Siegel metric and curvature of the moduli space of curves

Abstract

We study the curvature of the moduli space M g {M_g} of curves of genus g g with the Siegel metric induced by the period map j : M g → A g j:{ M_g}\rightarrow {A_g} . We give an explicit formula for the holomorphic sectional curvature of M g {M_g} along a Schiffer variation ξ P \xi _P , for P P a point on the curve X X , in terms of the holomorphic sectional curvature of A g {A_g} and the second Gaussian map. Finally we extend the Kähler form of the Siegel metric as a closed current on M ¯ g \overline {M}_g and we determine its cohomology class as a multiple of λ \lambda .