A bounded homogeneous domain and a projective manifold are not relatives

Abstract

Let $M_1$ and $M_2$ be two K\"ahler manifolds. One says that $M_1$ and $M_2$ are "relatives" if they share a non-trivial K\"ahler submanifold $S$, namely, if there exist two holomorphic and isometric immersions (K\"ahler immersions) $h_1: S -> M_1$ and $h_2: S -> M_2$. In this paper we show that a bounded homogeneous domain with a homogeneous K\"ahler metric and a projective K\"ahler manifold (i.e. a projective manifold endowed with the restriction of the Fubini-Study metric) are not relatives. Our result is a generalization of the result obtained by A. J. Di Scala and A. Loi (in "K\"ahler manifolds and their relatives", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 3 (2010), 495-501) for the Bergman metrics.