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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
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