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

Authors  
Keywords  Complex hyperbolic space form Holomorphic immersion Total geodesy Holomorphic isometry 
Issue Date  2011 
Publisher  Springer. The Journal's web site is located at http://www.springer.com/series/8806 
Citation  The Conference on Complex and Differential Geometry, Hannover, Germany, 1418 September 2009. In Springer Proceedings in Mathematics, 2011, v. 8, p. 323344 How to Cite? 
Abstract  In 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, CaoMok 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 ndimensional compact complex hyperbolic space form X into any mdimensional 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 complexanalytic subvarieties of large dimensions which are in some sense sufficiently deformable. While in the finitevolume case it is enough to apply the arguments of CaoMok, 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 AhlforsSchwarz Lemma, as motivated by recent work of KoziarzMok, 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. 
Description  Springer Proceedings in Mathematics v. 8 entitled: Complex and Differential Geometry: conference held at Leibniz Universitä, Hannover ... 2009 
Persistent Identifier  http://hdl.handle.net/10722/153394 
ISBN 
DC Field  Value  Language 

dc.contributor.author  Mok, N  en_US 
dc.date.accessioned  20120716T10:12:26Z   
dc.date.available  20120716T10:12:26Z   
dc.date.issued  2011  en_US 
dc.identifier.citation  The Conference on Complex and Differential Geometry, Hannover, Germany, 1418 September 2009. In Springer Proceedings in Mathematics, 2011, v. 8, p. 323344  en_US 
dc.identifier.isbn  9783642202995  en_US 
dc.identifier.uri  http://hdl.handle.net/10722/153394   
dc.description  Springer Proceedings in Mathematics v. 8 entitled: Complex and Differential Geometry: conference held at Leibniz Universitä, Hannover ... 2009   
dc.description.abstract  In 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, CaoMok 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 ndimensional compact complex hyperbolic space form X into any mdimensional 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 complexanalytic subvarieties of large dimensions which are in some sense sufficiently deformable. While in the finitevolume case it is enough to apply the arguments of CaoMok, 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 AhlforsSchwarz Lemma, as motivated by recent work of KoziarzMok, 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.language  eng  en_US 
dc.publisher  Springer. The Journal's web site is located at http://www.springer.com/series/8806  en_US 
dc.relation.ispartof  Springer Proceedings in Mathematics  en_US 
dc.rights  The original publication is available at www.springerlink.com   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.subject  Complex hyperbolic space form   
dc.subject  Holomorphic immersion   
dc.subject  Total geodesy   
dc.subject  Holomorphic isometry   
dc.title  On singularities of generically immersive holomorphic maps between complex hyperbolic space forms  en_US 
dc.type  Conference_Paper  en_US 
dc.identifier.openurl  http://library.hku.hk:4550/resserv?sid=HKU:IR&issn=9783642202995&volume=8&spage=323&epage=344&date=2011&atitle=On+singularities+of+generically+immersive+holomorphic+maps+between+complex+hyperbolic+space+forms  en_US 
dc.identifier.email  Mok, N: nmok@hku.hk  en_US 
dc.identifier.authority  Mok, N=rp00763  en_US 
dc.description.nature  postprint   
dc.identifier.doi  10.1007/9783642203008   
dc.identifier.hkuros  201001  en_US 
dc.identifier.volume  8  en_US 
dc.identifier.spage  323  en_US 
dc.identifier.epage  344  en_US 
dc.publisher.place  Germany   
dc.customcontrol.immutable  sml 140326   