On Seshadri constants of non-simple abelian varieties

TL;DR

This paper investigates conditions under which the Seshadri constant of a polarized abelian variety equals that of its subvarieties, providing explicit computations for certain non-simple abelian varieties.

Contribution

It establishes criteria relating Seshadri constants of abelian varieties and their subvarieties, and computes these constants for specific non-simple abelian threefolds.

Findings

Seshadri constant equals that of a subvariety under certain smallness conditions

Seshadri constant on abelian surfaces is computed by a unique curve when small

Explicit Seshadri constants are calculated for some non-simple abelian threefolds

Abstract

We prove that the Seshadri constant of a polarized abelian variety is equal to the Seshadri constant of its abelian subvariety if the Seshadri constant is relatively small with respect to its degree, or it contains an abelian divisor which has sufficiently small degree. As an application of these results, we show that the Seshadri constant of a polarized abelian surface is computed by the unique Seshadri curve if the Seshadri constant is small enough. As another application, we compute the Seshadri constants of some non-simple polarized abelian threefolds which contain principally polarized abelian surfaces.