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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).

##### MSC:
 32C25 Analytic subsets and submanifolds 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 13H15 Multiplicity theory and related topics
##### Citations:
Zbl 0613.32007; Zbl 0356.13008; Zbl 0553.14022