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Remarks on the geometry of moduli spaces

Author: Kefeng Liu
Journal: Proc. Amer. Math. Soc. 124 (1996), 689-695
MSC (1991): Primary 14H15, 53C55
MathSciNet review: 1291785
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Abstract: By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved.

References [Enhancements On Off] (What's this?)

  • 1. S. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Math. 35 (1971), no. 6, 1269--1293. MR 48:298
  • 2. A. Beauville, Le nombre minimun de fibres singulieres d'une courbe stable sur $P^1$, in Séminaire sur les pinceaux de courbes degenre au moins deux, ed. L. Szpiro, Asterisque 86 97--108.
  • 3. J. M. Bismut and J. B. Bost, Fibre determinant, metriques de Quillen et degenerescence des courbes, Acta Math. 165 (1990), 1--103. MR 91h:58122
  • 4. G. Faltings, Arakelov theorem for abelian varieties, Invent. Math. 73 (1983), 337--348. MR 85m:14061
  • 5. M. Gromov, Kahler hyperbolicity and $L_2$-Hodge theory, J. Differential Geom. 33 (1991), 263--292. MR 92a:58133
  • 6. G. Tian, Smoothness of the universal deformation space of compace Calabi-Yau manifolds and its Weil-Peterson metric, Mathematical Aspects of String Theory (S. T. Yau, ed.), World Scientific, Singapore, 1987, pp. 629--646. CMP 20:04
  • 7. G. Tian and S. T. Yau, Existence of Kahler-Einstein metrics on complete manifolds and their applications to algebraic geometry, Mathematical Aspects of String Theory (S. T. Yau, ed.), World Scientific, Singapore, 1987, pp. 574--628. CMP 20:04
  • 8. P. Vojta, Diophantine inequalities and Arakelov theory, in S. Lang, Introduction to Arakelov Theory, Springer-Verlag, 1988, pp. 155--178. MR 89m:11059
  • 9. S. Wolpert, On obtaining a positive line bundle from the Weil-Peterson class, Amer. J. Math. 107 (1985), 1485--1507.MR 87f:32058
  • 10. ------, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1985), 119--145. MR 87j:32070
  • 11. ------, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), 417--427. MR 91a:32030
  • 12. ------, Homology theory of moduli space of stable curves, Ann. of Math. (2) 118 (1983), 491--523. MR 86h:32036
  • 13. S. T. Yau, A general Schwarz lemma for Kahler manifolds, Amer. J. Math. 100 (1978), 197--203. MR 58:6370
  • 14. P. G. Zograf, The Liouville action on moduli spaces and uniformization on degenerating Riemann surfaces, Leningrad Math. J. 1 (1990), 941--965.MR 91c:32015
  • 15. P. Zograf and T. Takhtajan, A potential of the Weil-Peterson metric on the Torelli space, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI) 160 (1987).MR 89a:32022
  • 16. K. Ueno, Kodaira dimension of certain fiber spaces, Complex Analysis and Algebraic Geometry, Iwanami-Shoten, Tokyo, 1977, pp. 279--292.MR 56:5556
  • 17. W. Barth, C. Peters, and A. Ven de Ven, Compact complex surfaces, Springer-Verlag, Berlin and New York, 1984. MR 86c:32026
  • 18. A. N. Parshin, Algebraic curves over functional fields I, Math. USSR Izv. 2 (1968), 1145--1170. MR 41:1740
  • 19. E. D'Hoker and D. H. Phong, Geometry of quantum strings, Mathematical Aspects of String Theory (S. T. Yau, ed.), World Scientific, Singapore, 1987, pp. 29--59. CMP 20:04

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Additional Information

Kefeng Liu
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138-2901
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): June 7, 1994
Received by editor(s) in revised form: August 9, 1994
Communicated by: Peter Li
Article copyright: © Copyright 1996 American Mathematical Society