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Conference Paper: Analytic continuation on bounded symmetric domains and uniruled projective manifolds
Title | Analytic continuation on bounded symmetric domains and uniruled projective manifolds |
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Authors | |
Issue Date | 2015 |
Citation | 2015 Taipei Conference on Complex Geometry, National Taiwan University, Taipei, Taiwan, 19-23 December 2015 How to Cite? |
Abstract | Analytic continuation is a central issue in Several Complex Variables, starting with theHartogs Phenomenon. We examine techniques of analytic continuation for irreducible boundedsymmetric domains Ω and their dual Hermitian symmetric spaces of the compact typeS, andtheir generalizations to uniruled projective manifolds.As a starting point, for rank(S)≥2 we recall a proof using local differential geometry andthe Hartogs Phenomenon of a theorem of Ochiai (1970) for the analytic continuation of flatS-structures, and its generalization to the Cartan-Fubini extension principle of Hwang-Mok(2001) in the geometric theory of uniruled projective manifolds modeled on varieties of minimalrational tangents (VMRTs). Applying CR-geometry, Mok-Ng (2012) proved that under anondegeneracy assumption, a germ of measure-preserving holomorphic mapf: (Ω,λdμΩ; 0)→(Ω,dμΩ; 0)×···×(Ω,dμΩ; 0), wheredμΩdenotes the Bergman volume form andλ >0 is a realconstant, is necessarily a totally geodesic embedding, answering in the affirmative a questionof Clozel-Ullmo (2003) regarding commutants of Hecke correspondences. The proof involves anew Alexander-type extension theorem for irreducible bounded symmetric domains Ω of rank≥2.In another direction we explain the non-equidimensional Cartan-Fubini extension princi-ple of Hong-Mok (2010). We consider furthermore the problem of analytic continuation ofsubvarieties of uniruled projective manifolds (X,K) equipped with a VMRT-structure underthe assumption that the subvariety inherits a sub-VMRT structure defined by intersections ofVMRTs with projectivized tangent spaces, and establish a principle of analytic continuation(Mok-Zhang 2015) under auxiliary conditions by constructing a universal family of chains ofrational curves by an analytic process and proving its algebraicity by establishing a Thullenextension theorem on a paramentrized family of sub-VMRT structures along chains of rationalcurves. |
Description | Invited Lecture - Institute of Mathematics, Academia Sinica |
Persistent Identifier | http://hdl.handle.net/10722/270581 |
DC Field | Value | Language |
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dc.contributor.author | Mok, N | - |
dc.date.accessioned | 2019-05-31T03:46:45Z | - |
dc.date.available | 2019-05-31T03:46:45Z | - |
dc.date.issued | 2015 | - |
dc.identifier.citation | 2015 Taipei Conference on Complex Geometry, National Taiwan University, Taipei, Taiwan, 19-23 December 2015 | - |
dc.identifier.uri | http://hdl.handle.net/10722/270581 | - |
dc.description | Invited Lecture - Institute of Mathematics, Academia Sinica | - |
dc.description.abstract | Analytic continuation is a central issue in Several Complex Variables, starting with theHartogs Phenomenon. We examine techniques of analytic continuation for irreducible boundedsymmetric domains Ω and their dual Hermitian symmetric spaces of the compact typeS, andtheir generalizations to uniruled projective manifolds.As a starting point, for rank(S)≥2 we recall a proof using local differential geometry andthe Hartogs Phenomenon of a theorem of Ochiai (1970) for the analytic continuation of flatS-structures, and its generalization to the Cartan-Fubini extension principle of Hwang-Mok(2001) in the geometric theory of uniruled projective manifolds modeled on varieties of minimalrational tangents (VMRTs). Applying CR-geometry, Mok-Ng (2012) proved that under anondegeneracy assumption, a germ of measure-preserving holomorphic mapf: (Ω,λdμΩ; 0)→(Ω,dμΩ; 0)×···×(Ω,dμΩ; 0), wheredμΩdenotes the Bergman volume form andλ >0 is a realconstant, is necessarily a totally geodesic embedding, answering in the affirmative a questionof Clozel-Ullmo (2003) regarding commutants of Hecke correspondences. The proof involves anew Alexander-type extension theorem for irreducible bounded symmetric domains Ω of rank≥2.In another direction we explain the non-equidimensional Cartan-Fubini extension princi-ple of Hong-Mok (2010). We consider furthermore the problem of analytic continuation ofsubvarieties of uniruled projective manifolds (X,K) equipped with a VMRT-structure underthe assumption that the subvariety inherits a sub-VMRT structure defined by intersections ofVMRTs with projectivized tangent spaces, and establish a principle of analytic continuation(Mok-Zhang 2015) under auxiliary conditions by constructing a universal family of chains ofrational curves by an analytic process and proving its algebraicity by establishing a Thullenextension theorem on a paramentrized family of sub-VMRT structures along chains of rationalcurves. | - |
dc.language | eng | - |
dc.relation.ispartof | Taipei Conference on Complex Geometry, National Taiwan University | - |
dc.title | Analytic continuation on bounded symmetric domains and uniruled projective manifolds | - |
dc.type | Conference_Paper | - |
dc.identifier.email | Mok, N: nmok@hku.hk | - |
dc.identifier.authority | Mok, N=rp00763 | - |
dc.identifier.hkuros | 256590 | - |