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# Mathematical Research Letters

## Volume 2 (1995)

### Number 4

### Limits of complete holomorphic vector fields

Pages: 401 – 414

DOI: http://dx.doi.org/10.4310/MRL.1995.v2.n4.a3

#### Author

#### Abstract

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).