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Global minimality of generic manifolds and holomorphic extendibility of CR functions. (English) Zbl 0815.32007

Let \(M \subset \mathbb{C}^ n\) be a smooth generic submanifold. This paper deals with the property of CR functions on \(M\) to extend locally to manifolds with a boundary attached to \(M\) and holomorphically to generic wedges with edge \(M\). A. E. Tumanov showed [Duke Math. J. 73, No. 1, 1-24 (1994; Zbl 0801.32005)] that the direction of CR-extendibility moves parallelly with respect to a certain partial connection on a quotient bundle of the normal bundle to \(M\). Here the author gives a new and simplified presentation of the connection introduced by Tumanov. By the CR-orbit of a point \(z \in M\) we mean the set of points that can be reached by piecewise smooth integral curves of complex tangent vector fields. We say that \(M\) is globally minimal at \(z \in M\) if the CR-orbit of \(z\) contains a neighbourhood of \(z\) in \(M\). The author shows that the vector space generated by the directions of CR-extendibility of CR functions on \(M\) exchanges by the induced composed flow between top points in the same CR-orbit. As an application the main result is proved, which says that if \(M\) is globally minimal at \(z\) then the wedge extendibility of CR functions holds at every point in the CR-orbit of \(z\). This was conjectured by J.-M. Trepreau [Bull. Soc. Math. France 118, No. 4, 403-450 (1990; Zbl 0742.58053)].

MSC:

32V99 CR manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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