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Polynomial hull of a compact set of finite length and holomorphic chains with rectifiable boundary. (Enveloppe polynomiale d’un compact de longueur finie et chaînes holomorphes à bord rectifiable.) (French) Zbl 0942.32008
Let \(X\) be a complex variety of dimension \(n\), \(\Gamma\) a recifiable current of dimension \(2p-1\) of \(X\). Let \(\text{supp} \Gamma\) denote the support of \(\Gamma\). The boundary problem is the problem of finding necessary and sufficient conditions for \(\Gamma\) to be the boundary of a holomorphic \(p\)-chain of \(X\setminus\text{supp} \Gamma\) of locally finite mass, in the sense of currents on \(X\). For this problem to be solvable \(\Gamma\) must be closed and maximally complex. In \({\mathbb{C}}^n\), if \(\Gamma\) is a closed, oriented real \(C^1\) variety, Harvey and Lawson proved that the boundary problem is solvable if and only if for \(p>1\) the tangent space of \(\Gamma\) is maximally complex at every point, and, for \(p=1\), \(\Gamma\) satisfies the moment condition (that is, \((\Gamma,\phi)=0\) for every holomorphic \((1,0)\)-form \(\phi\) on \({\mathbb{C}}^n\)). The author generalizes the Harvey-Lawson theorem to the case of a closed, rectifiable, maximally complex current whose support satisfies the following condition \(A_{2p-1}\): \(\text{supp} \Gamma\) is \((H^{2p-1},2p-1)\)-rectifiable and the tangent cone of \(\text{supp} \Gamma\) at \(H^{2p-1}-\) almost all points is a real \((2p-1)\)-dimensional space. Here \(H^{2p-1}\) denotes Hausdorff measure. This result for \(p=1\) also generalizes the well known results of Wermer, Bishop, Stolzenberg, Alexander, and Lawrence on polynomal hulls of curves.
The proof combines the methods of Harvey-Lawson and Dolbeault-Henkin along with the author’s uniqueness theorem [C. R. Acad. Sci., Paris, Sér. I 322, No. 12, 1135-1140 (1996; Zbl 0865.32007)] to prove the case \(p\geq 2\) and \(n=p+1\). The general case follows from projecting from \({\mathbb{C}}^n\) to \({\mathbb{C}}^{n-p+1}\). The importance of the conditions \(A_1\) and \(A_{2p-1}\) are indicated by examples.
Finally, the author obtains a generalization of the result of P. Dolbeault and G. Henkin [Bull. Soc. Math. Fr. 125, No. 3, 383-445 (1997; see the paper above)] to the case of a rectifiable current \(\Gamma\). This result is valid when \(X\) is an \((n-p+1)\)-linearly concave open subset of \(\mathbb{C} P^n\). Explicit examples indicate the difference between the boundary problem in \(\mathbb{C} P^n\) and the problem in \({\mathbb{C}}^n\).
Reviewer: J.S.Joel (Kelly)

MSC:
32C30 Integration on analytic sets and spaces, currents
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
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