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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
##### Keywords:
positive closed currents; cluster set
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##### References:
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