Journal of the Mathematical Society of Japan

Residues of Chern classes

Tatsuo SUWA

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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

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

First available in Project Euclid: 5 December 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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

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