Advertisement

The Curvature of the Petersson-Weil Metric on the Moduli Space of Kähler-Einstein Manifolds

  • Georg Schumacher
Part of the The University Series in Mathematics book series (USMA)

Abstract

The Petersson-Weil metric is a main tool for investigating the geometry of moduli spaces. When A. Weil considered the classical Teichmüller space from the viewpoint of deformation theory, he suggested, in 1958, investigating the Petersson inner product on the space of holomorphic quadratic differentials. He conjectured that it induced a Kähler metric on the Teichmüller space. After proving this property, Ahlfors showed, in 1961, that the holomorphic sectional and Ricci curvatures were negative. Royden’s conjecture of a precise upper bound for the holomorphic sectional curvature was proven by Wolpert and Tromba in 1986 along with the negativity of the sectional curvature.

Keywords

Vector Field Modulus Space Horizontal Lift Holomorphic Sectional Curvature Holomorphic Vector Field 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Fujiki, Coarse moduli space for polarized compact Kähler manifolds, Publ. RIMS Kyoto Univ. 24, 141–168 (1988).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    A. Fujiki and G. Schumacher, The moduli space of compact extremal Kähler manifolds and generalized Petersson-Weil metrics, Publ. RIMS Kyoto Univ. 26, 101–183 (1990).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    N. Koiso, Einstein metrics and complex structure, Invent. Math. 73, 71–106 (1983).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    A. M. Nadel, Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Proc. Nat. Acad. Sci. USA 86, 7299–7300 (1989).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    A. M. Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, preprint (1991).Google Scholar
  6. 6.
    A. M. Nadel, The behaviour of multiplier ideal sheaves under morphisms, preprint (1990).Google Scholar
  7. 7.
    A. Nannicini, Weil-Petersson metric in the space of compact polarized Kähler-Einstein manifolds of zero first Chern class, Manuscripta Math. 54, 405–438 (1986).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    G. Schumacher, Construction of the coarse moduli space of compact polarized manifolds with c, = 0, Math. Ann. 264, 81–90 (1983).MathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Schumacher, Moduli of polarized Kähler manifolds, Math. Ann. 269, 137–144 (1984).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    G. Schumacher, On the geometry of moduli spaces, Manuscripta Math. 50, 229–267 (1985).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    G. Schumacher, The theory of Teichmüller spaces—a view towards moduli spaces of Kähler manifolds, in Encyclopedia of Mathematical Sciences (W. Barth and R. Narasimhan, eds.) (Springer Verlag, Berlin, 1990), Vol. 69, pp. 251–310.Google Scholar
  12. 12.
    Y.-T. Siu, Curvature of the Weil-Petersson metric in the moduli space of compact Kähler-Einstein manifolds of negative first Chern class, in Complex Analysis, Papers in Honour of Wilhelm Stoll (P.-M. Wong and A. Howard, eds.) (Vieweg, Braunschweig, 1986).Google Scholar
  13. 13.
    Y. T. Siu, The existence of Kähler-Einstein metrics on manifolds with positive anticanon-ical line bundle and a suitable finite symmetry group, Ann. of Math. 127, 585–627 (1988).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Tian Gang, On Kähler-Einstein metrics on certain Kähler manifolds with c 1 (M) > 0, Invent. Math. 89, 225–246 (1987).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Tian Gang and S. T. Yau, Kähler-Einstein metrics on complex surfaces with c x > 0, Comm. Math. Phys. 112, 175–203 (1987).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    S. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85, 119–145(1986).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Georg Schumacher
    • 1
  1. 1.Institut für MathematikRuhr-Universität BochumBochum 1Germany

Personalised recommendations