Holomorphic vector fields and rationality

TL;DR

This paper proves that complex projective varieties with certain holomorphic vector fields are rational and establishes a dimension-dependent upper bound on their Betti numbers, confirming conjectures and strengthening previous results.

Contribution

It demonstrates the rationality of varieties with specific holomorphic vector fields and provides a new, stronger upper bound on Betti numbers based solely on dimension.

Findings

Varieties with such vector fields are rational.

Betti numbers are bounded by a function of dimension.

Confirms Carrell's conjecture and improves previous bounds.

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.