Residues of Chern classes

Source abstract

If we have a finite number of sections of a complex vector bundle E over a manifold M, certain Chern classes of E are localized at the singular set S, i.e., the set of points where the sections fail to be linearly independent. When S is compact, the localizations define the residues at each connected component of S by the Alexander duality. If M itself is compact, the sum of the residues is equal to the Poincaré dual of the corresponding Chern class. This type of theory is also developed for vector bundles over a possibly singular subvariety in a complex manifold. Explicit formulas for the residues at an isolated singular point are also given, which express the residues in terms of Grothendieck residues relative to the subvariety.