On a numerical characterization of non-simple principally polarized abelian varieties

TL;DR

This paper develops a numerical framework to characterize non-simple principally polarized abelian varieties, linking their subvarieties' classes to intersection numbers and describing their moduli space explicitly over complex numbers.

Contribution

It introduces a canonical association of numerical classes to abelian subvarieties and characterizes these classes via intersection numbers, providing an explicit moduli space description.

Findings

Numerical classes are characterized by intersection numbers with the polarization.

Cycle classes of subvarieties are described in terms of their numerical divisor classes.

Explicit description of the moduli space for non-simple varieties with fixed numerical class.

Abstract

To every abelian subvariety of a principally polarized abelian variety $(A, \mathcal{L})$ we canonically associate a numerical class in the N\'eron-Severi group of $A$. We prove that these classes are characterized by their intersection numbers with $ \mathcal{L}$; moreover, the cycle class induced by an abelian subvariety in the Chow ring of $A$ modulo algebraic equivalence can be described in terms of its numerical divisor class. Over the field of complex numbers, this correspondence gives way to an explicit description of the (coarse) moduli space that parametrizes non-simple principally polarized abelian varieties with a fixed numerical class.