Inventiones mathematicae

, Volume 153, Issue 3, pp 631–678 | Cite as

Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Article

Abstract

Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M.: Riemann surfaces and spin structures. Ann. Sci. Éc. Norm. Supér., IV. Sér. 4, 47–62 (1971) Google Scholar
  2. 2.
    Calabi, E.: An intrinsic characterization of harmonic 1-forms. Global Analysis, Papers in Honor of K. Kodaira, D.C. Spencer and S. Iyanaga (eds.), 101–117 (1969) Google Scholar
  3. 3.
    Eskin, A., Masur, H., Zorich, A.: Moduli spaces of Abelian differentials: the principal boundary, counting problems and the Siegel–Veech constants, 88 pp. Submitted to Publ. Math., Inst. Hautes Étud. Sci.; electronic version in arXiv:math.DS/0202134 Google Scholar
  4. 4.
    Hubbard, J., Masur, H.: Quadratic differentials and foliations, Acta Math. 142, 221–274 (1979) Google Scholar
  5. 5.
    Johnson, D.: Spin structures and quadratic forms on surfaces. J. Lond. Math. Soc., II. Ser. 22, 365–373 (1980) Google Scholar
  6. 6.
    Katok, A.B.: Invariant measures of flows on oriented surfaces, Soviet Math. Dokl. 14, 1104–1108 (1973) MATHGoogle Scholar
  7. 7.
    Kontsevich, M., Zorich, A.: Lyapunov exponents and Hodge theory. Preprint IHES M/97/13, 1–16 electronic version: hep-th/9701164 Google Scholar
  8. 8.
    Lanneau, E.: Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities. To appear in Comment. Math. Helv.; electronic version in arXiv:math.GT/0210099 Google Scholar
  9. 9.
    Maier, A.G.: Trajectories on closed orientable surfaces. Sb. Math. 12, 71–84 (1943) (in Russian) Google Scholar
  10. 10.
    Masur, H.: The Jenkins–Strebel differentials with one cylinder are dense. Comment. Math. Helv. 54, 179–184 (1979) Google Scholar
  11. 11.
    Masur, H.: Interval exchange transformations and measured foliations. Ann. Math. 115, 169–200 (1982) MathSciNetMATHGoogle Scholar
  12. 12.
    Masur, H., Smillie, J.: Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms. Comment. Math. Helv. 68, 289–307 (1993) MathSciNetMATHGoogle Scholar
  13. 13.
    Milnor, J.: Remarks concerning spin manifolds. Differential and Combinatorial Topology (in Honor of Marston Morse). Princeton 1965 Google Scholar
  14. 14.
    Mumford, D.: Theta-characteristics of an algebraic curve. Ann. Sci. Éc. Norm. Supér., IV. Sér. 2, 181–191 (1971) Google Scholar
  15. 15.
    Rauzy, G.: Echanges d’intervalles et transformations induites. Acta Arith. 34, 315–328 (1979) MathSciNetMATHGoogle Scholar
  16. 16.
    Strebel, K.: Quadratic differentials. Springer 1984 Google Scholar
  17. 17.
    Veech, W.A.: Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 115, 201–242 (1982) MathSciNetMATHGoogle Scholar
  18. 18.
    Veech, W.A.: The Teichmüller geodesic flow. Ann. Math. 124, 441–530 (1986) MathSciNetMATHGoogle Scholar
  19. 19.
    Veech, W.A.: Moduli spaces of quadratic differentials. J. Anal. Math. 55, 117–171 (1990) MathSciNetMATHGoogle Scholar
  20. 20.
    Zorich, A.: How do the leaves of a closed 1-form wind around a surface. In: Pseudoperiodic Topology. AMS Translations, Ser. 2, 197, 135–178. Providence, RI: Am. Math. Soc. 1999 Google Scholar
  21. 21.
    Zorich, A.: Explicit construction of a representative of any extended Rauzy class. To appearGoogle Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  2. 2.Institut Mathématique de RennesUniversité Rennes-1RennesFrance

Personalised recommendations