Push-forwards of Chow groups of smooth ample divisors

TL;DR

This paper proposes a homological Lefschetz conjecture for Chow groups, illustrates it with examples, and proves the injectivity of the push-forward map induced by the Theta divisor embedding in a Jacobian.

Contribution

It introduces a new homological Lefschetz conjecture for Chow groups and proves a key injectivity result for the Theta divisor embedding.

Findings

Homological Lefschetz conjecture formulated and illustrated with examples

Injectivity of push-forward morphism for Theta divisor in Jacobian proved

Provides evidence supporting the conjecture through key cases

Abstract

We introduce a homological Lefschetz conjecture on (rational) Chow groups, which can be deduced from some well known conjectures, and illustrate it by a series of key examples. We then prove the injectivity of the push-forward morphism on Chow groups, induced by the closed embedding of the Theta divisor in it's Jacobian $J(C)$. Here $C$ is a smooth irreducible complex projective curve.