The Jacobian of a Riemann surface and the geometry of the cut locus of simple closed geodesics

Abstract

To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called a period Gram matrix. This paper provides upper and lower bounds for all the entries of the period Gram matrix with respect to a suitable homology basis. These bounds depend on the geometry of the cut locus of non-separating simple closed geodesics. Assuming that the cut loci can be calculated, a theoretical approach is presented followed by an example where the upper bound is sharp. Finally we give practical estimates based on the Fenchel-Nielsen coordinates of surfaces of signature (1,1), or Q-pieces. The methods developed here have been applied to surfaces that contain small non-separating simple closed geodesics in [BMMS].