Archiv der Mathematik

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

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



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 χ y genus for holomorphic symplectic manifolds due to George Thompson.

Mathematics Subject Classification (1991)



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


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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghaiChina

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