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A conjecture of Globevnik-Stout and a theorem of Morera for a holomorphic chain. (Conjecture de Globevnik-Stout et théorème de Morera pour une chaîne holomorphe.) (French) Zbl 0959.32020
Let \(D\Subset\mathbb C^n\) be a complex manifold of dimension \(p\geqslant 2\) with \(C^2\) boundary in \(\mathbb C^n.\) Let \(f\in C^1(\partial D)\) and \(V\) a generic and large enough family of complex \((n - p + 1)\)-planes.
The author proves the following: Suppose that for \(\nu\in V\) no connected component of \(\partial D\cap C_\nu^{n-p+1}\) is “almost” real analytic and that \(f\) is extended holomorphically into \(D\cap C_\nu^{n-p+1}.\) Then \(f\) is extended as a holomorphic function into \(D.\)
In a special case, this result gives a partial answer to a conjecture of J. Globevnik and E. L. Stout [Duke Math. J. 64, No. 3, 571-615 (1991; Zbl 0760.32002)]. By generalizing the theorem of Harvey-Lawson, it is proved a Morera type theorem for the boundary problem in \(\mathbb C^n\) which answers to a problem posed by Dolbeault and Henkin.

MSC:
32D15 Continuation of analytic objects in several complex variables
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