Holomorphic vector fields and rationality

Abstract

We show that a nonsingular complex projective variety admitting a holomorphic vector field with nonempty isolated zeroes, is rational using a key technique by Harvey-Lawson on finite volume flows. This statement was conjectured by J. Carrell. By the same technique, we obtain a uniform upper bound of Betti numbers of nonsingular complex projective variety admitting a holomorphic vector field with exact one zero point. Such an upper bound depends only on the dimension of the variety, which is a stronger version of a result of Akyildiz and Carrell.