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Intersection theory in complex analytic geometry. (English) Zbl 0911.32018
This paper contains the construction of an intersection product of irreducible complex analytic subsets $$X$$ and $$Y$$ of a complex manifold $$M$$ based on a pointwise defined intersection multiplicity. In order to assign to every point $$a\in M$$ an intersection multiplicity $$d(a)$$ of $$X$$ and $$Y$$ at $$a$$, the author uses the classical diagonal construction and the Stuckrad-Vogel intersection algorithm [see, e.g., W. Vogel, “Lectures on results on Bezout’s theorem” (1984; Zbl 0553.14022)], which is applied locally, in an open neighbourhood $$U$$ of $$c=(a,a)$$, to $$X\times Y$$ and smooth hypersurfaces $$H_1,\dots,H_m$$ of $$U$$ whose tangent spaces cut out the tangent space of the diagonal of $$M\times M$$. The resulting cycle, say $$T$$, has a unique decomposition $$T_m+\cdots+ T_0$$, where $$T_j$$ is a $$j$$-cycle. Let $$\nu(T_j,c)$$ denote the degree of $$T_j$$ at $$c$$. The open neighbourhood $$U$$ of $$c$$ and the smooth hypersurfaces $$H_j$$ of $$U$$ are chosen such that the so-called extended degree $$(\nu(T_m,c),\dots,\nu(T_0,c))$$ becomes minimal with respect to the lexicographic order. The multiplicity $$d(a)$$ of intersection of $$X$$ and $$Y$$ at the point $$a$$ is then defined to be $$\nu(T_m,c)+ \cdots+ \nu(T_0,c)$$ and gives the construction of the desired intersection cycle. The main difficulty is to prove that the function $$a\mapsto d(a)$$ is analytically constructible.
The author shows that his construction yields the classical intersection cycle if $$X$$ and $$Y$$ intersect properly [see R. N. Draper, Math. Ann. 180, 175-204 (1969; Zbl 0167.06903)] and, for an improper isolated point of intersection, it gives the multiplicity defined in an earlier paper of the author with T. Winiarski and the reviewer [Ann. Pol. Math. 51, 21-36 (1990; Zbl 0796.32006)].

MSC:
 32C25 Analytic subsets and submanifolds 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 32C99 Analytic spaces 32B10 Germs of analytic sets, local parametrization
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