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 xP , 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 K¨ahler form of the Siegel metric as a closed current on Mg and we determine its cohomology class as a multiple of lambda.