Holomorphic vector fields and Chow groups

TL;DR

This paper establishes an isomorphism between Chow groups and rational homology groups for certain smooth complex projective varieties with holomorphic vector fields, leading to verification of key conjectures.

Contribution

It demonstrates the isomorphism of Chow groups and rational homology for varieties with holomorphic vector fields and applies this to verify major conjectures.

Findings

Chow groups are isomorphic to rational homology groups under specified conditions.

Verification of Friedlander-Mazur and Generalized Hodge conjectures for these varieties.

Chow and Lawson homology groups with rational coefficients are explicitly described.

Abstract

We show that the chow group of $p$-cycles with rational coefficients are isomorphic to the corresponding rational homology groups for smooth complex projective varieties carrying a holomorphic vector field with an isolated zero locus. As applications, we obtain Chow groups and Lawson homology groups with rational coefficients and verify the Friedlander-Mazur conjecture and the Generalized Hodge conjecture for those varieties.