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postgraduate thesis: On the admissible pairs of rational homogeneous manifolds of Picard number 1 and geometric structures defined by their varieties of minimal rational tangents
Title  On the admissible pairs of rational homogeneous manifolds of Picard number 1 and geometric structures defined by their varieties of minimal rational tangents 

Authors  
Advisors  Advisor(s):Mok, N 
Issue Date  2014 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Zhang, Y. [张云鑫]. (2014). On the admissible pairs of rational homogeneous manifolds of Picard number 1 and geometric structures defined by their varieties of minimal rational tangents. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5317045 
Abstract  In a series of works, JunMuk Hwang and Ngaiming Mok have developed a geometric theory of uniruled projective manifolds, especially those of Picard Number 1, relying on the study of Varieties of Minimal Rational Tangents (VMRT) from both the algebrogeometric and the Gstructure perspectives. Based on this theory, Ngaiming Mok and Jaehyun Hong studied the standard embedding between two Rational Homogeneous Spaces (RHS) associated to long simple roots which are of different dimensions. In this thesis, I consider admissible pairs of RHS (X0, X) of Picard number 1 and locally closed complex submanifolds S ⊂ X inheriting VMRT substructures modeled on X0 = G0/P0 ⊂ X = G/P de_ned by taking intersections of VMRT of X with tangent space of S. Moreover, if any such S modeled on (X0, X) is necessarily the image of a standard embedding i : X0 → X, (X0, X) is said to be rigid. In this thesis, it is proved that an admissible pair (X0, X) is rigid whenever X is associated to a long simple root and X0 is nonlinear and de_ned by a marked Dynkin subdiagram. In the case of the pair (S0, S) of compact Hermitian Symmetric Spaces (cHSS), all the admissible pairs (S0, S) are completely classified. Based on this classification, a sufficient condition for the pair (S0, S) to be nonrigid is established through explicitly constructing a submanifold S ⊂ S such that S can never be obtained from the image of any standard embedding i : S0 → S. Besides, the term special pair is coined for those (S0; S) sorted out through classification, and the algebraicity of submanifolds modeled on special pairs is confirmed by checking a modified form of the nondegeneracy condition defined by Hong and Mok is satisfied. However, the question as to whether these special pairs are rigid, as pointed out in this thesis, remains to be investigated. Finally, pairs of hyperquadrics (Q^n, Q^m) are studied separately. Since nonrigidity is trivial, in these cases it is interesting to establish a characterization of the standard embedding i : Q^n→Q^m under some stronger condition. In this thesis, the latter problem is solved in terms of the partial vanishing of second fundamental forms. 
Degree  Doctor of Philosophy 
Subject  Functions of several complex variables Picard groups 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/206439 
HKU Library Item ID  b5317045 
DC Field  Value  Language 

dc.contributor.advisor  Mok, N   
dc.contributor.author  Zhang, Yunxin   
dc.contributor.author  张云鑫   
dc.date.accessioned  20141031T23:15:54Z   
dc.date.available  20141031T23:15:54Z   
dc.date.issued  2014   
dc.identifier.citation  Zhang, Y. [张云鑫]. (2014). On the admissible pairs of rational homogeneous manifolds of Picard number 1 and geometric structures defined by their varieties of minimal rational tangents. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5317045   
dc.identifier.uri  http://hdl.handle.net/10722/206439   
dc.description.abstract  In a series of works, JunMuk Hwang and Ngaiming Mok have developed a geometric theory of uniruled projective manifolds, especially those of Picard Number 1, relying on the study of Varieties of Minimal Rational Tangents (VMRT) from both the algebrogeometric and the Gstructure perspectives. Based on this theory, Ngaiming Mok and Jaehyun Hong studied the standard embedding between two Rational Homogeneous Spaces (RHS) associated to long simple roots which are of different dimensions. In this thesis, I consider admissible pairs of RHS (X0, X) of Picard number 1 and locally closed complex submanifolds S ⊂ X inheriting VMRT substructures modeled on X0 = G0/P0 ⊂ X = G/P de_ned by taking intersections of VMRT of X with tangent space of S. Moreover, if any such S modeled on (X0, X) is necessarily the image of a standard embedding i : X0 → X, (X0, X) is said to be rigid. In this thesis, it is proved that an admissible pair (X0, X) is rigid whenever X is associated to a long simple root and X0 is nonlinear and de_ned by a marked Dynkin subdiagram. In the case of the pair (S0, S) of compact Hermitian Symmetric Spaces (cHSS), all the admissible pairs (S0, S) are completely classified. Based on this classification, a sufficient condition for the pair (S0, S) to be nonrigid is established through explicitly constructing a submanifold S ⊂ S such that S can never be obtained from the image of any standard embedding i : S0 → S. Besides, the term special pair is coined for those (S0; S) sorted out through classification, and the algebraicity of submanifolds modeled on special pairs is confirmed by checking a modified form of the nondegeneracy condition defined by Hong and Mok is satisfied. However, the question as to whether these special pairs are rigid, as pointed out in this thesis, remains to be investigated. Finally, pairs of hyperquadrics (Q^n, Q^m) are studied separately. Since nonrigidity is trivial, in these cases it is interesting to establish a characterization of the standard embedding i : Q^n→Q^m under some stronger condition. In this thesis, the latter problem is solved in terms of the partial vanishing of second fundamental forms.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.subject.lcsh  Functions of several complex variables   
dc.subject.lcsh  Picard groups   
dc.title  On the admissible pairs of rational homogeneous manifolds of Picard number 1 and geometric structures defined by their varieties of minimal rational tangents   
dc.type  PG_Thesis   
dc.identifier.hkul  b5317045   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5317045   