MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

We say that a Kahler manifold is a Kahler group if it is also a Lie group. I would like to know which semi-simple Lie groups are also Kahler groups?

share|cite|improve this question
I was about to reply, but then I saw that you meant something quite different than what I thought. Kahler group usually means fundamental group of a compact Kahler manifold. But anyway, for the question you asked, do you want the metric to be invariant under the group action? – Donu Arapura Jul 22 '11 at 20:37
Yes, it should be invariant. – Jean Delinez Jul 24 '11 at 14:15
up vote 8 down vote accepted

Semisimple Lie groups admit bi-invariant metrics (although not necessarily positive-definite) and it is not hard to show that if a Lie group admits a bi-invariant metric and also a left-invariant Kähler structure, then the group is abelian, contradicting the assumption that it was semisimple. Hence no semisimple Lie group admits a left-invariant Kähler structure.

In the case where the Kähler structure is not left-invariant, the two structures do not talk to each other and hence you are asking whether a manifold which admits the structure of a semisimple Lie group could also admit a Kähler structure. The identity component of such a manifold is (rationally) homotopy equivalent to a product of odd spheres (of dimension at least 3), so $H^2$ vanishes and thus, if compact, they again cannot admit a Kähler structure.

I'm not sure about the noncompact case, though; but it looks unlikely to me at this time.

share|cite|improve this answer
In fact, it is enough for the group to be unimodular in order to deduce that if it has a left-invariant Kähler structure it is abelian. This is proved in a paper by Lichnerowicz and Medina ( – José Figueroa-O'Farrill Jul 22 '11 at 22:54

See and references therein.

share|cite|improve this answer
How does this answer the question? – José Figueroa-O'Farrill Jul 22 '11 at 22:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.