Convergence of holomorphic chains. (English) Zbl 0873.32005

A holomorphic \(p-\)chain in an open subset \(\varOmega\) of \(\mathbb C^n\) is a formal locally finite sum \(Z=\sum_{j\in J}k_jZ_j\) where \(Z_j\) are pairwise distinct irreducible analytic subsets of \(\varOmega\) of pure dimension \(p\) and \(k_j\in\mathbb Z\setminus\{0\}\) for \(j\in J.\) The set \(\mathcal G^p(\varOmega)\) of holomorphic \(p-\)chains in \(\varOmega\) is endowed with the structure of a free \(\mathbb Z-\)module.
The main aim of this note is to define a topology on \(\mathcal G^p(\varOmega)\) and to study some properties of this topological space. The result of this construction is second-countable, metrizable, and convergence in it coincides with the one defined in E. M. Chirka [‘Complex analytic sets’, Kluwer Acad. Publishers (1989; Zbl 0683.32002)]. The topology constructed here is useful in studying the intersections of analytic sets [P. Tworzewski, Ann. Polon. Math. 62, No. 2, 177-191 (1995)].


32B15 Analytic subsets of affine space
32C25 Analytic subsets and submanifolds
32C30 Integration on analytic sets and spaces, currents


Zbl 0683.32002
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