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Complete characterization of holomorphic chains of codimension one. (English) Zbl 0572.32005
The main result is the following characterization of holomorphic chains of codimension 1: If T is a closed, locally rectifiable (1,1)-current on a domain in $${\mathbb{C}}^ n$$, then T is a holomorphic chain. As a consequence, it is proved that if T is a closed, locally rectifiable current of bidimension (p,p) such that the Hausdorff $$(2p+3)$$-measure of Supp T vanishes, then T is a holomorphic chain. This gives an improvement of the characterization theorem of R. Harvey and the author [Ann. Math., II. Ser. 99, 553-587 (1974; Zbl 0287.32008)]. The proof uses the following new Hartogs-type theorem on almost-everywhere separately meromorphic functions: Let $$\Delta^ n$$ denote the unit n-disk in $${\mathbb{C}}^ n$$. Suppose $$f: S\to {\mathbb{C}}$$ where $$S\subset \Delta^ n$$. Write $$f^ z_ j(t)=f(z_ 1,...,z_{j-1},t,z_ j,...,z_{n-1})$$ for $$z=(z_ 1,...,z_{n-1})\in \Delta^{n-1}$$, $$1\leq j\leq n$$, and let $$S^ z_ j\subset \Delta$$ denote the domain of $$f^ z_ j$$. Suppose that for $$1\leq j\leq n$$ and for almost all $$z\in \Delta^{n-1}$$, $$\Delta -S^ z_ j$$ has Lebesgue measure 0 and $$f^ z_ j$$ extends to a meromorphic function on $$\Delta$$. Then there exists a meromorphic function $$\tilde f$$ on $$\Delta^ n$$ such that $$f=\tilde f$$ almost everywhere on S. In the above result, neither f nor S is assumed to be measurable. However, the following corollary for measurable functions is given: Let $$f: \Delta$$ $${}^ n\to {\mathbb{C}}$$ be a measurable function such that $$f^ z_ j$$ is equal to a meromorphic function on $$\Delta$$ for almost all $$z\in \Delta^{n-1}$$, for $$1\leq j\leq n$$. Then f is almost everywhere equal to a meromorphic function on $$\Delta^ n$$. These results are extensions of results of J. Siciak [Ann. Pol. Math. 22, 145-171 (1969; Zbl 0185.152) and 39, 175-211 (1981; Zbl 0477.32018)] and M. V. Kazaryan [Mat. Sbornik 125(167), No.3, 384-397 (1984)].

##### MSC:
 32C30 Integration on analytic sets and spaces, currents 32D15 Continuation of analytic objects in several complex variables
##### Keywords:
current; holomorphic chain; Hartogs theorem
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##### References:
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