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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
Citations:
Zbl 0167.06903
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