Some remarks on homogeneous K\"ahler manifolds

TL;DR

This paper proves that simply-connected homogeneous K"ahler manifolds with an integral K"ahler form can be holomorphically and isometrically embedded into complex projective space, confirming a conjecture and extending previous results.

Contribution

It provides a positive answer to a conjecture about embeddings of homogeneous K"ahler manifolds into projective space, expanding the understanding of their geometric structure.

Findings

Confirmed the conjecture for simply-connected homogeneous K"ahler manifolds.

Extended results from homogeneous bounded domains to all homogeneous K"ahler manifolds.

Established conditions under which such manifolds admit holomorphic isometric immersions.

Abstract

In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous K\"ahler manifold M endowed with an integral K\"ahler form $\mu\omega$, admits a holomorphic isometric immersion in the complex projective space, for a suitable $\mu>0$. This result has two corollaries which extend to homogeneous K\"ahler manifolds the results obtained by the authors in [8] and in [12] for homogeneous bounded domains.