Asymptotics of the Bergman Kernel for Positive Line Bundles

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曾遠涵(Yuan-Han Tseng) 查詢紙本館藏

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論文名稱

(Asymptotics of the Bergman Kernel for Positive Line Bundles)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ Bergman kernel

關鍵字(英)

★ Bergman kernel

論文目次

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

參考文獻

[1] Banyaga, A., and Houenou, D. F., A Brief Introduction to Symplectic and Contact Manifolds, Singapore:
World Scientific, (2017).
[2] Berman, R., Berndtsson, B., Sjöstrand, J., A direct approach to Bergman kernel asymptotics for positive
line bundles, Ark. Mat., 46 (2008), 197-217.
[3] Catlin, D, The Bergman Kernel and a Theorem of Tian. In: Komatsu, G., Kuranishi, M. (eds) Analysis
and Geometry in Several Complex Variables, (1999). Trends in Mathematics, pp.1-23, Birkhäuser Boston.
[4] Charles, L., Berezin-Toeplitz Operators, a Semi-Classical Approach, Comm. Math. Phys., 239 (2003),
1-28.
[5] Fefferman, C., The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent Math
26, 1–65 (1974).
[6] Friedlander, F. G. (Friedrich Gerard), and Joshi, M. S., Introduction to the Theory of Distributions. 2nd
ed., F.G. Friedlander, with additional material by M.S. Joshi, Cambridge, UK ;: Cambridge University
Press, (1999).
[7] Krantz, S. G., Geometric Analysis of the Bergman Kernel and Metric, New York: Springer-Verlag, (2013).
[8] Le Floch, Y., A Brief Introduction to Berezin-Toeplitz Operators on Compact Kahler Manifolds, Cham:
Springer International Publishing, (2018).
[9] Ma, X. and Marinescu, G., Holomorphic Morse Inequalities and Bergman Kernels Basel: Birkhäuser,
(2007).
[10] Boutet de Monvel, L., Sjöstrand, J., Johannes, Sur la singularité des noyaux de Bergman et de Szegö,
Journées équations aux dérivées partielles (1975), Astérisque 34-35 pp. 123-164.
[11] Cannas da Silva, A.,Lectures on Symplectic Geometry. 1st ed. 2008, Berlin, Heidelberg: Springer Berlin
Heidelberg, (2008).
[12] Tian, G., On a set of polarized Kähler metrics on algebraic manifolds, Journal of Differential Geometry
32.1, 99-130 (1990).
[13] Tu, W. L., An Introduction to Manifolds. 2nd ed. 2011, New York, NY: Springer New York, (2011).
[14] Zelditch, S., Szegö kernels and a theorem of Tian, International Mathematics Research Notices, Issue 6,
1998, pp. 317–331, (1998).

指導教授

黃榮宗(Rung-Tzung Hung)

審核日期

2022-6-15

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