Remarks on the geometry of moduli spaces

Liu, Kefeng

doi:10.1090/S0002-9939-96-03046-8

Remarks on the geometry of moduli spaces
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by Kefeng Liu

Proc. Amer. Math. Soc. 124 (1996), 689-695

DOI: https://doi.org/10.1090/S0002-9939-96-03046-8

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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

S. Ju. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293 (Russian). MR 0321933
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.
J.-M. Bismut and J.-B. Bost, Fibrés déterminants, métriques de Quillen et dégénérescence des courbes, Acta Math. 165 (1990), no. 1-2, 1–103 (French). MR 1064578, DOI 10.1007/BF02391902
G. Faltings, Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983), no. 3, 337–347. MR 718934, DOI 10.1007/BF01388431
M. Gromov, Kähler hyperbolicity and $L_2$-Hodge theory, J. Differential Geom. 33 (1991), no. 1, 263–292. MR 1085144
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.
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.
Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124, DOI 10.1007/978-1-4612-1031-3
Scott A. Wolpert, On obtaining a positive line bundle from the Weil-Petersson class, Amer. J. Math. 107 (1985), no. 6, 1485–1507 (1986). MR 815769, DOI 10.2307/2374413
Scott A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119–145. MR 842050, DOI 10.1007/BF01388794
Scott A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), no. 2, 417–472. MR 1037410
Scott Wolpert, On the homology of the moduli space of stable curves, Ann. of Math. (2) 118 (1983), no. 3, 491–523. MR 727702, DOI 10.2307/2006980
Shing Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), no. 1, 197–203. MR 486659, DOI 10.2307/2373880
P. G. Zograf, Liouville action on moduli spaces and uniformization of degenerate Riemann surfaces, Algebra i Analiz 1 (1989), no. 4, 136–160 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 4, 941–965. MR 1027464
P. G. Zograf and L. A. Takhtadzhyan, Potential of the Weil-Petersson metric on Torelli space, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160 (1987), no. Anal. Teor. Chisel i Teor. Funktsiĭ. 8, 110–120, 299 (Russian); English transl., J. Soviet Math. 52 (1990), no. 3, 3077–3085. MR 906849, DOI 10.1007/BF02342926
K. Ueno, Kodaira dimensions for certain fibre spaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 279–292. MR 0447241
W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
A. N. Paršin, Algebraic curves over function fields. I, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1191–1219 (Russian). MR 0257086
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.