An obstruction to homogeneous manifolds being Kähler
Annales de l'Institut Fourier, Volume 55 (2005) no. 1, p. 229-241

Let G be a connected complex Lie group, H a closed, complex subgroup of G and X:=G/H. Let R be the radical and S a maximal semisimple subgroup of G. Attempts to construct examples of noncompact manifolds X homogeneous under a nontrivial semidirect product G=SR with a not necessarily G-invariant Kähler metric motivated this paper. The S-orbit S/SH in X is Kähler. Thus SH is an algebraic subgroup of S [4]. The Kähler assumption on X ought to imply the S-action on the base Y of any homogeneous fibration XY is algebraic too. Natural considerations allow a reduction to the case where H=Γ is a discrete subgroup and there is a homogeneous fibration X=G/ΓG/I=:Y with I an abelian, normal subgroup of G and the fiber I /(I Γ) a Cousin group. An algebraic condition does hold in the homogeneous manifold Y=G ^/Γ ^, where G ^:=G/I and Γ ^:=I/I , namely, an element g ^Γ ^ of infinite order lying in a semisimple subgroup S ^ of G ^ is an obstruction to the existence of a Kähler metric on X. So X Kähler implies S ^Γ ^ finite.

Soit G un groupe de Lie complexe, H un sous-groupe complexe fermé de G, et X:=G/H. Soit R le radical et S un sous-groupe semi-simple maximal de G. La construction d’exemples de variétés non compactes X homogènes d’un produit semi-direct G=SR, possédant une métrique kählérienne pas nécessairement invariante par G, a suscité ce travail. L’orbite S/SH de S dans X est kählérienne. Donc SH est un sous-groupe algébrique de S [4]. La présence d’une structure kählérienne sur X devrait impliquer que l’action de S sur la base Y de chaque fibration homogène XY soit algébrique. Des considérations naturelles permettent de se placer dans le cas d’un sous-groupe discret H=Γ et d’une fibration homogène X=G/ΓG/I=:Y, où le sous-groupe I est abélien et normal dans G et la fibre I /(I Γ) est un groupe de Cousin. Une telle condition algébrique existe alors dans cet espace homogène Y=G ^/Γ ^, où G ^:=G/I et Γ ^:=I/I . Ceci signifie que l’existence d’un élément g ^Γ ^ d’ordre infini appartenant à un sous-groupe semi-simple S ^ de G ^ est une obstruction à l’existence d’une métrique kählérienne sur X. Ainsi X kählérien implique que S ^Γ ^ fini.

Classification:  32M10,  32Q15
Keywords: homogeneous complex manifolds, Kähler manifolds
     author = {Gilligan, Bruce},
     title = {An obstruction to homogeneous manifolds being K\"ahler},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     pages = {229-241},
     doi = {10.5802/aif.2097},
     zbl = {1070.32017},
     mrnumber = {2141697},
     language = {en},
     url = {}
Gilligan, Bruce. An obstruction to homogeneous manifolds being Kähler. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 229-241. doi : 10.5802/aif.2097.

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