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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)

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