Theta divisors of abelian varieties and push-forward homomorphism at the level of Chow groups

TL;DR

This paper proves that for abelian varieties embedded in Jacobians, the inclusion of certain theta divisors induces an injective push-forward on Chow groups, extending to all principally polarized abelian varieties.

Contribution

It establishes the injectivity of the push-forward homomorphism at the Chow group level for theta divisors in a broad class of abelian varieties.

Findings

Injective push-forward homomorphism for theta divisors in abelian varieties.

Extension of results to all principally polarized abelian varieties.

Connection between theta divisors and Chow group embeddings.

Abstract

In this text we prove that if an abelian variety $A$ admits of an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $\Th_A$ to be the divisor $\Th_C\cap A$, where $\Th_C$ denotes the theta divisor of $J(C)$, then the embedding of $\Th_A$ into $A$ induces an injective push-forward homomorphism at the level of Chow groups. We show that this is the case for every principally polarized abelian varieties.