Homogeneous Kähler Manifolds

Abstract

In this chapter we survey what is known about the existence of Kähler immersions of homogeneous Kähler manifolds into complex space forms . Recall that a homogeneous Kähler manifold is a Kähler manifold on which the group of holomorphic isometries $\mathop {\mathrm {Aut}} (M)\cap \mathrm {Isom}(M, g)$ acts transitively on M (here $\mathop {\mathrm {Aut}} (M)$ denotes the group of biholomorphisms of M). In the first two sections we summarize the results of Di Scala et al. (Asian J Math (3) 16:479–488, 2012) about Kähler immersion of homogeneous Kähler manifolds into complex Euclidean and hyperbolic spaces . Section 3.1 is devoted to proving that the only homogeneous bounded domains which are projectively induced for all positive multiples of their metrics are given by the product of complex hyperbolic spaces. This result, combined with the solution of Dorfmeister and Nakajima (Acta Math 161(1–2):23–70, 1988) of the fundamental conjecture on homogeneous Kähler manifolds (Theorem 3.2), will be applied in Sect. 3.2 to classify homogeneous Kähler manifolds admitting a Kähler immersion into $\mathbb {C}\mathrm {H}^N$ or $\mathbb {C}^N$, N ≤∞ (Theorem 3.3). In the last three sections we consider Kähler immersions of homogeneous Kähler manifolds into $\mathbb {C}\mathrm {P}^N$, N ≤∞. The general case is discussed in Sect. 3.3, while in Sects. 3.4 and 3.5 we detail the case of Käher immersions of bounded symmetric domains into $\mathbb {C}\mathrm {P}^\infty$.