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Conference Paper: From transcendence to algebraicity: techniques of analytic continuation on bounded symmetric domains and their dual compact Hermitian symmetric spaces
Title  From transcendence to algebraicity: techniques of analytic continuation on bounded symmetric domains and their dual compact Hermitian symmetric spaces 

Authors  
Issue Date  2015 
Citation  International Conference on Complex Geometry and Several Complex Variables, Shanghai, China, 1115 May 2015 How to Cite? 
Abstract  Analytic continuation is a central issue in Several Complex Variables, starting with the Hartogs Phenomenon. We examine the applicationsof techniques of analytic continuation in Complex Geometry for irreduciblebounded symmetric domains Ω and their dual Hermitian symmetric spaces ofthe compact typeS, and their ramifications to the geometric theory of uniruled projective manifolds. As a starting point, in the case where rank(S)≥2 we recall a proof of Ochiai’s theorem (1970) for analytic continuation of flatSstructure using Hartogs extension, and its generalization to the CartanFubini extension principle of HwangMok (2001) in the geometric theory ofuniruled projective manifolds basing on varieties of minimal rational tangents(VMRTs). Applying methods of algebraic extension in CRgeometry of Webster and Huang, and Ochiai’s theorem, we give the proof of MokNg (2012)that under a nondegeneracy assumption, a germ of measurepreserving holomorphic mapf: (Ω,λdμΩ; 0)→(Ω,dμΩ; 0)×···×(Ω,dμΩ; 0), wheredμΩdenotes the Bergman volume form andλ >0 is a real constant, is necessarilya totally geodesic diagonal embedding, answering in the affirmative a problemof ClozelUllmo stemming from a problem in Arithmetic Dynamics regarding Hecke correspondences. The proof involves Alexander’s Theorem for thecomplex unit ballBn,n≥2, in the rank1 case and a new Alexandertype extension theorem for the case of irreducible bounded symmetric domains Ω ofrank≥2 for germs of holomorphic maps preserving the regular part Reg(∂Ω)of the boundary. In another direction we explain the nonequidimensionalCartanFubini extension principle of HongMok (2010) and its application tothe characterization of smooth Schubert varieties in rational homogeneousmanifolds of Picard number 1 (HongMok 2013). Finally, we consider theproblem of analytic continuation of subvarieties of uniruled projective manifolds (X,K) equipped with a VMRTstructure (e.g. irreducible Hermitiansymmetric spacesSof the compact type) under the assumption that thesubvariety inherits a subVMRT structure by taking intersections of VMRTswith tangent spaces, and establish a principle of analytic continuation (MokZhang 2015) by a parametrized Thullen extension of subVMRT structuresalong chains of rational curves. 
Description  Plenary Lecture  Venue: East China Normal University 
Persistent Identifier  http://hdl.handle.net/10722/269928 
DC Field  Value  Language 

dc.contributor.author  Mok, N   
dc.date.accessioned  20190516T03:34:20Z   
dc.date.available  20190516T03:34:20Z   
dc.date.issued  2015   
dc.identifier.citation  International Conference on Complex Geometry and Several Complex Variables, Shanghai, China, 1115 May 2015   
dc.identifier.uri  http://hdl.handle.net/10722/269928   
dc.description  Plenary Lecture  Venue: East China Normal University   
dc.description.abstract  Analytic continuation is a central issue in Several Complex Variables, starting with the Hartogs Phenomenon. We examine the applicationsof techniques of analytic continuation in Complex Geometry for irreduciblebounded symmetric domains Ω and their dual Hermitian symmetric spaces ofthe compact typeS, and their ramifications to the geometric theory of uniruled projective manifolds. As a starting point, in the case where rank(S)≥2 we recall a proof of Ochiai’s theorem (1970) for analytic continuation of flatSstructure using Hartogs extension, and its generalization to the CartanFubini extension principle of HwangMok (2001) in the geometric theory ofuniruled projective manifolds basing on varieties of minimal rational tangents(VMRTs). Applying methods of algebraic extension in CRgeometry of Webster and Huang, and Ochiai’s theorem, we give the proof of MokNg (2012)that under a nondegeneracy assumption, a germ of measurepreserving holomorphic mapf: (Ω,λdμΩ; 0)→(Ω,dμΩ; 0)×···×(Ω,dμΩ; 0), wheredμΩdenotes the Bergman volume form andλ >0 is a real constant, is necessarilya totally geodesic diagonal embedding, answering in the affirmative a problemof ClozelUllmo stemming from a problem in Arithmetic Dynamics regarding Hecke correspondences. The proof involves Alexander’s Theorem for thecomplex unit ballBn,n≥2, in the rank1 case and a new Alexandertype extension theorem for the case of irreducible bounded symmetric domains Ω ofrank≥2 for germs of holomorphic maps preserving the regular part Reg(∂Ω)of the boundary. In another direction we explain the nonequidimensionalCartanFubini extension principle of HongMok (2010) and its application tothe characterization of smooth Schubert varieties in rational homogeneousmanifolds of Picard number 1 (HongMok 2013). Finally, we consider theproblem of analytic continuation of subvarieties of uniruled projective manifolds (X,K) equipped with a VMRTstructure (e.g. irreducible Hermitiansymmetric spacesSof the compact type) under the assumption that thesubvariety inherits a subVMRT structure by taking intersections of VMRTswith tangent spaces, and establish a principle of analytic continuation (MokZhang 2015) by a parametrized Thullen extension of subVMRT structuresalong chains of rational curves.   
dc.language  eng   
dc.relation.ispartof  International Conference on Complex Geometry and Several Complex Variables   
dc.title  From transcendence to algebraicity: techniques of analytic continuation on bounded symmetric domains and their dual compact Hermitian symmetric spaces   
dc.type  Conference_Paper   
dc.identifier.email  Mok, N: nmok@hku.hk   
dc.identifier.authority  Mok, N=rp00763   
dc.identifier.hkuros  243586   