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Semiglobal extension of maximally complex submanifolds. (English) Zbl 1239.32028
The problem considered by the authors is the boundary problem, that is the problem to characterize boundaries of complex analytic varieties in \({\mathbb{C}}^n\). The compact case was solved for dimension one by J. Wermer [Ann. Math. (2) 68, 550–561 (1958; Zbl 0084.33402)] and for higher dimensions by F. R. Harvey and H. B. Lawson jun. [Ann. Math. (2), 106, 213–238 (1977; Zbl 0361.32010)]. The authors consider the non-compact version of this problem in a semi-global setting.
That is, let \(\Omega\) be a weakly pseudoconvex domain and let \(A \subset b\Omega\) be a domain. The authors prove that there is a subdomain \(E\) of \({\mathbb{C}}^n\) depending only \(\Omega\) and \(A\) such that a smooth, closed, maximally complex \(M \subset A\) is the boundary of a complex subvariety of \(E\). The authors also give a generalization to analytic sets of depth 4.

MSC:
32V25 Extension of functions and other analytic objects from CR manifolds
32T15 Strongly pseudoconvex domains
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