Homogeneous complex manifold

A complex manifold $ M $ whose automorphism group acts transitively. All simply-connected one-dimensional complex manifolds — the Riemann sphere, the complex plane and the upper half-plane — are homogeneous. The manifold $ G / H $ of cosets of a complex Lie group $ G $ by a closed complex subgroup $ H $ is a homogeneous complex manifold.

Among the compact homogeneous manifolds there are the complex flag manifolds, which include all compact Hermitian symmetric spaces (cf. Symmetric space) [8]. Complex flag manifolds can be characterized as simply-connected compact Kähler manifolds (cf. Kähler manifold) [4], and also as manifolds $ G / P $ where $ G $ is a semi-simple complex Lie group (cf. Lie group, semi-simple) and $ P $ is a parabolic subgroup. Every compact homogeneous complex manifold admits a structure of a homogeneous holomorphic fibre bundle over a flag manifold with fibre isomorphic to the manifold of cosets of a complex Lie group by a discrete subgroup (see Tits bundle, and also [6], [9]).

Another important class of homogeneous complex manifolds is formed by the homogeneous bounded domains (cf. Homogeneous bounded domain), which include in particular the symmetric domains dual to compact Hermitian symmetric spaces.

Flag manifolds and homogeneous bounded domains represent special cases of homogeneous Kähler manifolds (i.e. Kähler manifolds on which a group of analytic automorphisms preserving the Kähler metric acts transitively). There is a conjecture [2] that every homogeneous Kähler manifold admits a structure of a homogeneous holomorphic fibre bundle with as base a homogeneous bounded domain and as fibre the direct product of a flag manifold and the manifold of cosets of a complex vector space by a discrete subgroup. This conjecture has been proved for homogeneous Kähler manifolds admitting a semi-simple [1] or completely solvable [2] transitive group of automorphisms, for compact homogeneous Kähler manifolds, which are thus isomorphic to direct products of flag manifolds and complex tori, [10], and recently in full generality [12].

In the theory of homogeneous complex manifolds, the canonical Hermitian form $ h _ \mu $ plays an important role. For any measure $ \mu $ on a complex manifold $ M $ given by the exterior differential form

$$ i ^ {n} K d z _ {1} \wedge \dots \wedge d z _ {n} \wedge \ d \overline{z} _ {1} \wedge \dots \wedge d \overline{z} _ {n} , $$

the Hermitian differential form

$$ h _ \mu = d ^ \prime d ^ {\prime\prime} \mathop{\rm ln} K = \ \sum _ { i,j } \frac{\partial ^ {2} \mathop{\rm ln} K }{\partial z _ {i} \partial \overline{z} _ {j} } \ d z _ {i} d \overline{z} _ {j} $$

can be constructed (it is degenerate in general). It is independent of the choice of the coordinate system, and does not change under multiplication of the measure $ \mu $ by a constant. If the measure $ \mu $ is defined in the standard way with respect to some Kähler metric on $ M $, then the form $ ( - h _ \mu ) $ is the same as the Ricci form of this metric [7]. If one requires $ \mu $ to be invariant with respect to some transitive group of automorphisms of $ M $, then it is uniquely determined by this group up to multiplication by a constant, and the form $ h _ \mu $ is uniquely determined. In the case when $ M $ is a homogeneous bounded domain, the Hermitian form $ h _ \mu $ so defined is positive definite and coincides with the Bergman metric. For flag manifolds, the form $ h _ \mu $ is negative definite.

The canonical Hermitian form of a homogeneous complex manifold can be computed in terms of the corresponding Lie algebra [3]. This is fundamental for the algebraization of the theory of homogeneous bounded domains and other homogeneous complex manifolds.

One direction in the theory of homogeneous complex manifolds is the study of holomorphic functions on them with the aid of the apparatus of linear representations of Lie groups. For example, it has been proved in this way [5] that the manifold $ G / H $ of cosets of a semi-simple complex Lie group $ G $ by a connected closed complex subgroup $ H $ is a Stein manifold if and only if $ H $ is reductive (cf. Reductive group).

There exist homogeneous complex manifolds that do not admit a transitive Lie group of automorphisms [11].

References

[1]	A. Borel, "Kählerian coset spaces of semisimple Lie groups" Proc. Nat. Acad. Sci. USA , 40 (1954) pp. 1147–1151
[2]	E.B. Vinberg, S.G. Gindikin, "Kählerian manifolds admitting a transitive solvable automorphism group" Math. USSR-Sb. , 3 (1967) pp. 333–351 Mat. Sb. , 74 : 3 (1967) pp. 357–377
[3]	J.L. Koszul, "Sur la forme Hermitienne canonique des espaces homogènes complexes" Canad. J. Math. , 7 : 4 (1955) pp. 562–576
[4]	A. Lichnerowicz, "Espaces homogènes Kähleriens" , Géométrie différentielle. Colloq. Internat. , CNRS (1953) pp. 171–184
[5]	A.L. Onishchik, "Complex hulls of compact homogeneous spaces" Soviet Math. Dokl. , 1 (1960) pp. 88–90 Dokl. Akad. Nauk SSSR , 130 (1960) pp. 726–729
[6]	J. Tits, "Espaces homogènes complexes compacts" Comment. Mat. Helv. , 37 (1962–1963) pp. 111–120
[7]	B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian)
[8]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[9]	H.-C. Wang, "Closed manifolds with homogeneous complex structure" Amer. J. Math. , 76 : 1 (1954) pp. 1–32
[10]	A. Borel, R. Remmert, "Ueber kompakte homogene Kählersche Mannigfaltigkeiten" Math. Ann. , 145 : 5 (1962) pp. 429–439
[11]	W. Kaup, "Reelle Transformationsgruppen und invariante Metriken auf komplexe Räume" Invent. Math. , 3 : 1 (1967) pp. 43–70
[12]	J. Dorfmeister, K. Nakajima, "The fundamental conjecture for homogeneous Kähler manifolds" Acta Math. , 161 : 1–2 (1988) pp. 23–70