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研究生： 廖欣瑩
Sin-Ying Liao
論文名稱： Riemann-Roch-Hirzebruch Theorem For Circle Bundle Over A Riemann Surface
指導教授： 黃榮宗
Rung-Tzung Huang
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2017
畢業學年度： 105
語文別： 英文
論文頁數： 30
中文關鍵詞： 黎曼面
外文關鍵詞： Hirzebruch-Riemann-Roch
相關次數： 點閱：10  下載：0 
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    • 鄭老師、蕭老師和蔡老師最近有一個定理，是在有 circle action 的 CR manifolds 上的 CR Riemann-Roch-Hirzebruch 定理。我們先複習在 Riemann surfaces 上的 Riemann-Roch-Hirzebruch 定理，然後再討論鄭老師、蕭老師和蔡老師在 CR manifolds 上 circle bundle 的 CR Riemann-Roch-Hirzebruch 定理。

    In a recent paper of Cheng, Hsiao and Tsai, they obtained a CR Riemann-Roch-Hirzebruch Theorem for CR manifolds with a circle action. In this thesis we first review the Riemann-Roch-Hirzebruch Theorem for Riemann surfaces. Then we discuss CR Riemann-Roch-Hirzebruch Theorem of Cheng, Hsiao and Tsai for a circle bundle over a Riemann surface in details.

    1 Introduction..............1 2 Riemann-Roch-Hirzebruch Theorem for a Riemann Surface.....................2 2.1 Riemann-Roch Theorem...2 2.2 Characteristic Classes....6 2.3 Riemann-Roch-Hirzebruch Theorem for a Riemann surface....8 3 Riemann-Roch-Hirzebruch Theorem for a circle bundle over a Riemann surface...............11 3.1 Set up and terminology...............11 3.2 Riemann-Roch-Hirzebruch theorem On Circle Bundles........................13 3.2.1 Tangential Character Classes for circle bundle over a Riemann Surface.........25 3.2.2 Main Theorem............28 4 參考文獻……………………………………………………………………………………………………30

    [1] M.-S. Baouendi and L.-P. Rothschild and F.-Treves, \textit{CR structures with group action and extendability of CR functions}, Invent. Math., 83(1985), 359-396.
    [2] J.-H. Cheng, C.-Y. Hsiao, I.-H. Tsai, \textit{Heat Kernel, Asymptotics And A Local Index Theorem For CR Manifolds With $S^1$ Action}, arXiv:1511.00063v2.
    [3] X. Dai, \textit{Lectures on Dirac Operators and Index Theory}, January 7, 2015.
    [4] X. Ma and G. Marinescu, \textit{Holomorphic Morse inequalities and Bergman kernels}, Progress in Math., vol. 254, Birkh$\ddot{a}$user, Basel, 2007, 422 pp.
    [5] D. Varolin, \textit{Riemann Surfaces by Way of Complex Analytic Geometry}, 2011.

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