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Multiplicities for improper intersections of analytic subsets. (English) Zbl 0785.32005
If \(M\) is a complex analytic manifold and \(X\), \(Y\) are pure-dimensional analytic subsets of \(M\), it is well-known how to define the intersection multiplicity \(i(X \cdot Y;C)\) for a proper irreducible component \(C\) of \(X \cap Y\) [cf. E. Selder, Rev. Roum. Math. Pures Appl. 29, No. 5, 417-432 (1984; Zbl 0613.32007)]. The aim of the note under review is to extend this to the case of improper irreducible components \(C\) of \(X\cap Y\) of arbitrary dimension.
The authors proceed by first defining \(i_ a(X,Y)\) for every point \(a \in X \cap Y\), and then showing that this number is the same for all \(a \in \overline C\), a dense Zariski-open subset of \(C\). By using the method of compact semianalytic Stein neighbourhoods, they derive from a result in algebraic geometry [cf. Satz 3.13 in M. Herrmann, R. Schmidt and W. Vogel, Theorie der normalen Flachheit (1977; Zbl 0356.13008)] that a certain Samuel multiplicity does the job.
In the algebraic case, \(i(X \cdot Y;C)\) coincides with the multiplicity defined by J. Stückrad and W. Vogel [cf. W. Vogel, Lectures on results on Bezout’s theorem (1984; Zbl 0553.14022)]. The case of embedded components \(C\) of \(X \cap Y\) is dealt with the authors’ paper in Manuscr. Math. 80, 291-308 (1993).

32C25 Analytic subsets and submanifolds
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
13H15 Multiplicity theory and related topics