Roberto Mossa

A bounded homogeneous domain and a projective manifold are not relatives

Pages: 55-59
Received: 8 March 2012   
Accepted in revised form: 13 June 2012
Mathematics Subject Classification (2010): 53D05, 53C55.

Keywords: Diastasis, homogeneous bounded domains, Kähler metric, relatives.

Abstract: Let M₁ and M₂ be two Kähler manifolds. Following [4] one says that M₁ and M₂ are relatives if they share a non-trivial Kähler submanifold S, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) h₁ : S → M₁ and h₂ : S → M₂ . Our main results in this paper is Theorem 1.2 where we show that a bounded homogeneous domain with a homogeneous Kähler metric and a projective Kähler 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 in [4] for the Bergman metrics.

 
[4] A. J. Di Scala, A. Loi, Kähler manifolds and their relatives, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 3, 495-501