On the closed embedding of the product of theta divisors into product of Jacobians and Chow groups

TL;DR

This paper extends the injectivity of push-forward homomorphisms from symmetric powers of curves to higher-dimensional varieties and their products, with applications to theta divisors and Jacobians.

Contribution

It generalizes known results on Chow group injectivity for symmetric powers to higher dimensions and products, including applications to theta divisors in Jacobians.

Findings

Proves injectivity of push-forward on Chow groups for symmetric powers of higher-dimensional varieties.

Establishes analogous results for products of symmetric powers.

Demonstrates injectivity for embeddings of theta divisors into Jacobians.

Abstract

In this article we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of $\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self product of theta divisor into the self product of the Jacobian of a smooth projective curve.