Some remarks on homogeneous Kähler manifolds

Abstract

In this paper we provide a positive answer to a conjecture due to Di Scala et al. (Asian J Math, 2012, Conjecture 1) claiming that a simply-connected homogeneous Kähler manifold M endowed with an integral Kähler form $$\mu _0\omega $$ , admits a holomorphic isometric immersion in the complex projective space, for a suitable $$\mu _0>0$$ . This result has two corollaries which extend to homogeneous Kähler manifolds the results obtained by the authors Loi and Mossa (Geom Dedicata 161:119–128, 2012) and Mossa (J Geom Phys 86:492–496, 2014) for homogeneous bounded domains.