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Book Chapter: Uniqueness Modulo Reduction of Bergman Meromorphic Compactifications of Canonically Embeddable Bergman Manifolds

TitleUniqueness Modulo Reduction of Bergman Meromorphic Compactifications of Canonically Embeddable Bergman Manifolds
Authors
Issue Date2014
PublisherWorld Scientific
Citation
Uniqueness Modulo Reduction of Bergman Meromorphic Compactifications of Canonically Embeddable Bergman Manifolds. In Ge, ML ... (et al) (Eds.), Frontiers in differential geometry, partial differential equations and mathematical physics: in memory of Gu Chaohao, p. 271-289. Singapore; New Jersey: World Scientific, 2014 How to Cite?
AbstractA 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.
Persistent Identifierhttp://hdl.handle.net/10722/205255
ISBN

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_US
dc.date.accessioned2014-09-20T02:07:07Z-
dc.date.available2014-09-20T02:07:07Z-
dc.date.issued2014en_US
dc.identifier.citationUniqueness Modulo Reduction of Bergman Meromorphic Compactifications of Canonically Embeddable Bergman Manifolds. In Ge, ML ... (et al) (Eds.), Frontiers in differential geometry, partial differential equations and mathematical physics: in memory of Gu Chaohao, p. 271-289. Singapore; New Jersey: World Scientific, 2014en_US
dc.identifier.isbn9789814578073-
dc.identifier.urihttp://hdl.handle.net/10722/205255-
dc.description.abstractA 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.-
dc.languageengen_US
dc.publisherWorld Scientificen_US
dc.relation.ispartofFrontiers in differential geometry, partial differential equations and mathematical physics: in memory of Gu Chaohao-
dc.titleUniqueness Modulo Reduction of Bergman Meromorphic Compactifications of Canonically Embeddable Bergman Manifoldsen_US
dc.typeBook_Chapteren_US
dc.identifier.emailMok, N: nmok@hku.hken_US
dc.identifier.authorityMok, N=rp00763en_US
dc.identifier.doi10.1142/9789814578097_0017en_US
dc.identifier.hkuros237309en_US
dc.identifier.spage271en_US
dc.identifier.epage289en_US
dc.publisher.placeSingapore; New Jerseyen_US

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