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Conference Paper: On singularities of generically immersive holomorphic maps between complex hyperbolic space forms

TitleOn singularities of generically immersive holomorphic maps between complex hyperbolic space forms
Authors
KeywordsComplex hyperbolic space form
Holomorphic immersion
Total geodesy
Holomorphic isometry
Issue Date2011
PublisherSpringer. The Journal's web site is located at http://www.springer.com/series/8806
Citation
The Conference on Complex and Differential Geometry, Hannover, Germany, 14-18 September 2009. In Springer Proceedings in Mathematics, 2011, v. 8, p. 323-344 How to Cite?
AbstractIn 1965, Feder proved using a cohomological identity that any holomorphic immersion t: Pn→Pm between complex projective spaces is necessarily a linear embedding whenever m < 2n. In 1991, Cao-Mok adapted Feder’s identity to study the dual situation of holomorphic immersions between compact complex hyperbolic space forms, proving that any holomorphic immersion f : X→Y from an n-dimensional compact complex hyperbolic space form X into any m-dimensional complex hyperbolic space form Y must necessarily be totally geodesic provided that m < 2n. We study in this article singularity loci of generically injective holomorphic immersions between complex hyperbolic space forms. Under dimension restrictions, we show that the open subset U over which the map is a holomorphic immersion cannot possibly contain compact complex-analytic sub-varieties of large dimensions which are in some sense sufficiently deformable. While in the finite-volume case it is enough to apply the arguments of Cao-Mok, the main input of the current article is to introduce a geometric argument that is completely local. Such a method applies to f: X→Y in which the complex hyperbolic space form X is possibly of infinite volume. To start with we make use of the Ahlfors-Schwarz Lemma, as motivated by recent work of Koziarz-Mok, and reduce the problem to the local study of contracting leafwise holomorphic maps between open subsets of complex unit balls. Rigidity results are then derived from a commutation formula on the complex Hessian of the holomorphic map.
DescriptionSpringer Proceedings in Mathematics v. 8 entitled: Complex and Differential Geometry: conference held at Leibniz Universitä, Hannover ... 2009
Persistent Identifierhttp://hdl.handle.net/10722/153394
ISBN

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_US
dc.date.accessioned2012-07-16T10:12:26Z-
dc.date.available2012-07-16T10:12:26Z-
dc.date.issued2011en_US
dc.identifier.citationThe Conference on Complex and Differential Geometry, Hannover, Germany, 14-18 September 2009. In Springer Proceedings in Mathematics, 2011, v. 8, p. 323-344en_US
dc.identifier.isbn978-3-642-20299-5en_US
dc.identifier.urihttp://hdl.handle.net/10722/153394-
dc.descriptionSpringer Proceedings in Mathematics v. 8 entitled: Complex and Differential Geometry: conference held at Leibniz Universitä, Hannover ... 2009-
dc.description.abstractIn 1965, Feder proved using a cohomological identity that any holomorphic immersion t: Pn→Pm between complex projective spaces is necessarily a linear embedding whenever m < 2n. In 1991, Cao-Mok adapted Feder’s identity to study the dual situation of holomorphic immersions between compact complex hyperbolic space forms, proving that any holomorphic immersion f : X→Y from an n-dimensional compact complex hyperbolic space form X into any m-dimensional complex hyperbolic space form Y must necessarily be totally geodesic provided that m < 2n. We study in this article singularity loci of generically injective holomorphic immersions between complex hyperbolic space forms. Under dimension restrictions, we show that the open subset U over which the map is a holomorphic immersion cannot possibly contain compact complex-analytic sub-varieties of large dimensions which are in some sense sufficiently deformable. While in the finite-volume case it is enough to apply the arguments of Cao-Mok, the main input of the current article is to introduce a geometric argument that is completely local. Such a method applies to f: X→Y in which the complex hyperbolic space form X is possibly of infinite volume. To start with we make use of the Ahlfors-Schwarz Lemma, as motivated by recent work of Koziarz-Mok, and reduce the problem to the local study of contracting leafwise holomorphic maps between open subsets of complex unit balls. Rigidity results are then derived from a commutation formula on the complex Hessian of the holomorphic map.-
dc.languageengen_US
dc.publisherSpringer. The Journal's web site is located at http://www.springer.com/series/8806en_US
dc.relation.ispartofSpringer Proceedings in Mathematicsen_US
dc.rightsThe original publication is available at www.springerlink.com-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectComplex hyperbolic space form-
dc.subjectHolomorphic immersion-
dc.subjectTotal geodesy-
dc.subjectHolomorphic isometry-
dc.titleOn singularities of generically immersive holomorphic maps between complex hyperbolic space formsen_US
dc.typeConference_Paperen_US
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=978-3-642-20299-5&volume=8&spage=323&epage=344&date=2011&atitle=On+singularities+of+generically+immersive+holomorphic+maps+between+complex+hyperbolic+space+formsen_US
dc.identifier.emailMok, N: nmok@hku.hken_US
dc.identifier.authorityMok, N=rp00763en_US
dc.description.naturepostprint-
dc.identifier.doi10.1007/978-3-642-20300-8-
dc.identifier.hkuros201001en_US
dc.identifier.volume8en_US
dc.identifier.spage323en_US
dc.identifier.epage344en_US
dc.publisher.placeGermany-
dc.customcontrol.immutablesml 140326-

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