Limits of complete holomorphic vector fields

Abstract

Recently Buzzard and Fornaess answered the question, raised in [8], as to whether there exist holomorphic vector fields on C for n ≥ 2 which can not be approximated by complete holomorphic vector fields. They showed for instance that the quadratic vector field on C, V (z, w) = z(z−1) ∂ ∂z −w ∂ ∂w , is not a limit of complete holomorphic fields; the same is true for the field z ∂ ∂z on C 2 (private communication, January 1995). Their method proves that the set of complete holomorphic vector fields is nowhere dense in the set of all entire vector fields on C, in the topology of uniform convergence on compact sets. It was clear from the results of [8] that most Hamiltonian holomorphic vector fields can not be approximated by complete Hamiltonian fields. This phenomenon is in strong contrast with the situation for smooth vector fields which can always be made complete by a modification supported outside a given compact set in M . In this article we obtain qualitative information on the fundamental domain in complex time of those holomorphic vector fields V on a Stein manifold M which are limits of complete fields (Theorem 1.1). We apply this to show that several specific types of vector fields on C are not limits of complete fields (uniformly on compacts).