On the Zeroes of Meromorphic Vector-Fields

  • Paul F. Baum
  • Raoul Bott

Abstract

Let M be a compact complex analytic manifold and let x be a holomorphic vector-field on M. In an earlier paper by one of us (see [2]) it was shown that the behavior of x near its zeroes determined all the Chern numbers of M and the nature of this determination was explicitly given where x had only nondegenerate zeroes. The primary purpose of this note is to extend this result to meromorphic fields, or equivalently to sections s of TL where T is the holomorphic tangent bundle to M and L is a holomorphic line bundle. We will also drop the non-degeneracy assumption of the zeroes of s, but we treat only the case where s vanishes at isolated points {p}.

Keywords

Line Bundle Invariant Form Holomorphic Section Characteristic Ring Local Invariant 
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.

Bibliography

  1. [1]
    Atiyah, M.: Complex analytic connections in fiber bundles. Trans. Amer. Math. Soc. 85, 181–207 (1957).MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Bott, R.: Vector fields and characteristic numbers. Mich. Math. J. 14, 231–244 (1967).MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Bott, R.: A residue formula for holomorphic vector-fields. Differential Geometry 1/4, 311–330(1967).Google Scholar
  4. [4]
    Bott, R.: On a topological obstruction to integrability. To be published in Berkeley Symposium, 1968.Google Scholar
  5. [5]
    Hartshorne, R.: Residues and Duality. Berlin-Heidelberg-New York: Springer, Lecture Notes in Mathematics 20 (1966).Google Scholar
  6. [6]
    Lubkin, S.: A p-adic proof of Weils conjectures. Ann. of Math. 87/1, 105–194 (1968); 87 /2, 195–255 (1968).MathSciNetCrossRefGoogle Scholar
  7. [7]
    Eilenberg, S., and N. Steenrod: Foundations of Algebraic Topology. Princeton: Princeton University Press 1952.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1970

Authors and Affiliations

  • Paul F. Baum
  • Raoul Bott

There are no affiliations available

Personalised recommendations