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研究生：	曾遠涵 Yuan-Han Tseng
論文名稱：	Asymptotics of the Bergman Kernel for Positive Line Bundles
指導教授：	黃榮宗 Rung-Tzung Hung
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：	2022
畢業學年度：	110
語文別：	英文
論文頁數：	52
中文關鍵詞：	Bergman kernel
外文關鍵詞：	Bergman kernel
相關次數：	點閱：6 下載：0
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基於 Charles 以及 Berman, Berndtsson 和 Sj\"{o}strand 的論文，我們展示了一個在$L^k$上 Bergman kernel 漸近展開式的證明，其中$L$是緊緻的凱勒流形上的正線叢。最後我們給出了在複射影空間上的 Bergman kernel 漸進展開式的詳細計算。

We present a proof of an asymptotic expansion in power of $k$ on the Bergman kernel to $L^k$, where $L$ is a positive line bundle over a compact K\"{a}hler manifold, based on the papers of Charles and Berman, Berndtsson and Sj\"{o}strand. We give an explicit computation of the Bergman kernel on complex projective spaces.

中文摘要 i
ABSTRACT iii
Explanation of Symbols v
Chapter 1. Introduction 1
Chapter 2. Complex Manifold 3
2.1. Bilinear forms 3
2.2. Almost complex structure 4
2.3. Compatible almost complex structure 5
2.4. Compatibility 6
Chapter 3. Dolbeault Theorem 7
3.1. The complexified tangent bundle 7
3.2. Decomposition of forms 7
3.3. Dolbeault cohomology 8
Chapter 4. Kähler Manifold 11
4.1. Kähler form 11
4.2. Kähler potential 12
4.3. Some properties 13
Chapter 5. Prequantum Line Bundle 15
5.1. Complex line bundle 15
5.2. Connections on line bundle 16
5.3. Quantum spaces 19
Chapter 6. Asymptotics of the Projector 21
6.1. The section E 21
6.2. Schwartz kernel 22
6.3. The Bergman kernel 24
6.4. The proof of the projector asymptotics 26
Chapter 7. The Bergman Kernel on Complex Projective Space 31
7.1. The Kähler potential 31
7.2. The prequantum line bundle 32
7.3. The Liouville volume form 33
7.4. The section E 34
7.5. The Bergman kernel 36
Bibliography 39
                                

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Journées équations aux dérivées partielles (1975), Astérisque 34-35 pp. 123-164.
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