Singularities and Chern-Weil theory. II: Geometric atomicity. (English) Zbl 1049.32030

Harvey, F. Reese; Lawson, H. Blaine jun.

Duke Math. J. 119, No. 1, 119-158 (2003).

Summary: 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 \(\Phi\) of \(E\) or \(F\) satisfies an equation of the form \[ \Phi=L+dT, \] where \(L\) is an explicit localization of \(\Phi\) along the singularities of \(\alpha\) 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, the authors, Asian J. Math. 4, 71–95 (2000; Zbl 0981.58003)]. 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 singularity sets of \(\alpha\) have dimensions greater than those of the generic map. The method yields explicit calculations in this general context. A number of examples are worked out in detail.

MSC:

32S20	Global theory of complex singularities; cohomological properties
57R45	Singularities of differentiable mappings in differential topology
57R20	Characteristic classes and numbers in differential topology
53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
53B15	Other connections

Citations:

Zbl 0981.58003

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