Siegel metric and curvature of the moduli space of curves

We study the curvature of the moduli space Mg of curves of genus g with the Siegel metric induced by the period map j : Mg→ Ag .We give an explicit formula for the holomorphic sectional curvature of Mg along a Schiffer variation εP,for P a point on the curve X, in terms of the holomorphic sectional curvature of Ag and the second Gaussian map. Finally we extend the Kâhler form of the Siegel metric as a closed current on M̄g and we determine its cohomology class as a multiple of λ.