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Limits of complete holomorphic vector fields. (English) Zbl 0859.32006

Let \(M\) be a Stein manifold and \(V\) a holomorphic vector field on \(M\). It was shown recently by Buzzard and Fornaess that it is not always possible to approximate \(V\) by complete holomorphic vector fields. Here the author shows that, if \(V\) can be approximated by complete holomorphic vector fields, uniformly on compact subsets of \(M\), then the fundamental domain of \(V\) is single-sheeted and pseudoconvex, and has simply-connected fibres. Moreover every complex orbit of \(V\) has connectivity at most one. These results are used to exhibit several classes of holomorphic vector fields which are not limits of complete fields.

MSC:

32E10 Stein spaces
34M99 Ordinary differential equations in the complex domain
32L05 Holomorphic bundles and generalizations
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