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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 蕭欽玉 | zh_TW |
| dc.contributor.advisor | Chin-Yu Hsiao | en |
| dc.contributor.author | 蔣岳霖 | zh_TW |
| dc.contributor.author | Yueh-Lin Chiang | en |
| dc.date.accessioned | 2023-08-15T17:42:58Z | - |
| dc.date.available | 2023-11-09 | - |
| dc.date.copyright | 2023-08-15 | - |
| dc.date.issued | 2023 | - |
| dc.date.submitted | 2023-08-08 | - |
| dc.identifier.citation | [1] Robert Berman. Bergman kernels and local holomorphic Morse inequalities, Math. Z. 248 (2004), no. 2, 325–344.
[2] Robert Berman, Bo Berndtsson, Johannes Sjöstrand. A direct approach to Bergman kernel asymptotics for positive line bundles, Ark. Math. 46 (2008), no. 2,197–217. [3] Robert Berman, Johannes Sjöstrand. Asymptotics for Bergman-Hodge kernels for high powers of complex line bundles, Ann. Fac. Sci. Toulouse Math. (6), 16 (2007), No.4, 719-771. [4] Martin Bordemann, Eckhard Meinrenken, Martin Schlichenmaier. Toeplitz quantization of Kähler manifolds and gl(N), N→1 limits, Comm. Math. Phys., 165(1994), No.2, 281-296. [5] Thierry Bouche. Convergence de la métrique de Fubini-Study d’un fibré linéaire positif. Annales de l’Institut Fourier, Volume 40 (1990) no. 1, pp. 117-130. doi : 10.5802/aif.1206. [6] David Catlin. The Bergman kernel and a theorem of Tian. In Analysis and geometry in several complex variables (Katata, 1997), Trends Math., pages 1-23. Birkhäuser Boston, Boston, MA, 1999. [7] Xiuxiong Chen, Simon Donaldson, Song Sun. Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Amer. Math. Soc., 28 (2015), No.1, 183-197. [8] Xiuxiong Chen, Simon Donaldson, Song Sun. Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof, J. Amer. Math. Soc., 28 (2015), No.1, 235-278. [9] Xianzhe Dai, Kefeng Liu, Xiaonan Ma. On the asymptotic expansion of Bergman kernel, C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 193–198. [10] Xianzhe Dai, Kefeng Liu, Xiaonan Ma. On the asymptotic expansion of Bergman kernel, J. Differential Geom., 72, (2006), no. 1, 1–41. [11] Edward Davies. Spectral theory and differential operators, volume 42 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. [12] Jean-Pierre Demailly. Complex analytic and algebraic geometry. https://www-fourier.ujfgrenoble.fr/demailly/manuscripts/agbook.pdf, 2012. [13] Simon Donaldson. Scalar curvature and projective embeddings. I, J. Differential Geom., 59(2001), No.3, 479-522. [14] Simon Donaldson, Song Sun. Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math., 213 (2014), No.1, 63-106. [15] Matthew Gaffney. Hilbert space methods in the theory of harmonic integrals. Trans. Amer. Math. Soc., 78:426–444, 1955. [16] Peter Gilkey. Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, Publish or Perish. [17] Phillip Griffiths, Joseph Harris. Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience [JohnWiley and Sons], New York, 1978. [18] Lars Hörmander. The analysis of linear partial differential operators. I. Classics in Mathematics. Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. [19] Yu-Chi Hou. Asymptotic expansion of the Bergman kernel via semiclassical symbolic calculus. Bull. Inst. Math. Acad. Sin. (N.S.) 17(1), 1–51 (2022). [20] Chin-Yu Hsiao. Bergman kernel asymptotics and a pure analytic proof of the Kodaira embedding theorem, In Complex analysis and geometry, volume 144 of Springer Proc. Math. Stat., pages 161-173. Springer, Tokyo, 2015. [21] Chin-Yu Hsiao, George Marinescu. Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles, Comm. Anal. Geom., 22 (2014), No.1, 1-108. [22] Chin-Yu Hsiao, George Marinescu. Berezin–Toeplitz quantization for lower energy forms, Communications in Partial Differential Equations (2017), 42:6, 895-942. [23] Xiaonan Ma, George Marinescu. Berezin-Toeplitz quantization on Kähler manifolds, J. Reine Angew. Math., 662 (2012), 1-56. [24] Xiaonan Ma, George Marinescu. The first coefficients of the asymptotic expansion of the Bergman kernel of the spin^{c} Dirac operator, Internat. J. Math., 17 (2006), No.6, 737-759. [25] Xiaonan Ma, George Marinescu. Holomorphic Morse inequalities and Bergman kernels, volume 254 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2007. [26] Raymond O. Wells Jr.. Differential Analysis on Complex Manifolds, third edition, Graduate Texts in Mathematics, 65, Springer, New York, 2008. [27] Martin Schlichenmaier. Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results. Adv. Math. Phys., pages Art. ID 927280, 38, 2010. [28] Gang Tian. On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., 32 (1990), No.1, 99-130. [29] Edward Witten. Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661-692. [30] Kosaku Yosida. Functional analysis. Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition. | - |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/88770 | - |
| dc.description.abstract | 在這篇論文中，我們發展了一種新的伸縮方法，用於研究複流形線叢之高階張量冪的譜核與柏格曼核於局部譜間隙條件下的行為。特別的，我們給出了譜核和柏格曼核的逐點漸進性質的簡單證明。作為一個新結果，在純函數而不帶有形式的情況下，我們在具有指數衰減的譜間隙條件下得到了柏格曼核的主要項。此外，在 (0,q)-形式的一般情況下，即使線叢的曲率退化，漸進性質仍然成立。 | zh_TW |
| dc.description.abstract | In this thesis, we develop a new scaling method to study spectral and Bergman kernels for the k-th tensor power of a line bundle over a complex manifold under local spectral gap condition. In particular, we establish a simple proof of the pointwise asymptotics of spectral and Bergman kernels. As a new result, in the function case, we obtain the leading term of Bergman kernel under spectral gap with exponential decay. Moreover, in the general cases of (0,q)-forms, the asymptotics remain valid while the curvature of the line bundle is degenerate. | en |
| dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2023-08-15T17:42:58Z No. of bitstreams: 0 | en |
| dc.description.provenance | Made available in DSpace on 2023-08-15T17:42:58Z (GMT). No. of bitstreams: 0 | en |
| dc.description.tableofcontents | Verification Letter from the Oral Examination Committee i
Acknowledgements ii 摘要 iii Abstract iv Contents v Chapter I Introduction ...p.1 I.1 Set-up and the main results ...p.4 Chapter II Preliminaries and terminology ...p.10 II.1 Standard notations ...p.10 II.2 Complex geometry and Hermitian holomorphic line bundle ...p.12 II.3 The spectral and Bergman kernels ...p.16 II.4 The Sobolev and Gårding inequalities ...p.19 Chapter III The local uniform bounds for scaled spectral and Bergman kernels ...p.23 III.1 The scaled bundles ...p.23 III.2 The Laplacians ...p.28 III.3 The uniform bounds ...p.34 Chapter IV Asymptotics of spectral and Bergman kernels ...p.43 IV.1 The model case ...p.44 IV.2 Mapping properties of the approximated integral operator ...p.50 IV.3 Asymptotic of the function case ...p.54 IV.4 The spectral gap of the extended Laplacian...p.60 IV.5 Asymptotics of the general (0,q)-forms cases ...p.67 References ...p.79 | - |
| dc.language.iso | en | - |
| dc.title | (0, q)-形式的柏格曼核與譜核之半經典漸進 | zh_TW |
| dc.title | Semi-Classical Asymptotics of Bergman and Spectral Kernels for (0,q)-forms | en |
| dc.type | Thesis | - |
| dc.date.schoolyear | 111-2 | - |
| dc.description.degree | 碩士 | - |
| dc.contributor.oralexamcommittee | 李瑩英;郭庭榕;楊劼之;黃榮宗 | zh_TW |
| dc.contributor.oralexamcommittee | Yng-Ing Lee;Ting-Jung Kuo ;Ryosuke Takahashi;Rung-Tzung Huang | en |
| dc.subject.keyword | 柏格曼核,複幾何,半經典分析,複分析,譜核,譜間隙, | zh_TW |
| dc.subject.keyword | Bergman Kernel,Complex Geometry,Semi-Classical Analysis,Complex Analysis,Spectral Kernel,Spectral Gap, | en |
| dc.relation.page | 82 | - |
| dc.identifier.doi | 10.6342/NTU202302926 | - |
| dc.rights.note | 同意授權(限校園內公開) | - |
| dc.date.accepted | 2023-08-09 | - |
| dc.contributor.author-college | 理學院 | - |
| dc.contributor.author-dept | 數學系 | - |
| dc.date.embargo-lift | 2024-08-01 | - |
| 顯示於系所單位： | 數學系 | |
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