Residues of Chern classes

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.