2019/08/30 by Andersson, Mats, Eriksson, Dennis, Kalm, Håkan Samuelsson +2
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.11759
In this article we develop intersection theory in terms of the 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 B-classes have well-defined multiplicities at each point. We focus on a B-analogue of the intersection theory based on the Stückrad-Vogel procedure and the join construction in projective space. Our approach provides global B-classes which satisfy a Bézout theorem and have the expected local intersection numbers. An essential feature is that we take averages, over various auxiliary choices, by integration. We also introduce B-analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a B-variant of van Gastel's formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.