Geometric Approach to the Weil-Petersson Symplectic Form

Abstract

In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.