Archiv der Mathematik

, Volume 101, Issue 5, pp 495–499 | Cite as

Poincaré polynomial and \({SL(2, \mathbb{C})}\) -representation on Kähler manifolds

Article

Abstract

The cohomology ring of any compact Kähler manifold gives rise to an \({SL(2, \mathbb{C})}\) -representation. In this short note, we show that the character of this representation essentially is the Poincaré polynomial of this Kähler manifold, which gives a natural interpretation of the Poincaré polynomial for Kähler manifolds. Our result is an analogue to an interpretation of the χy genus for holomorphic symplectic manifolds due to George Thompson.

Mathematics Subject Classification (1991)

53C55 

Keywords

Kähler manifold Poincaré polynomial \({SL(2, \mathbb{C})}\)-representation Character 

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References

  1. 1.
    A. Fujiki, On the de Rham cohomology group of a compact Kähler symplectic manifold, Algebraic geometry, Sendai, 1985, 105–165, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987.Google Scholar
  2. 2.
    W. Fulton and J. Harris, Representation theory, a first course, Springer-Verlag, New York, 1991.Google Scholar
  3. 3.
    P. Griffiths and J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley, New York, 1978.Google Scholar
  4. 4.
    Thompson G.: A geometric interpretation of the χy genus on hyper-Kähler manifolds. Comm. Math. Phys. 212, 649–652 (2000)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    R. O. Wells, Differential Analysis on Complex Manifolds, third edition, Springer- Verlag, New York, 2008.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghaiChina

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