On improper isolated intersection in complex analytic geometry.

*(English)*Zbl 0796.32006The authors define the intersection multiplicity for locally analytic subsets (of pure dimension) of a complex analytic manifold at an isolated point of the intersection. The approach is analytic; using as main tools the intersection multiplicity theory for the proper isolated intersections [R. Draper, Math. Ann. 180, 175–204 (1969; Zbl 0157.40502)] and the diagonal construction [P. Samuel, J. Math. Pures Appl. (9) 30, 159–274 (1951; Zbl 0044.02701) and A. Weil, Foundations of algebraic geometry. New York: AMS (1946; Zbl 0063.08198); rev. ed. (1962; Zbl 0168.18701)] the authors extend the definition to the improper case. They also compare this definition with the algebraic approach by Samuel’s multiplicity.

First they define the intersection multiplicity between a locally analytic subset \(X\) of pure dimension \(k\) and a submanifold \(N\) (both in an analytic manifold \(M\) of dimension \(m)\) at the isolated point \(a\), \(\tilde i (X \cdot N,a)\), as the infimum of the intersection multiplicities of \(X\) with locally analytic subsets \(V\) of dimension \(m - k\) containing the germ of \(N\) at \(a\) and with \(a\) isolated point of \(X \cap V\) \((V\) intersect properly \(X\) in \(a)\). Also the subsets \(V\) for which \(\tilde i (X \cdot N,a)\) is reached is characterized in terms of the relative tangent cone of \(X\) and \(N\) and the tangent spaces of the subsets involved. After that the intersection multiplicity between two locally analytic subsets of \(X\) and \(Y\) is defined as \(i(X \cdot Y,a) = \tilde i((X \times Y) \cdot \Delta, (a,a))\). Some properties like additivity on the components are proved. Also the formula \(i(X \cdot Y,a) \geq \deg_aX \deg_aY\) is given, characterizing the equality by the condition that the tangent cones have trivial intersection. At the end the comparison with the Samuel’s theory of multiplicity is given, the intersection multiplicity above defined is equal to the multiplicity of the primary ideal of the diagonal \(\Delta\) in the local ring \({\mathcal O}_{x \times Y,(a,a)}\).

First they define the intersection multiplicity between a locally analytic subset \(X\) of pure dimension \(k\) and a submanifold \(N\) (both in an analytic manifold \(M\) of dimension \(m)\) at the isolated point \(a\), \(\tilde i (X \cdot N,a)\), as the infimum of the intersection multiplicities of \(X\) with locally analytic subsets \(V\) of dimension \(m - k\) containing the germ of \(N\) at \(a\) and with \(a\) isolated point of \(X \cap V\) \((V\) intersect properly \(X\) in \(a)\). Also the subsets \(V\) for which \(\tilde i (X \cdot N,a)\) is reached is characterized in terms of the relative tangent cone of \(X\) and \(N\) and the tangent spaces of the subsets involved. After that the intersection multiplicity between two locally analytic subsets of \(X\) and \(Y\) is defined as \(i(X \cdot Y,a) = \tilde i((X \times Y) \cdot \Delta, (a,a))\). Some properties like additivity on the components are proved. Also the formula \(i(X \cdot Y,a) \geq \deg_aX \deg_aY\) is given, characterizing the equality by the condition that the tangent cones have trivial intersection. At the end the comparison with the Samuel’s theory of multiplicity is given, the intersection multiplicity above defined is equal to the multiplicity of the primary ideal of the diagonal \(\Delta\) in the local ring \({\mathcal O}_{x \times Y,(a,a)}\).

Reviewer: F.Delgado (Valladolid)