Holomorphic vector fields and Chow groups

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.