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Global convexity properties of some families of three-dimensional compact Levi-flat hypersurfaces. (English) Zbl 0761.32010
Assume, that on a complex manifold \(M\) a real analytic hypersurface \(S\) is given together with a foliation \(F\) of \(S\) into complex hypersurfaces (relative \(M\)). One may expect some interplay between properties of the foliation and the way, \(S\) is embedded in \(M\) (properties of neighbourhoods of \(S\) in \(M\)). The author tries to attack this problem at the hand of two quite special types in the case \(\dim_ \mathbb{C} M=2\), \(S\) compact (extending also an example of Ohsawa). One type is of the form \(S=T^ 3\) (torus) with some special real analytic CR-structure, and \(M\) being the complexification of this. He proves, that five (respectively six) of certain properties of \(F\) and \(S\subset M\) are equivalent. He also points out that in different situations, even if \(dim_ \mathbb{C} M=2\), these equivalences do not hold in general. And in cases \(dim_ \mathbb{C} M>2\) the situation still may be much more involved. The referee proposes, also to take the leaf-structure \(S/F\) into account.
Reviewer: K.Spallek (Bochum)

MSC:
32V40 Real submanifolds in complex manifolds
32T99 Pseudoconvex domains
57R30 Foliations in differential topology; geometric theory
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