Singularities and Chern-Weil theory, II: Geometric atomicity

Abstract

This paper introduces a general method for relating characteristic classes to singularities of a bundle map. The method is based on the notion of geometric atomicity. This is a property of bundle maps $\alpha :E\to F$ which universally guarantees the existence of certain limits arising in the theory of singular connections. Under this hypothesis, each characteristic form $\text{Φ}$ of E or F satisfies an equation of the form

$$\text{Φ}=L+dT$$,

where L is an explicit localization of Φ along the singularities of α and T is a canonical form with locally integrable coefficients. The method is constructive and leads to explicit calculations. For normal maps (those transversal to the universal singularity sets) it retrieves classical formulas of R. MacPherson at the level of forms and currents; see Part I ([HL4]). It also produces such formulas for direct sum and tensor product mappings. These are new even at the topological level. The condition of geometric atomicity is quite broad and holds in essentially every case of interest, including all real analytic bundle maps. An important aspect of the theory is that it applies even in cases of "excess dimension," that is, where the the singularity sets of α have dimensions greater than those of the generic map. The method yields explicit number of examples are worked out detail.