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Zbl 1160.32023
Gilligan, Bruce; Oeljeklaus, Karl
Two remarks on Kähler homogeneous manifolds. (English)
[J] Ann. Fac. Sci. Toulouse, Math. (6) 17, No. 1, 73-80 (2008). ISSN 0240-2963

If $G$ is a complex Lie group, $H$ is a closed complex subgroup of $G$ and $X:=G/H$ is the corresponding complex homogeneous manifold, the authors discuss two facts related to whether $X$ is Kähler or not. First, they prove that if $G$ is solvable and $X$ is Kähler (this object is called ``solvmanifold" by the authors), then the holomorphic reduction of $X$ is, up to a finite covering, a principal bundle with a complex abelian Lie group admitting no nonconstant holomorphic functions as fiber. Next, if $G$ admits a Levi-Malcev decomposition $G=S \cdot R$ where $S$ is semisimple, $R$ is the radical of $G$ and both factors have positive dimension, and $X$ is such that $\cal O(X)\cong \Bbb{C}$ and if one assumes that $H$ is not contained in any proper parabolic subgroup of $G$ and $R\cap H$ is Zariski dense in $R$, then $X$ is not Kähler. Several interesting examples are also presented.
[Eugen Pascu (Montréal)]

MSC 2000:: *32Q15 Kähler manifolds
32M05 Automorphism groups of complex spaces
32M10 Homogeneous complex manifolds

Keywords: complex Lie group; Kähler manifold; homogeneous space; solvmanifold

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