General expansion for period maps of Riemann surfaces

In this paper, we get the full expansion for period map from the moduli space $ mathcal{M}_g $ of curves to the coarse moduli space $ mathcal{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 Π([X]) = $ sqrt { - 1} I_g $ and the Christoffel symbols of L ²-induced Bergman metric on $ mathcal{M}_g $ .