Siegel metric and curvature of the moduli space of curves

Colombo, Elisabetta; Frediani, Paola

doi:10.1090/S0002-9947-09-04845-4

Siegel metric and curvature of the moduli space of curves
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by Elisabetta Colombo and Paola Frediani

Trans. Amer. Math. Soc. 362 (2010), 1231-1246

DOI: https://doi.org/10.1090/S0002-9947-09-04845-4

Published electronically: October 19, 2009

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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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