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Riemann surfaces of given boundary in \(\mathbb{C} P^ n\). (Surfaces de Riemann de bord donnĂ© dans \(\mathbb{C} P^ n\).) (French) Zbl 0821.32008
Skoda, Henri (ed.) et al., Contributions to complex analysis and analytic geometry. Based on a colloquium dedicated to Pierre Dolbeault, Paris, France, June 23-26, 1992. Braunschweig: Vieweg. Aspects Math. E 26, 163-187 (1994).
Let \(\gamma\) be a closed 1-chain of class \(C^ 2\) in \(\mathbb{C} P_ n\) \((n > 1)\) and let us assume that \(d\gamma = b \gamma = 0\). Then the following main result is established
Theorem: The following two conditions are equivalent
(i) there exists a holomorphic 1-chain \(S\) in \(\mathbb{C} P_ n \backslash \text{supp} (\gamma)\), having a simple extension to \(\mathbb{C} P_ n\) such that \(bS = \gamma\)
(ii) there exists a point \((x_ 0, y_ 0) \in \mathbb{C} \times \mathbb{C}^{n - 1}\) in the neighborhood of which the function \[ G(x,y) : = 1/2 \pi i \int_ \gamma \mu {dg \over g}, \] where \(\mu : = (z_ 1, \dots, z_{n - 1}) \in \mathbb{C}^{n - 1}\) and \(g : = z_ n - x - y \mu\) with \(x \in \mathbb{C}\) and \(y : = (y_ 1, \dots, y_{n - 1}) \in \mathbb{C}^{n - 1}\), is equal to \[ \sum^{N^ +}_{j = 1} f^ +_ j (x,y) - \sum^{N^ - }_{j = 1} f^ -_ j (x,y) \] where the functions \(f_ j\), with scalar components \(f_{jk} (1 \leq k \leq n - 1)\) are holomorphic functions in \((x,y)\) and satisfy the following relations \[ f_{jk} {\partial f_ j \over \partial x} = {\partial f_ j \over \partial y_ k}. \] This result was established in C. R. Acad. Sci., Paris, Ser. I 316, No. 1, 27- 32 (1993; Zbl 0776.32008) when \(n = 2\). In this paper the authors provide complete proofs (resp. some proof sketch) for the implication (i)\(\Rightarrow\)(ii) (resp. (ii)\(\Rightarrow\)(i)) in section 5 (resp. section 6).
For the entire collection see [Zbl 0811.00006].

MSC:
32C30 Integration on analytic sets and spaces, currents
Keywords:
currents
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