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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
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