Limits of complete holomorphic vector fields

Let $V$ be a holomorphic vector field\ on a Stein manifold $M$. If $V$ can be approximated by complete \holomorphic vector field s, uniformly on compacts in $M$, we prove that the fundamental domain of $V$ is single sheeted, pseudoconvex, and it has simply connected fibers. Moreover, every complex orbit of $V$ has connectivity at most one (Theorem 1.1). We then find several explicit classes of \holomorphic vector field s on ${\Bbb C}^2$ which are not limits of complete fields (Corollaries 1.4–1.6).

Full Text (PDF format)

Published 1 January 1995