Let $G$ be a compact complex Lie group and $M$ be a compact Kähler manifold. Does there exist any example of a holomorphic principal $G$-bundle over $M$ admitting Kähler structures?
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2$\begingroup$ Are there many compact complex Lie groups? Aren't they all tori? If $G$ is K\"ahler itself, what about the trivial bundle $M\times G\to M$: it is holomorphic and admits a K\"ahler metric. $\endgroup$– Alex DegtyarevCommented Apr 2, 2014 at 17:48
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4$\begingroup$ As Alex says, a compact complex Lie group is a complex torus. A result of Atyiah says that if the total space of a torus bundle over a simply connected Kahler manifold is Kahler, then the bundle is trivial. Looking at such bundles is actually one of the main methods of constructing non-Kahler manifolds. $\endgroup$– Gunnar Þór MagnússonCommented Apr 2, 2014 at 18:29
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2$\begingroup$ A compact complex Lie group is an extension of a finite group by a complex torus. $\endgroup$– Ben McKayCommented Apr 2, 2014 at 20:10
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$\begingroup$ What's a "principle" bundle? $\endgroup$– Alex SuciuCommented Apr 3, 2014 at 0:05
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$\begingroup$ compact coadjoint orbits? $\endgroup$– user21574Commented Jun 11, 2014 at 17:13
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1 Answer
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A connected compact complex Lie group is a torus, hence the question is apparently about principal torus bundles. Of course, product of a torus and a compact Kahler manifold is Kahler, giving a trivial answer to your question. For non-trivial examples, one may take a finite quotient of a trivial torus fibration by a finite group acting freely and compatible with the fibration structure.
answered Apr 10, 2014 at 15:37