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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 |
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Authors | |
Issue Date | 2015 |
Citation | International Conference on Complex Geometry and Several Complex Variables, Shanghai, China, 11-15 May 2015 How to Cite? |
Abstract | Analytic continuation is a central issue in Several Complex Vari-ables, 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 unir-uled 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 flatS-structure using Hartogs extension, and its generalization to the Cartan-Fubini extension principle of Hwang-Mok (2001) in the geometric theory ofuniruled projective manifolds basing on varieties of minimal rational tangents(VMRTs). Applying methods of algebraic extension in CR-geometry of Web-ster and Huang, and Ochiai’s theorem, we give the proof of Mok-Ng (2012)that under a nondegeneracy assumption, a germ of measure-preserving holo-morphic 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 Clozel-Ullmo stemming from a problem in Arithmetic Dynamics regard-ing Hecke correspondences. The proof involves Alexander’s Theorem for thecomplex unit ballBn,n≥2, in the rank-1 case and a new Alexander-type ex-tension 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 non-equidimensionalCartan-Fubini extension principle of Hong-Mok (2010) and its application tothe characterization of smooth Schubert varieties in rational homogeneousmanifolds of Picard number 1 (Hong-Mok 2013). Finally, we consider theproblem of analytic continuation of subvarieties of uniruled projective man-ifolds (X,K) equipped with a VMRT-structure (e.g. irreducible Hermitiansymmetric spacesSof the compact type) under the assumption that thesubvariety inherits a sub-VMRT structure by taking intersections of VMRTswith tangent spaces, and establish a principle of analytic continuation (Mok-Zhang 2015) by a parametrized Thullen extension of sub-VMRT 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 |
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dc.contributor.author | Mok, N | - |
dc.date.accessioned | 2019-05-16T03:34:20Z | - |
dc.date.available | 2019-05-16T03:34:20Z | - |
dc.date.issued | 2015 | - |
dc.identifier.citation | International Conference on Complex Geometry and Several Complex Variables, Shanghai, China, 11-15 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 Vari-ables, 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 unir-uled 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 flatS-structure using Hartogs extension, and its generalization to the Cartan-Fubini extension principle of Hwang-Mok (2001) in the geometric theory ofuniruled projective manifolds basing on varieties of minimal rational tangents(VMRTs). Applying methods of algebraic extension in CR-geometry of Web-ster and Huang, and Ochiai’s theorem, we give the proof of Mok-Ng (2012)that under a nondegeneracy assumption, a germ of measure-preserving holo-morphic 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 Clozel-Ullmo stemming from a problem in Arithmetic Dynamics regard-ing Hecke correspondences. The proof involves Alexander’s Theorem for thecomplex unit ballBn,n≥2, in the rank-1 case and a new Alexander-type ex-tension 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 non-equidimensionalCartan-Fubini extension principle of Hong-Mok (2010) and its application tothe characterization of smooth Schubert varieties in rational homogeneousmanifolds of Picard number 1 (Hong-Mok 2013). Finally, we consider theproblem of analytic continuation of subvarieties of uniruled projective man-ifolds (X,K) equipped with a VMRT-structure (e.g. irreducible Hermitiansymmetric spacesSof the compact type) under the assumption that thesubvariety inherits a sub-VMRT structure by taking intersections of VMRTswith tangent spaces, and establish a principle of analytic continuation (Mok-Zhang 2015) by a parametrized Thullen extension of sub-VMRT 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 | - |