Seshadri constants and periods of polarized abelian varieties

Consider a polarized abelian variety $(A,L)$ over the field of complex numbers. Following Demailly, one can associate to $(A,L)$ a real number $\epsilon(A,L)$, its {\em Seshadri constant}, which in effect measures how much of the positivity of $L$ can be concentrated at any given point of $A$. There has been considerable recent interest in finding bounds on the Seshadri constants of abelian varieties and on smooth projective varieties in general. In the present paper we first generalize an approach of Buser and Sarnak to give a lower bound on the minimal period length of $(A,L)$ in terms of the type of the polarization, which by a recent result of Lazarsfeld leads to a lower bound on the Seshadri constant of $(A,L)$. Secondly, we consider Prym varieties and show that they have small Seshadri constants and therefore unusually small periods. Finally, we obtain refined results for the case of abelian surfaces, which imply in particular the surprising fact that Seshadri constants are always rational in this case.