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Non-compact boundaries of complex analytic varieties. (English) Zbl 1140.32025
The authors prove the following theorem: Let \(\Omega\) be a possibly unbounded domain in \(\mathbb C^n\) (\(n\geq3\)) with smooth boundary \(b\Omega\). Let \(M\) be a maximally complex closed \((2m+1)\)-dimensional real submanifold (\(m\geq1\)) of \(b\Omega\). Assume that \(b\Omega\) is weakly pseudoconvex and its Levi-form has at least \(n-m\) positive eigenvalues at every point of \(M\), and that the closure of \(M\) in \(\mathbb P^n\) does not intersect an algebraic hypersurface in \(\mathbb P^n\). Then there exists a unique \((m+1)\)-dimensional complex analytic subvarity \(W\) of \(\Omega\) such that \(bW=M\). Moreover, the singular locus of \(W\) is discrete and the closure of \(W\) in \(\overline\Omega \setminus \text{Sing}(W)\) is a smooth submanifold with boundary \(M\). When \(\Omega\) is bounded, the result is due to F. R. Harvey and H. B. Lawson, jun. [Ann. Math. (2) 102, 223–290 (1975; Zbl 0317.32017); Ann. Math. (2)106, 213–238 (1977; Zbl 0361.32010)].

32V25 Extension of functions and other analytic objects from CR manifolds
32V15 CR manifolds as boundaries of domains
32T15 Strongly pseudoconvex domains
Full Text: DOI arXiv
[1] DOI: 10.5802/afst.931 · Zbl 0959.32020 · doi:10.5802/afst.931
[2] DOI: 10.1007/978-3-663-14196-9_6 · doi:10.1007/978-3-663-14196-9_6
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