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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)].

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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