A result of Borel-Remmert in 1961/1962 published in Math. Ann. states that a compact homogeneous Kähler manifold must be the product of a complex torus and a projective-rational manifold. This implies that a simply-connected compact homogeneous Kähler manifold has nonzero Euler characteristic.


My question is, if a simply-connected compact homogeneous complex manifold has nonzero Euler characteristic, can we conclude that it admits a Kähler metric? Of course, if so, it must be a projective-rational manifold by the above-mentioned result. Or is there any counterexample?


It seems that the results of a 1954 paper of H.C. Wang published in Amer. J. Math. should be related to this question. But I cannot obtain the answer from this paper. 


Many thanks in advance!



A question about simply-connected homogeneous compact complex manifold
Yes, it appears to be true. An argument is written up in a recent preprint by Ping Li:


Nonnegative Hermitian holomorphic vector bundles and Chern numbers <https://arxiv.org/pdf/1702.01701.pdf>



> 
> Theorem 4.6.
>  Suppose $M^n$ is a compact connected homogeneous complex manifold. Then
>  we have the following implications:
> 
> 
> The Chern number $c\_1^n\neq 0 \Leftrightarrow$ Some Chern numbers of $M$ is nonzero $\Rightarrow$ $M$ is a projective algebraic manifold
> 
> 
> 


