Kähler immersions of homogeneous Kähler manifolds into complex space forms

Abstract

Abstract. In this paper we study the homogeneous Kähler manifolds (h.K.m.) which can be Kähler immersed into finite or infinite dimensional complex space forms. On the one hand we completely classify the h.K.m. which can be Kähler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. On the other hand, we extend known results about Kähler immersions into the finite dimensional complex projective space to the infinite dimensional setting.

Two conjectures on Ricci-flat Kähler metrics

“…Indeed one can prove (see, e.g. [12]) that, given a compact simply connected homogeneous Kähler (Einstein) manifold M with integral Kähler form ω, then the Kodaira map k : M → CP N suitably normalized is a Kähler immersion. One can see [12] that the assumptions of simply-connectedness and compactness of M and the integrality of ω are necessary (this excludes for example the compact flat torus to be projectively induced).…”

“…[12]) that, given a compact simply connected homogeneous Kähler (Einstein) manifold M with integral Kähler form ω, then the Kodaira map k : M → CP N suitably normalized is a Kähler immersion. One can see [12] that the assumptions of simply-connectedness and compactness of M and the integrality of ω are necessary (this excludes for example the compact flat torus to be projectively induced). Moreover, if the Kähler form ω of a compact and simply-connected homogeneous Kähler manifold (M, g) is integral then there exists a positive integer k such that kg is projectively induced (see [37] for a proof based on semisimple Lie groups and Dynkin diagrams).…”

On holomorphic isometric immersions of nonhomogeneous Kähler–Einstein manifolds into the infinite dimensional complex projective space

“…Kähler-Einstein submanifolds with codimension 1 and 2 in the finite dimensional complex projective space have been completely classified by Chern [7] and Tsukada [31], respectively. Homogeneous Kähler submanifolds of finite or infinite dimensional complex Euclidean form or hyperbolic space have been classified by Scala, Ishi, Loi in [10].…”

The unitary representations parametrized by the Wallach set for a homogeneous bounded domain

“…Let D be an irreducible symmetric bounded domain of rank r. In [5,Theorem 15], the assertion (i) is generalized as follows: which is not necessarily an arithmetic sequence. Moreover, we see from [2,Theorem 4] that the discrete part consists of one point (i.e. d D 1) if and only if D is holomorphically equivalent to a direct product of complex unit balls (see the last paragraph of [2]).…”

“…Moreover, we see from [2,Theorem 4] that the discrete part consists of one point (i.e. d D 1) if and only if D is holomorphically equivalent to a direct product of complex unit balls (see the last paragraph of [2]). …”