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Proper intersection multiplicity and regular separation of analytic sets. (English) Zbl 0847.32008
Let $$M$$ be a complex manifold and let $$X,Y$$ be pure dimensional analytic subsets of $$M$$. The main theorem is: if $$X$$ and $$Y$$ intersect properly on $$M$$ (i.e. $$\dim (X \cap Y) = \dim (X) + \dim (Y) - \dim (M))$$, $$a \in X \cap Y$$, $$p = \deg_a (X \cdot Y)$$ $$(X \cdot Y$$ means the intersection product of $$X$$ and $$Y$$ in the sense of R. Draper [Math. Ann. 180, 175-204 (1969; Zbl 0167.06903)], then $$X$$ and $$Y$$ are $$p$$-separated at $$a$$ i.e. $\rho (z,X) + \rho (z,Y) \geq c \rho (z,X \cap Y)^p$ in a neighbourhood of $$a$$ for some $$c > 0$$ (the using of the usual distance function $$\rho (\cdot, Z)$$ to the set $$Z$$ has sense because the property of $$p$$-separation is local and invariant with respect to biholomorphisms).

##### MSC:
 32B99 Local analytic geometry 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
##### Keywords:
analytic set; separation; proper intersection
Zbl 0167.06903
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