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Hulls and positive closed currents. (English) Zbl 0958.32004
In a previous paper [Duke Math. J. 79, No. 2, 487-513 (1995; Zbl 0838.32006)], the authors gave a characterization of the polynomial (resp. rational) hull in terms of compactly positive currents.
In this paper, in section 1, they show that for any totally real \(n\)-dimensional submanifold \(S\) in \(\mathbb{C}^n\), \({\mathcal C}_{1,1} (S)\neq\emptyset\) where \({\mathcal C}_{1,1}(S)\) denotes the set of nonzero compactly supported positive currents \(T\) of bidimension (1,1) such that \(dT=j_*V\) where \(V\) is a current of dimension 1 on \(S\) and \(j:S\hookrightarrow\mathbb{C}^n\) is the inclusion map.
In the next section they show that such a conclusion is false if the dimension of \(S\) is less than \(n\) by explicitly exhibiting a counterexample.
Inspired by a result of H. Alexander [Invent. Math. 125, No. 1, 135-148 (1996; Zbl 0853.32003)] for any bounded holomorphic function \(f:D\to\mathbb{C}^n\), satisfying certain conditions, its cluster set \({\mathcal C}_f\) is characterised. Finally in the last section, they show that if \(S\) is a totally real torus in \(\mathbb{C}^2\) and \(P\) is a polynomial whose restriction to \(S\) is of rank 1, then there exists a Riemann surface with boundary on \(S\).

MSC:
32C30 Integration on analytic sets and spaces, currents
32D10 Envelopes of holomorphy
32V40 Real submanifolds in complex manifolds
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