An obstruction to homogeneous manifolds being Kähler
[Une obstruction à l'existence de métriques kählériennes sur les espaces homogènes]
Gilligan, Bruce1
1 University of Regina, department of Mathematics and Statistics , Regina, S4S 0A2 (Canada)
Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 229-241.

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=S⋉R, possédant une métrique kählérienne pas nécessairement invariante par G, a suscité ce travail. L’orbite S/S∩H de S dans X est kählérienne. Donc S∩H 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 X→Y 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/Γ→G/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.

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=S⋉R with a not necessarily G-invariant Kähler metric motivated this paper. The S-orbit S/S∩H in X is Kähler. Thus S∩H 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 X→Y 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/Γ→G/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.

DOI : 10.5802/aif.2097
Classification : 32M10, 32Q15
Keywords: homogeneous complex manifolds, Kähler manifolds
Mot clés : espaces homogènes complexes, espaces kählériens
Gilligan, Bruce 1

1 University of Regina, department of Mathematics and Statistics , Regina, S4S 0A2 (Canada) 
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Gilligan, Bruce. An obstruction to homogeneous manifolds being Kähler. Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 229-241. doi : 10.5802/aif.2097. https://www.numdam.org/articles/10.5802/aif.2097/

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