General expansion for period maps of Riemann surfaces

In this paper, we get the full expansion for period map from the moduli space M(g) of curves to the coarse moduli space A(g) of g-dimensional principally polarized abelian varieties in Bers coordinates. This generalizes fully the famous Rauch's variational formula. As applications, we compute the curvature of Siegel metric at point [X] with Pi([X]) = root-1I(g) and the Christoffel symbols of L(2)-induced Bergman metric on M(g).