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Cycles of zeroes of holomorphic mappings. (English) Zbl 0759.32015
This paper is written in the spirit of the classical intersection theory in analytic geometry. The following interesting theorem is proved: If $$f:M\to\mathbb{C}^ d$$ is a holomorphic map, and $$A$$ is an irreducible analytic subset of the $$d$$-dimensional complex manifold $$M$$, such that the intersection $$f^{-1}(0)\cap A$$ is a finite set, then if $$\varphi:N\to A$$ is an analytic cover with covering number $$s$$ $$(s$$- parametrization of $$A)$$ the following equality holds $s\cdot\deg(Z_ f\cdot A)=\deg Z_{f\cdot\varphi}.$ Here by $$Z_ f$$ is denoted an $$O$$- cycle on $$M$$ and by $$Z_ f\cdot A$$ the intersection product of $$Z_ f$$ and $$A$$.
Reviewer: S.Dimiev (Sofia)

##### MSC:
 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 32S20 Global theory of complex singularities; cohomological properties