Hartogs Type Domains

Abstract

Hartogs type domains are a class of domains of $\mathbb {C}^{n+m}$ characterized by a Kähler metric described locally by a Kähler potential of the form $\varPhi (z,w)=H(z)-\log \left (F(z)-|w|{ }^2\right )$, for suitable functions H and F. They have been studied under several points of view and represent a large class of examples in Kähler geometry (the reader finds precise references inside each section). The first section describes Cartan–Hartogs domains. Proposition 5.1 discusses the existence of a Kähler immersion into the infinite dimensional complex projective space in terms of the Cartan domains they are based on, and Theorem 5.2 proves they represent a counterexample for Conjecture 4.2 when the ambient space is infinite dimensional. Section 5.2 extends some of these results when the base domain is not symmetric but just a bounded homogeneous domain. Finally, in Sect. 5.3 we discuss the existence of a Kähler immersion for a large class of Hartogs domains whose Kähler potentials are given locally by $-\log \left (F(|z_0|{ }^2)-||z||{ }^2\right )$ for suitable function F (see Proposition 5.2).