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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)

32C30 Integration on analytic sets and spaces, currents
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
Full Text: DOI
[1] Alexander, H., Polynomial, approximation and hulls in sets of finite linear measure in Cn.Amer. J. Math., 93 (1971), 65–74. · Zbl 0221.32011
[2] –, The polynomial hull of a set of finite linear measure in Cn.J. Analyse Math., 47 (1986), 238–242. · Zbl 0615.32009
[3] –, Polynomial hulls and linear measure, dansComplex Analysis, II (College Park, MD, 1985/86), pp. 1–11. Lecture Notes in Math., 1276, Springer-Verlag, Berlin-New York, 1987.
[4] –, The polynomial hull of a rectifiable curve in Cn.Amer. J. Math., 110 (1988), 629–640. · Zbl 0659.32017
[5] Bishop, E., Analyticity in certain Banach algebras.Trans. Amer. Math. Soc., 102 (1962), 507–544. · Zbl 0112.07301
[6] –, Holomorphic completions, analytic continuations and the interpolation of semi-norms.Ann. of Math., 78 (1963), 468–500. · Zbl 0131.30901
[7] –, Conditions for the analyticity of certain sets.Michigan Math J., 11 (1964) 289–304. · Zbl 0143.30302
[8] Chabat, B.,Introduction à l’analyse complexe I. ”Mir”, Moscou, 1990.
[9] Dinh, T. C., Chaînes holomorphes à bord rectifiable.C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1135–1140. · Zbl 0865.32007
[10] Dolbeault, P. &Henkin, G., Surfaces de Riemann de bord donné dansCP n, dansContributions to Complex Analysis and Analytic Geometry, pp. 163–187. Aspects of Math., E 26. Vieweg, Braunschweig, 1994. · Zbl 0821.32008
[11] – Chaînes holomorphes de bord donné dansCP n.Bull. Soc. Math. France, 125 (1997), 383–445.
[12] Dolbeault, P. & Poly, J. B. Variations sur le problème des bords dansCP n. Prépublication, 1995.
[13] Federer, H.,Geometric Measure Theory. Grundlehren Math. Wiss. 153. Springer-Verlag, New York, 1969. · Zbl 0176.00801
[14] Goluzin, G. M.,Geometric Theory of Functions of a Complex Variable. Transl. Math. Monographs, 26. Amer. Math. Soc., Providence, RI, 1969. · Zbl 0183.07502
[15] Harvey, R., Holomorphic chains and their boundaries.Proc. Sympos. Pure Math., 30:1 (1977), 309–382. · Zbl 0374.32002
[16] Harvey, R. &Lawson, B., On boundaries of complex analytic varieties I.Ann. of Math., 102 (1975), 233–290. · Zbl 0317.32017
[17] King, J., The currents defined by analytic varieties.Acta Math., 127 (1971), 185–220. · Zbl 0224.32008
[18] King, J., Open problems in geometric function theory, dansProceedings of the Fifth International Symposium, p. 4. Division of Math., The Taniguchi Foundation, 1978.
[19] Lawrence, M. G., Polynomial hulls of rectifiable curves.Amer. J. Math., 117 (1995), 405–417. · Zbl 0827.32012
[20] Lewrence, M. G., Polynomial hulls of sets of finite length in strictly convex boundaries. Manuscrit.
[21] Pommerenke, Ch., On analytic, functions with cluster sets of finite linear measure.Michigan Math. J., 34 (1987), 93–99. · Zbl 0621.30028
[22] Stolzenberg, G., Uniform approximation on smooth curvesActa Math., 115 (1966), 185–198. · Zbl 0143.30005
[23] Wermer, J., Function rings and Riemann surfaces.Ann. of Math. 67 (1958), 45–71. · Zbl 0081.32902
[24] –, The hull of a curve inC n.Ann. of Math., 68 (1958), 550–561. · Zbl 0084.33402
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