## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 55, Number 1 (2003), 269-287.

### 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.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 55, Number 1 (2003), 269-287.

**Dates**

First available in Project Euclid: 5 December 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1196890854

**Digital Object Identifier**

doi:10.2969/jmsj/1196890854

**Mathematical Reviews number (MathSciNet)**

MR1939197

**Zentralblatt MATH identifier**

1094.14500

**Subjects**

Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 32A27: Local theory of residues [See also 32C30] 57R20: Characteristic classes and numbers

Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 32S05: Local singularities [See also 14J17]

**Keywords**

Chern classes frames of vector bundles localization Grothendieck residues relative to subvarieties

#### Citation

SUWA, Tatsuo. Residues of Chern classes. J. Math. Soc. Japan 55 (2003), no. 1, 269--287. doi:10.2969/jmsj/1196890854. https://projecteuclid.org/euclid.jmsj/1196890854