Journal of Mathematical Sciences

, Volume 160, Issue 6, pp 697–716 | Cite as

Moduli space of complex structures



We investigate the moduli space of complex structures on the Riemann sphere with marked points using signature formulas.


Modulus Space Riemann Surface Complex Manifold Euler Characteristic Mapping Class Group 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. V. Anosov and A. A. Bolibruch, “The Riemann-Hilbert problem, Asp. Math., Vieweg, Braunschweig, Wiesbaden (1994).MATHGoogle Scholar
  2. 2.
    W. Baily and A. Borel, “Compactification of arithmetic quatients of bounded symmetric domains,” Ann. Math., 84, 442–528 (1966).CrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Bruce, “Euler characteristics of real varieties,” +. Soc., 22, 213–219 (1990).MathSciNetGoogle Scholar
  4. 4.
    P. Deligne and G. D. Mostow, “Monodromy of hypergeometric functions and nonlattice integral monodromy,” Publ. IHES, 63, 5–89 (1986).MATHMathSciNetGoogle Scholar
  5. 5.
    S. K. Donaldson, “A new proof of a theorem of Narasimhan and Seshadri,” J. Differ. Geom., 18, 269–277 (1983).MATHMathSciNetGoogle Scholar
  6. 6.
    D. Eisenbud and H. Levine, “An algebraic formula for the degree of a C map germ,” Ann. Math., 106, 19–44 (1977).CrossRefMathSciNetGoogle Scholar
  7. 7.
    G. Giorgadze, “Quadratic mappings and configuration spaces,” Banach Center Publ., 62, 73–86 (2004).MathSciNetGoogle Scholar
  8. 8.
    G. Giorgadze, “Euler characteristic of certain moduli spaces,” Bull. Georgian Acad. Sci., 169, No. 2, 245–249 (2004).MathSciNetGoogle Scholar
  9. 9.
    J. Harer and D. Zagier, “The Euler characteristic of the emoduli space of curves,” Invent. Math., 85, 457–485 (1986).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Gunning, “Lectures on Riemann Surfaces, Princeton Univ. Press, Princeton (1966).MATHGoogle Scholar
  11. 11.
    Yi Hu, “Moduli spaces of stable polygons and symplectic structures on \(\bar{M}\) 0,n,” Compos. Math., 118, No. 2, 159–187 (1999).MATHCrossRefGoogle Scholar
  12. 12.
    K. Iwasaki, “Fuchsian moduli on a Riemann surface, its Poisson structure and Poincaré–Lefschetz duality,” Pac. J. Math., 155, No. 2, 319–340 (1992).MATHMathSciNetGoogle Scholar
  13. 13.
    K. Iwasaki, “Moduli and deformation for Fuchsian projective connection on a Riemann surface,” J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 38, 431–531 (1991).MathSciNetGoogle Scholar
  14. 14.
    K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida, From Gauss to Painlevé. A Modern Theory of Special Functions, Vieweg, Braunschweig (1991).MATHGoogle Scholar
  15. 15.
    M. Kapovich and J. Millson, “On the moduli space of polygons in the Euclidean plane,” J. Differ. Geom., 42, 430–464 (1995).MATHMathSciNetGoogle Scholar
  16. 16.
    R. Kaufmann, Yu. Manin, and D. Zagier, “Higher Weil–Petersson volumes of spaces of stable n-pointed curves,” Commun. Math. Phys., 181, 763–787 (1996).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    M. Kashivara and T. Kawai, Hodge structures and holonomic systems,” Proc. Jpn. Acad. Ser. A, 62, 1–4 (1986).CrossRefGoogle Scholar
  18. 18.
    S. Keel, “Intersection theory of moduli space of stable N-pointed curves of genus zero,” Trans. Amer. Math. Soc., 330, 545–574 (1992).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    G. Khimshiashvili, “Signature formulae for topological invariants,” Proc. A. Razmadze Math. Inst., 125, 1–121 (2001).MATHMathSciNetGoogle Scholar
  20. 20.
    G. Khimshiashvili, “On the local degree of a smooth mapping,” Bull. Acad. Sci. Georgian SSR, 85, 309–312 (1977).MATHGoogle Scholar
  21. 21.
    M. Kita, “On the number of parameters of linear differential equations with regular singularities on a compact Riemann surface,” Tokyo J. Math., 10, 69–75 (1987).MATHMathSciNetGoogle Scholar
  22. 22.
    K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer-Verlag, Berlin–Heidelberg (1986).MATHGoogle Scholar
  23. 23.
    E. Looijienga, “Smooth Deligne–Mumford compactifications by means of Prym level structures,” J. Algebraic Geom., 3, 283–293 (1994).MathSciNetGoogle Scholar
  24. 24.
    Yu. I. Manin, “Generating functions in algebraic geometry and sums over trees,” in: The Moduli Spaces of Curves (R. Dijkgraaf, C. Faber, and G. van der Geer, eds.), Prog. Math., 129, Birkhäuser (1995), pp. 401–418.Google Scholar
  25. 25.
    Yu. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Preprint Max-Planck-Institut für Mathematik (1994).Google Scholar
  26. 26.
    J. Milnor and J. Stasheff, Characterictic Classes, Princeton Univ. Press (1974).Google Scholar
  27. 27.
    J. Millson and J. Poritz, “Around polygons in ℝ3 and \(\mathbb{S}^3\),” Commun. Math. Phys., 218, 315–331 (2001).MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    D. Mumford, Geometric Invariant Theory, Springer-Verlag, Berlin (1965).MATHGoogle Scholar
  29. 29.
    M. Namba, Branched Coverings and Algebraic Functions, New York (1987).Google Scholar
  30. 30.
    M. S. Narasimhan and T. R. Seshadri, Stable and unitary vector bundles on a compact Riemann surface,” Ann. Math., 82, 540–564 (1965).CrossRefMathSciNetGoogle Scholar
  31. 31.
    M. Ohtsuki, “On the number of apparent singularites of a linear differential equation,” Tokyo J. Math., 5, No. 1, 23–29 (1982).MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    J. Poritz, “Parabolic vector bundles and Hermitian Yang–Mills connections over a Riemann surface,” Int. J. Math., 4, No. 3, 467–501 (1993).MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    A. Pressley and G. Segal, Loop Groups, Clarendon Press, Oxford (1984).Google Scholar
  34. 34.
    A. Ramanathan, “Stable principal bundles on a compact Riemann surface,” Math. Ann., 213, 129–152 (1975).MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    B. Riemann, “Theorie der Abel’schen Funktionen,” J. Reine Angew. Math., 54, 115–155 (1857).MATHGoogle Scholar
  36. 36.
    A. Saeki, “On the number of apparent singularities,” Proc. Jpn. Acad., Ser. A, 66, 209–213 (1990).MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    T. Sasai and S. Tsuchiya, “On a fourth-order Fuchsian differential equation of Okubo type,” Funkc. Ekvac., 34, 211–221 (1991).MATHMathSciNetGoogle Scholar
  38. 38.
    I. Satake, “On representations and compactifications of symmetric Riemannian space,” Ann. Math., 71, 77–110 (1960).CrossRefMathSciNetGoogle Scholar
  39. 39.
    Z. Szafraniec, “On the Euler characteristic of analytic and algebraic sets,” Topology, 25, 235–238 (1987).CrossRefMathSciNetGoogle Scholar
  40. 40.
    W. Thurston, “Shapes of polyhedra and triangulations of the sphere,” Geom. Topol. Monogr. 1, 511–549 (1998).CrossRefMathSciNetGoogle Scholar
  41. 41.
    K. Uhlenbeck, “Connections with L p bounds on curvature,” Commun. Math. Phys., 83, No. 1, 31–42 (1982).MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    R. Vakil, “The moduli space of curves and its tautological ring,” Notes Amer. Math. Soc., 50, No. 6, 647–658 (2003).MATHMathSciNetGoogle Scholar
  43. 43.
    V. Vargas, “The index of nonalgebraically isolated singularities,” Bol. Soc. Math. Mexica, 8, 141–147 (2002).MATHGoogle Scholar
  44. 44.
    C. T. C.Wall, “Rational Euler characteristics,” Proc. Cambridge Philos. Soc., 57, 182–300 (1961).MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    M. Yoshida, “On the number of apparent singularities of the Riemann–Hilbert problem on Riemann surfaces,” J. Math. Soc. Jpn., 49, No. 1, 145–159 (1997).MATHCrossRefGoogle Scholar
  46. 46.
    H. Zieschang, E. Vogt, and H.-D. Goldwey, “Surfaces and planar decontinuous groups,” Lect. Notes Math., 835, Springer-Verlag (1980).Google Scholar
  47. 47.
    P. G. Zograf and L. A. Takhtajan, On geometry of the moduli spaces of vector bundles over a Riemann surface, Preprint LOMI P-5-87 (1987).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Tbilisi State UniversityTbilisiGeorgia

Personalised recommendations