Improper intersections in complex analytic geometry.

*(English)*Zbl 1095.13508This dissertation is devoted to the study of improper intersections in complex analytic geometry. In the case of algebraic geometry there are two approaches to improper intersections. One is by W. Fulton and R. MacPherson [in: Real and complex Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 179–197 (1977; Zbl 0385.14002)], and the other is by J. Stückrad and W. Vogel [Curves Semin. at Queen’s, Vol. 2, Queen’s Pap. Pure Appl. Math. 61, 1–32 (1982; Zbl 0599.14003)]. The relation between the two approaches was clarified by the work of L. J. van Gastel [Invent. Math. 103, No. 1, 197–222 (1991; Zbl 0733.14002)]. The main point of the present paper is to investigate the analytic intersection algorithm (as initiated by Stückrad and Vogel) from the perspective of the method of deformation to the normal cone.

Since in the analytic case the author cannot deal with equivalence classes, the basic technical tool is the notion of a filter regular sequence of elements in a local ring. This is related to the notion of a “generic” collection of divisors related to these elements. Under a certain condition of filter regularity the author is able to apply the deformation to the normal cone twice and get a deformation of the initial analytic set to an algebraic bicone so that the extended degree of the result of the intersection algorithm (for improper intersections) coincides with the degree of that bicone. In this way he reduces the general problem of analytic improper intersections to that of an algebraic bicone intersecting linear hyperplanes. That is, there is a linearization procedure from which many consequences concerning extended intersection index and intersection multiplicity for improper intersections in the complex analytic case are derived. This extends results shown by R. Achilles and M. Manaresi [Math. Ann. 309, No. 4, 573–591 (1997; Zbl 0894.14005)] in the local algebraic case.

Making use of properties of normal cones the author generalizes some of his earlier results [Bull. Pol. Acad. Sci., Math. 48, 121–130 (2000; Zbl 0981.13015); ibid. 48, No.2, 131–140 (2000; Zbl 0981.13016)]. Furthermore, he proves the upper semicontinuity of the intersection multiplicity function in the analytic Zariski topology. The linearization procedure for the analytic intersection algorithm allows him to compare the generalized intersection index with the so-called Segre numbers intoduced by T. Gaffney and R. Gassler [J. Algebr. Geom. 8, 695–736 (1999; Zbl 0971.13021)]. There is also a generalization of the classical reduction theorem for the diagonal procedure to the case of analytic improper intersections.

The author notes in the supplement that his version of Bézout’s theorem for cones was first given by P. Tworzewski at a seminar by T. Winiarski in 1998. The author emphasizes that Bézout’s theorem fails to be true because improper intersections concern only the intersection product studied therein. The intersection theories by Fulton-MacPherson and Stückrad-Vogel provide versions of Bézout’s theorem in the general case of improper intersections.

Since in the analytic case the author cannot deal with equivalence classes, the basic technical tool is the notion of a filter regular sequence of elements in a local ring. This is related to the notion of a “generic” collection of divisors related to these elements. Under a certain condition of filter regularity the author is able to apply the deformation to the normal cone twice and get a deformation of the initial analytic set to an algebraic bicone so that the extended degree of the result of the intersection algorithm (for improper intersections) coincides with the degree of that bicone. In this way he reduces the general problem of analytic improper intersections to that of an algebraic bicone intersecting linear hyperplanes. That is, there is a linearization procedure from which many consequences concerning extended intersection index and intersection multiplicity for improper intersections in the complex analytic case are derived. This extends results shown by R. Achilles and M. Manaresi [Math. Ann. 309, No. 4, 573–591 (1997; Zbl 0894.14005)] in the local algebraic case.

Making use of properties of normal cones the author generalizes some of his earlier results [Bull. Pol. Acad. Sci., Math. 48, 121–130 (2000; Zbl 0981.13015); ibid. 48, No.2, 131–140 (2000; Zbl 0981.13016)]. Furthermore, he proves the upper semicontinuity of the intersection multiplicity function in the analytic Zariski topology. The linearization procedure for the analytic intersection algorithm allows him to compare the generalized intersection index with the so-called Segre numbers intoduced by T. Gaffney and R. Gassler [J. Algebr. Geom. 8, 695–736 (1999; Zbl 0971.13021)]. There is also a generalization of the classical reduction theorem for the diagonal procedure to the case of analytic improper intersections.

The author notes in the supplement that his version of Bézout’s theorem for cones was first given by P. Tworzewski at a seminar by T. Winiarski in 1998. The author emphasizes that Bézout’s theorem fails to be true because improper intersections concern only the intersection product studied therein. The intersection theories by Fulton-MacPherson and Stückrad-Vogel provide versions of Bézout’s theorem in the general case of improper intersections.

Reviewer: Peter Schenzel (MR 2002i:32007b)