceedings of the Edinburgh Mathematical Society26 (1983), 121–138.MATHMathSciNetGoogle Scholar
  3. [3]
    M. F. Atiyah and I. M. Singer,The index of elliptic operators: III, Annals of Mathematics87 (1968), 546–604.CrossRefMathSciNetGoogle Scholar
  4. [4]
    G. E. Bredon,Introduction to Compact Transformation Groups (Pure and Applied Mathematics46), Academic Press, New York, 1972.MATHGoogle Scholar
  5. [5]
    R. Connelly and D. W. Henderson,A convex 3-complex not simplicially isomorphic to a strictly convex complex, Proceedings of the Cambridge Philosophical Society88 (1980), 299–306.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    V. I. Danilov and A. G. Khovanskii,Newton polyhedra and an algorithm for computing Hodge-Deligne numbers, Izvestiya Akademii Nauk SSSR50 (1986), (Russian); English transl.: Mathematics of the USSR Izvestiya29:2 (1987), 279–298.Google Scholar
  7. [7]
    A. Degtyarev and V. Kharlamov,Topological properties of real algebraic varieties: Rokhlin’s way, Uspekhi Matematicheskikh Nauk55 (2000), no. 4(334), 129–212 (Russian).MathSciNetGoogle Scholar
  8. [8]
    J. A. de Loera and F. J. Wicklin,On the need of convexity in patchworking, Advances in Applied Mathematics20 (1998), 188–219.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    I. Fary,Cohomologie des variétés algébriques, Annals of Mathematics65 (1957), 21–73.CrossRefMathSciNetGoogle Scholar
  10. [10]
    S. Fiedler-Le Touzé and S. Yu. Orevkov,A flexible affine M-sextic non-realizable algebraically, Journal of Algebraic Geometry11 (2002), 293–310.MATHMathSciNetGoogle Scholar
  11. [11]
    M. H. Freedman and F. Quinn,Topology of 4-manifolds (Princeton Mathematical Series39), Princeton University press, Princeton, NJ, 1990.MATHGoogle Scholar
  12. [12]
    W. Fulton,Introduction to Toric Varieties (Annals of Mathematics Studies131), Princeton University Press, Princeton, NJ, 1993.MATHGoogle Scholar
  13. [13]
    I. M. Gelfand, M. M. Kapranov and A. V. Zelevinski,Discriminants Resultants and Multidimensional Determinants, Birkhäuser, Boston 1994.MATHGoogle Scholar
  14. [14]
    M. Gromov,Pseudo-holomorphic curves in sympletic manifolds, Inventiones Mathematicae82 (1985), 307–347.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    D. A. Gudkov,Topology of real projective algebraic varieties, Russian Mathematical Surveys29 (1974), 3–79.CrossRefMathSciNetGoogle Scholar
  16. [16]
    B. Haas,The Ragsdale conjecture for maximal T-curves, Preprint, Universität Basel, 1997.Google Scholar
  17. [17]
    F. Hirzebruch,The signature of ramified coverings, inGlobal Analysis (Papers in honor of K. Kodaira), University of Tokyo Press, Tokyo, 1969, pp. 253–265.Google Scholar
  18. [18]
    F. Hirzebruch,The signature theorem: reminiscences and recreation, inProspects in Mathematics (Annals of Mathematics Studies,70), Princeton University Press, Princeton, 1971, pp. 3–31.Google Scholar
  19. [19]
    I. Itenberg,Counterexamples to Ragsdale conjecture and T-curves, inContemporary, Mathematics 182 (Proc. Conf. Real Alg. Geom., December 17–21, 1993, Michigan, ed.S. Akbulut), AMS, Providence, RI, 1995, pp. 55–72.Google Scholar
  20. [20]
    I. Itenberg and E. Shustin,Combinatorial patchworking of real pseudo-holomorphic curves. Turkish Journal of Mathematics26 (2002), 27–51.MATHMathSciNetGoogle Scholar
  21. [21]
    I. Itenberg and O. Viro,Patchworking algebraic curves disproves the Ragsdale conjecture, The Mathematical Intelligencer18 (1996), 19–28.MATHMathSciNetGoogle Scholar
  22. [22]
    V. Kharlamov,New congruences for the Euler characteristic of real algebraic manifolds, Functional Analysis and its Applications7 (1973), 147–150.MATHCrossRefGoogle Scholar
  23. [23]
    V. Kharlamov,Additional conruences for the Euler characteristic of even-dimensional real algebraic manifolds, Functional Analysis and its Applications9 (1975), 134–141.MATHCrossRefGoogle Scholar
  24. [24]
    S. Yu. Orevkov,Link theory and oval arragements of real algebraic curves, Topology38 (1999), 779–810.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    S. Yu. Orevkov and E. Shustin,Flexible, algebraically unrealizable curves: Rehabilitation of Hilbert-Rohn-Gudkov approach, Journal für die reine und angewandte Mathematik (2002), to appear.Google Scholar
  26. [26]
    P. Parenti,Combinatorics of dividing T-curves, Ph.D. Thesis, Università di Pisa, 1999.Google Scholar
  27. [27]
    J.-J. Risler,Construction d’hypersurfaces réelles [d’après Viro],Séminaire N. Bourbaki, no. 763, Vol. 1992–93, Novembre 1992.Google Scholar
  28. [28]
    V. A. Rokhlin,Congruences modulo 16 in Hilbert’s 16th, problem, Functional Analysis and its Applications6 (1972), 301–306.MATHCrossRefGoogle Scholar
  29. [29]
    V. A. Rokhlin,Complex topological characteristics of real algebraic curves, Russian Mathematical Surveys33 (1978), 85–98.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    F. Santos,Improved counterexamples to the Ragsdale conjecture, Preprint, Universidad de Cantabria, 1994.Google Scholar
  31. [31]
    E. Shustin,Topology of real plane algebraic curves, inProc. Intern. Conf. Real Algebraic Geometry, Rennes, June 24–29 1991, Lecture Notes in Mathematics1524, Springer, Berlin, 1992, pp. 97–109.Google Scholar
  32. [32]
    O. Ya. Viro,Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves, inProc. Leningrad Int. Topological Conf., Leningrad, Aug. 1982, Nauka, Leningrad, 1983, pp. 149–197 (Russian).Google Scholar
  33. [33]
    O. Ya. Viro,Gluing of plane real algebraic curves and construction of curves of degrees 6 and 7, Lecture Notes in Mathematics1060, Springer, Berlin, 1984, pp. 187–200.Google Scholar
  34. [34]
    O. Ya. Viro,Real plane curves of degrees 7 and 8: new prohibitions, Mathematics of the USSR Izvestia23 (1984), 409–422.MATHCrossRefGoogle Scholar
  35. [35]
    O. Ya. Viro,Progress in the topology of real algebrraic varieties over the last six years, Russian Mathematical Surveys41 (1986), 55–82.MATHCrossRefGoogle Scholar
  36. [36]
    O. Ya. Viro,Real algebraic plane curves: constructions with controlled topology, Leningrad Mathematical Journal1 (1990), 1059–1134.MATHMathSciNetGoogle Scholar
  37. [37]
    O. Ya. Viro,Mutual position of hypersurfaces in projective space, inGeometry of Differential Equations (American Mathematical Society Transl., Ser. 2,186), American Mathematical Society, Providence, RI, 1998, pp. 161–176.Google Scholar
  38. [38]
    O. Ya. Viro,Patchworking real algebraic varieties. Available at http://www.math.uu.se/∼oleg/pw.ps.Google Scholar
  39. [39]
    J.-Y. Welschinger, Courbes algébriques rèelles et courbes flexibles sur les surfaces réglées de base ℂP 1, Proceedings of the London Mathematical Society, (3)85 (2002), 367–392.MATHCrossRefMathSciNetGoogle Scholar
  40. [40]
    G. Wilson,Hilbert’s sixteenth problem, Topology17 (1978), 53–73.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    G. Ziegler,Lectures of Polytopes (Graduate Texts in Mathematics152), Springer-Verlag, Berlin, 1995.Google Scholar

Copyright information

© Hebrew University 2003

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique de RennesCNRSRennes CedexFrance
  2. 2.School of Mathematical SciencesTel Aviv University Ramat AvivTel AvivIsrael

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