This paper is devoted to study multiplicity and regularity of complex analytic sets. We present an equivalence for complex analytical sets, named blow-spherical equivalence and we obtain several applications with this new approach. For example, we reduce to homogeneous complex algebraic sets a version of Zariski’s multiplicity conjecture in the case of blow-spherical homeomorphism, we give some partial answers to the Zariski’s multiplicity conjecture, we show that a blow-spherical regular complex analytic set is smooth and we give a complete classification of the complex analytic curves.

中文翻译：

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复杂解析集的多重性，规则性和打击球面等价

本文致力于研究复杂解析集的多重性和规律性。我们提出了一个复杂的分析集的等效项，称为吹球等效项，并且使用这种新方法获得了一些应用。例如，在吹球形同胚的情况下，我们将Zariski多重猜想的一个形式简化为齐次复代数集，对Zariski多重猜想给出了部分答案，我们证明了一个吹球形正则复分析集是光滑的，并且我们给出了复杂分析曲线的完整分类。