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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)].

##### MSC:
 32V25 Extension of functions and other analytic objects from CR manifolds 32V15 CR manifolds as boundaries of domains 32T15 Strongly pseudoconvex domains
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##### References:
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