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An Integral Formula for the Chern from of a Hermitian Bundle

Published online by Cambridge University Press:  22 January 2016

Hideo Omoto
Hideo Omoto*
Affiliation:
Mathematical Institute, Nagoya University
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We shall consider a Hermitian n-vector bundle E over a complex manifold X. When X is compact (without boundary), S.S. Chern defined in his paper [3] the Chern classes (the basic characteristic classes of E) Ĉi(E), i = 1, · · ·, n, in terms of the basic forms Φi on the Grassmann manifold H(n, N) and the classifying map f of X into H(n, N). Moreover he proved ([3], [4]) that if Ek denotes the k-general Stiefel bundle associated with E, the (n — k + 1)-th Chern class Ĉn-k+1(E) coincides with the characteristic class C(Ek) of Ek defined as follows: Let K be a simplicial decomposition of X and K2(n-k)+1 the 2(n — k) + 1 — shelton of K.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 42 , March 1971 , pp. 135 - 172
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Bott, R. and Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math., 114 (1965), 71–112.CrossRefGoogle Scholar
[2] Levine, H.I.: A theorem on holomorphic mappings into complex projective space. Ann. Math., 71 (1960), 529–535.CrossRefGoogle Scholar
[3] Chern, S.S.: Characteristic classes of Hermitian manifold. Ann. Math., 47 (1946), 85–121.CrossRefGoogle Scholar
[4] Steenrod, N.: The topology of fibre bundles. Princeton University Press (1951).CrossRefGoogle Scholar
[5] Akizuki, Y.: The theory of harmonic integrals. Iwanami (Japan), (1955–1956).Google Scholar
[6] Sternberg, S.: Lectures on Differential Geometry. New Jersey, Prentice-Hall, Inc., (1964).Google Scholar