zbMATH — the first resource for mathematics

Existence d’une structure kählérienne sur les variétés homogènes semi-simples. (Existence of a kähler metric on semi-simple homogeneous manifolds). (French) Zbl 0635.32019
Suppose G is a Lie group which is algebraic over \({\mathbb{C}}\) and H is an algebraic subgroup. Then the coset space G/H has a natural structure of a quasi-projective algebraic variety and, in particular, has a Kähler metric. The purpose of this note is to prove that if G is a complex semi- simple Lie group and G/H is Kähler, then H is algebraic. The methods rely on previous work on invariant plurisubharmonic functions, e.g. see the author’s dissertation [Fonctions plurisousharmoniques et kählériennité des variétés homogènes semi-simples, Thèse de Doctorat d’Université, U.S.T.L. Flandres Artois (1987)] or the author’s paper in J. Anal. Math. 48, 267-276 (1987; see the review above), and this result generalizes a previous result of the author and K. Oeljeklaus in the case of discrete isotropy [see “Invariant plurisubharmonic functions and hypersurfaces on semi-simple complex Lie groups”, Math. Ann., to appear].
Incidentally, for the case of solv-manifolds one should consult the paper of K. Oeljeklaus and W. Richthofer [On the structure of complex solv-manifolds, J. Diff. Geom., to appear]. Also related is the recent paper of W. Richthofer [Currents in homogeneous manifolds, to appear].
Reviewer: B.Gilligan

32M10 Homogeneous complex manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
22E10 General properties and structure of complex Lie groups
32U05 Plurisubharmonic functions and generalizations
31C10 Pluriharmonic and plurisubharmonic functions