Abstract
We develop intersection theory in terms of the {{mathscr {B}}}-group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the {{mathscr {B}}}-classes have well-defined multiplicities at each point. We focus on a {{mathscr {B}}}-analogue of the intersection theory based on the Stückrad–Vogel procedure and the join construction in projective space. Our approach provides global {{mathscr {B}}}-classes which satisfy a Bézout theorem and have the expected local intersection numbers. We also introduce {{mathscr {B}}}-analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a {{mathscr {B}}}-variant of van Gastel’s formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.
Highlights
Let Y be a smooth manifold of dimension n
That is, if codim V = κ, there is a well-defined intersection cycle μ1 ·Y . . . ·Y μr = m j Vj, where Vj are the irreducible components of V and m j are integers
Following Fulton–MacPherson, see [12], there is an intersection product μ1 ·Y · · · ·Y μr, which is an element in the Chow group An−κ (V ); that is, the product is represented by a cycle on V of dimension n − κ that is determined up to rational equivalence
Summary
Let Y be a smooth manifold of dimension n. The main objective of this paper is to introduce a product of cycles in P n that at each point carries the local intersection numbers and at the same time have reasonable global properties, such as respecting the Bézout formula. To this end we must extend the class of cycles, and our construction is based on the Z-module GZ(X ) of generalized cycles on a (reduced) analytic space X introduced in [4].
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