On Seshadri constants of non-simple abelian varieties

Abstract

Let (A, L) be a polarized abelian variety of dimension n. We prove two results for the Seshadri constants of abelian varieties. The first one is that if the Seshadri constant \(\epsilon (A,L)\) is relatively small with respect to \(L^n\), there exists a proper abelian subvariety B such that \(\epsilon (A,L)\) is equal to \(\epsilon (B,L|_B)\). Second, we prove that if there exists a codimension one abelian subvariety D satisfying certain numerical conditions, then \(\epsilon (D,L|_D)\) is equal to \(\epsilon (A,L)\). As an application of these results, we investigate the structure of low dimensional polarized abelian varieties whose Seshadri constants is sufficiently small with respect to \(L^n\).