published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

透過使用調和映射的Bochner技巧, Siu[15, 16]證明了對於複維數≥2 時不可約對稱域緊致商空間的複結構的強剛定理. 其後在[9]中, Mok 證明了在任何秩≥2 的不可約對稱域緊致商空間上, 所有具備非正全純雙截曲率的Hermitian 度量必然和典範度量相差一個常數因子. 由這個定理和Siu 的定理可以得出Mostow 剛性定理[14]在特殊情形下的推廣.本論文會對Mok的結果作出研究. / By using Bochner technique of harmonic maps, Siu[15, 16] proved a strong rigidity theorem concerning the complex structure of compact quotients of irreducible bounded symmetric domain of complex dimension≥ 2. Later in [9], Mok proved a metric rigidity theorem which asserts that any Hermitian metric of seminegative holomorphic bisectional curvature on a compact quotient of an irreducible bounded symmetric domain of rank≥ 2 is necessarily a constant multiple of the canonical metric. This theorem together with the theorem of Siu yields a generalization of a special case of Mostow's rigidity theorem[14]. This thesis is an exposition of Mok's results. / Detailed summary in vernacular field only. / Li, Ka Fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 102-104). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Symmetric Space --- p.5 / Chapter 2.1 --- Riemannian Symmetric Spaces --- p.5 / Chapter 2.2 --- Lie Groups and Lie Algebras --- p.10 / Chapter 2.3 --- Riemannian Symmetric Spaces of Compact and Non-compact type --- p.11 / Chapter 2.4 --- Hermitian Symmetric Spaces --- p.16 / Chapter 2.5 --- Duality --- p.19 / Chapter 3 --- Some Embedding Theorems --- p.22 / Chapter 3.1 --- The Borel Embedding Theorem --- p.22 / Chapter 3.2 --- Root Space Decomposition and Root System --- p.24 / Chapter 3.3 --- The Polydisc Theorem --- p.28 / Chapter 3.4 --- The Harish-Chandra Embedding Theorem --- p.36 / Chapter 4 --- Bounded Symmetric Domains --- p.42 / Chapter 4.1 --- Classical Bounded Symmetric Domains --- p.42 / Chapter 4.2 --- The Bergman metric --- p.57 / Chapter 5 --- Projective and Characteristic Bundle --- p.65 / Chapter 5.1 --- Projectivization of Hermitian Vector Bundle --- p.65 / Chapter 5.2 --- Characteristic bundle --- p.69 / Chapter 6 --- The Hermitian Metric Rigidity Theorem --- p.83 / Chapter 7 --- Appendix --- p.100 / Bibliography --- p.102

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

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Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (M. Phil.)--University of Hong Kong, 2006. / Also available online.

In this thesis, we study three problems in complex geometry. In the first part, we study the behavior of the Kahler-Ricci flow on complete non-compact manifolds with negative holomorphic curvature. We show that Kahler-Ricci flow converges to a Kahler-Einstein metric when the initial manifold admits a suitable exhaustion function, thus improving upon a result of D. Wu and S.T. Yau. These results are partly obtained in joint work with S. Huang, M.-C. Lee and L.-F. Tam. In the second part of this thesis, we introduce a new Kodaira-Bochner type formula for closed (1, 1)-form in non-Kahler geometry. Based on this new formula, We propose a new curvature positivity condition in non-Kahler manifolds and proved a strong rigidity type theorem for manifolds satisfying this curvature positivity condition. We also find interesting examples non-Kahler manifolds satisfying the curvature positivity condition in a class of manifolds called Vaisman manifolds. In the third part of this thesis, we study the degenerations of asymptotically conical Calabi-Yau manifolds as the Kahler class degenerates to a non-Kahler class. Under suitable hypothesis, we prove the convergence of asymptotically conical Calabi-Yau metrics to a singular asymptotically conical Calabi-Yau current with compactly supported singularities. Using this, we construct singular asymptotically conical Calabi-Yau metrics on non-compact singular varieties and identify the topology of these singular metrics with the singular variety. We also give some interpretations of these asymptotically conical Calabi-Yau metrics from the point of view of physics. These results are obtained in joint work with T. Collins and B. Guo.

The thesis mainly studies two problems in Algebraic Geometry and Hodge Theory. The first problem deals with the geometric realizations of certain Hermitian symmetric domains as moduli space of algebraic varieties, notably the Abelian varieties and Calabi-Yau varieties. The study of the first problem occupies most of the thesis. In section 1.3; we study the second problem, namely, the L2 Higgs cohomology of polarized variation of Hodge structures over Hermitian symmetric domains. / Sheng Mao. / "December 2005." / Advisers: Shing-Tung Yau; Nai-Chung Conan Leung. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6442. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 108-113). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves [86]-88).