Nonproper intersection products and generalized cycles

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.