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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!

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  • $\begingroup$ There are many non-simply connected examples, e.g., Iwasawa manifolds. By the classification of complex Lie groups, I believe (but should double-check) that every compact, simply-connected, homogeneous complex manifold is projective and Fano. $\endgroup$
    – Jason Starr
    Commented Jan 6, 2017 at 16:35
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    $\begingroup$ There are homogeneous complex structures on $S^{2n-1}\times S^{2m-1}$ (Calabi-Eckmann manifolds), so not every homogeneous compact simply-connected complex manifold is projective. $\endgroup$
    – Yury Ustinovskiy
    Commented Jan 6, 2017 at 17:18
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    $\begingroup$ Indeed the Calabi-Eckmann manifolds are not K\"ahler, because they have $b_2=0$. $\endgroup$
    – abx
    Commented Jan 6, 2017 at 20:17

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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

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