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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
##### Keywords:
boundary problem; holomorphic chains
##### Citations:
Zbl 0942.32007; Zbl 0865.32007
Full Text:
##### References:
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