Uniqueness Modulo Reduction of Bergman Meromorphic Compactifications of Canonically Embeddable Bergman Manifolds

Abstract

A complex manifold is said to be a Bergman manifold if the Bergman kernel form induces in the standard way a Kähler metric on the manifold. A Bergman manifold is said to be canonically embeddable if the canonical map into a possibly infinite-dimensional projective space defined using the Hilbert space of square-integrable holomorphic n-forms is a holomorphic embedding. In this article we define for a canonically embeddable Bergman manifold X the notion of Bergman meromorphic compactifications i: X ↪ Z into compact complex manifolds Z characterized in terms of extension properties concerning the Bergman kernel form on X, and define the notion of minimal elements among such compactifications. We prove that such a compact complex manifold Z is necessarily Moishezon. When X is given, assuming the existence of Bergman meromorphic compactifications i: X ↪ Z we prove the existence of a minimal element among them. More precisely, starting with any Bergman meromorphic compactification i: X ↪ Z we construct reductions of the compactification, and show that any reduction necessarily defines a minimal element. We show that up to a certain natural equivalence relation the minimal Bergman meromorphic compactification is unique. Examples of such compactifications include Borel embeddings of bounded symmetric domains into their compact dual manifolds and also those arising from canonical realizations of bounded homogeneous domains as Siegel domains or as bounded domains on Euclidean spaces and hence as domains on projective spaces.