If we have a finite number of sections of a complex vector bundle over a manifold , certain Chern classes of are localized at the singular set , i.e., the set of points where the sections fail to be linearly independent. When is compact, the localizations define the residues at each connected component of by the Alexander duality. If 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.
Tatsuo SUWA. "Residues of Chern classes." J. Math. Soc. Japan 55 (1) 269 - 287, January, 2003. https://doi.org/10.2969/jmsj/1196890854