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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Siegel metric and curvature of the moduli space of curves


Authors: Elisabetta Colombo and Paola Frediani
Journal: Trans. Amer. Math. Soc. 362 (2010), 1231-1246
MSC (2000): Primary 14H10, 14H15, 14K25, 53C42, 53C55
Published electronically: October 19, 2009
MathSciNet review: 2563728
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Abstract: We study the curvature of the moduli space $ {M_g}$ of curves of genus $ g$ with the Siegel metric induced by the period map $ j:{ M_g}\rightarrow {A_g}$. We give an explicit formula for the holomorphic sectional curvature of $ {M_g}$ along a Schiffer variation $ \xi_P$, for $ P$ a point on the curve $ X$, in terms of the holomorphic sectional curvature of $ {A_g}$ and the second Gaussian map. Finally we extend the Kähler form of the Siegel metric as a closed current on $ \overline{M}_g$ and we determine its cohomology class as a multiple of $ \lambda$.


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Additional Information

Elisabetta Colombo
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133, Milano, Italy
Email: elisabetta.colombo@unimi.it

Paola Frediani
Affiliation: Dipartimento di Matematica, Università di Pavia, via Ferrata 1, I-27100 Pavia, Italy
Email: paola.frediani@unipv.it

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04845-4
PII: S 0002-9947(09)04845-4
Received by editor(s): July 19, 2007
Published electronically: October 19, 2009
Additional Notes: The authors thank Gilberto Bini and Pietro Pirola for several fruitful suggestions and discussions on the subject. The present research took place in the framework of the PRIN 2005 MIUR: “Spazi dei moduli e teoria di Lie” and PRIN 2006 of MIUR: “Geometry of algebraic varieties”.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.