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Riemann surfaces of given boundary in $$\mathbb{C} P^ n$$. (Surfaces de Riemann de bord donné 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
currents