Divisors and Euler Sparks of Atomic Sections

The purpose of this paper is twofold. The rst purpose is to give a simple treatment of the basic theory of atomic sections and their zero currents or divisors. This theory (see HS,HL1]) places the classical theory of characteristic classes on a new and more general analytic footing based on de Rham currents and geometric measure theory. The second purpose is to discuss an instance of the Cheeger-Simons theory of diierential characters from this analytic perspective. A vector bundle isomorphism can be used to transplant a connection from one of the bundles to the other bundle. A vector bundle map (which is not necessarily an isomor-phism) can be used to transplant a connection from one of the bundles to a singular connection on the other vector bundle. A Chern-Weil theory for such singular connections was introduced in HL1]. One of the crucial ingredients in the development of this theory was the concept of the divisor, or zero current, denoted by Div(), of a section of a vector bundle. Sections which have divisors are called atomic sections. The precise definitions of atomic section and divisor current were introduced in HS]. An abundance of examples of this Chern-Weil current theory were presented as geometric residue theorems in HL2]. These examples all reduced to the Euler case (i.e. to the case of zero currents of sections of vector bundles) using methods from intersection theory (see Fu]), but in a C 1 context. More will be said about the Euler case later, as it is the primary case discussed in this paper. In the process of proving the existence of a characteristic current for a general (geometrically atomic) bundle map the method of \\nite volume ows"was introduced in HL3]. Such ows provide a new, straightforward development of Morse Theory. The special ow which occurs in the Chern-Weil current theory is just the homothety of the bundle map. Analysis of this particular nite volume ow yields the beautiful classical cohomology formulas of MacPherson but at the local level of forms and currents, HL4]. The purpose of this paper is twofold. The rst aim is to extract and simplify some of the basic results of HS] and HL1] concerning sections of real and complex vector bundles, and to make them more accessible. The simpliications are achieved by emphasizing the role of the homothety ow in the bers of the vector bundle. This enables the fundamental current …