Advertisement

Vector Fields and Cohomology of G/B

  • James B. Carrell
Part of the Progress in Mathematics book series (PM, volume 14)

Abstract

The topic I will discuss today is one which arose from a question which I believe Professor Matsushima originally asked: namely, if one is given a holomorphic vector field V on a projective manifold X, is it true that X has no nontrivial holomorphic p-forms if p > dimC zero (V)? Alan Howard answered this question affirmatively in [H] and later, D. Lieberman and I discovered other relationships between zeros of holomorphic vector fields and topology. Perhaps the most interesting of these is that if one has a holomorphic vector field V on a compact Kaehler manifold X with isolated zeros, then the whole cohomology ring of X can be calculated on the zeros of V. Although holomorphic vector fields with isolated zeros are not abundant, they do exist on a fundamental class of spaces, namely the algebraic homogeneous spaces. In the one example that has been carefully analyzed, the Grassmannians, the calculation of the cohomology ring on the zeros of V gives a new insight on the connection between Schubert calculus and the theory of symmetric functions [C].

Keywords

Vector Field Spectral Sequence Weyl Group Algebra Homomorphism Cohomology Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    Akyildiz, E., “A vector field on G/P with one zero,” Proc. Amer. Math. Soc. 67, 32–34 (1977).MathSciNetzbMATHGoogle Scholar
  2. [B]
    Borel, A., “Sur la cohomologie des espaces fibres principaux et des espaces homogenes des groupes de Lie compacts,” Ann. of Math.(2), 57, 115–207 (1953).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [C]
    Carrell, J.B., “Chern classes of the Grassmannians and Schubert calculus,” Topology 17, 177–182 (1978).MathSciNetzbMATHCrossRefGoogle Scholar
  4. [C-L1]
    Carrell, J.B. and Lieberman, D.I., “Holomorphic vector fields and Kaehler manifolds,” Invent. Math. 21, 303–309 (1974).MathSciNetCrossRefGoogle Scholar
  5. [C-L2]
    Carrell, J.B. and Lieberman, D.I., “Vector fields and Chern numbers,” Math. Annalen 225, 263–273 (1977).MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Ch]
    Chevalley, C., “Invariants of finite groups generated by reflections,” Amer. J. Math. 67, 778–782 (1955).MathSciNetCrossRefGoogle Scholar
  7. [H]
    Howard, A., “Holomorphic vector fields on proiective manifolds,” J. Math. 94, 1282–1290 (1972).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • James B. Carrell
    • 1
  1. 1.The University of British ColumbiaVancouverCanada

Personalised recommendations