Some remarks on Homogeneous Kaehler manifolds

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 μ0ω, admits a holomorphic isometric immersion in the complex projective space, for a suitable >0μ0>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.