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題名 K 穩定性與熱帶幾何之研究
On K Stability and Tropical Geometry
作者 李威德
Li, Wei De
貢獻者 蔡炎龍
Tsai, Yen Lung
李威德
Li, Wei De
關鍵詞 K穩定性
熱帶幾何
法諾超平面
K stability
tropical geometry
Fano hypersurface
日期 2011
上傳時間 30-Oct-2012 16:27:29 (UTC+8)
摘要 在這篇論文中，我們從K energy的角度探討緊緻法諾超平面上的K穩定性。首先，我們給K energy一個較明確的型式，接著再透過分析的手法求解其導函數。後續，我們引進熱帶幾何的結構來重新分析主要的結果，最後給一些法諾超平面的實例，驗證我們所得到的公式。
In this thesis, we analyze K stability on compact Fano hypersurfaces from K energy. We first represent the K energy into an explicitly formula. Then we compute the derivative by using some analytic techniques. Furthermore, we introduce some structures of tropical geometry to analyze the main result. Finally, we give some examples of compact Fano hypersurface to test and verify the formula we get.
參考文獻 [1] T. Aubin. Equations du type de Monge-Ampére sur les variétés Kähleriennes compactes. C. R. Acad. Sci. Paris. 283: 119-121, 1976.
[2] D. Burns and P. De Bartolomeis. Stability of vector bundles amd extremal metrics. Inventions Mathematicae. 92(2):403–407, 1988.
[3] W. Y. Ding and G. Tian. Kähler-Einstein metrics and the generalized Futaki invariant. Inventions Mathematicae. 110: 315–335, 1992.
[4] M. Einsiedler, M. Kapranov and D. Lind. Non-Archimedean amoebas and tropical varieties. ArXiv preprint:math.AG/0408311, 2004.
[5] S. K. Donaldson. Scalar curvature and stability of toric varieties. Journal of Differential Geometry. 62(2): 289–349, 2002.
[6] A. Futaki. An obstruction to the existence of Einstein- Kähler metrics. Inventions Mathematicae. 73: 437–443, 1983.
[7] A. Gathmann. Tropical algebraic geometry. Jahresbericht der Deutschen Mathematiker-Vereinigung. 108(1): 3–32, 2006.
[8] Y. J. Hong. Gauge-fixing constant scalar curvature equations on ruled manifolds and the Futaki invariants. Journal of Differential Geometry. 60(3): 389–453, 2002.
[9] M. Kapranov. Amoebas over non-archimedean fields. Preprint. 2000.
[10] Z. Lu. On the Futaki invariants of complete intersections. Duke Mathematical Journal. 100(2): 359–372, 1999.
[11] Z. Lu. K energy and K stability on hypersurfaces. Communications in Analysis and Geometry. 12(3): 599-628, 2004.
[12] T. Mabuchi. K energy maps integrating Futaki invariants. Tohoku Mathematical Journal. 38: 245–257, 1986.
[13] Y. Matsushima. Sur la structure du group d`homeomorphismes analytiques d`une certaine varietie Kahleriennes. Nagoya Mathematical Journal. 11: 145–150, 1957.
[14] D. H. Phong and J. Sturm. Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Annals of Mathematics II. 152(1): 277–329, 2000.
[15] J. Ross and R. Thomas. A study of the Hilbert-Mumford criterion for the stability of projective varieties, Journal of Differential Geometry. 16(2): 201–255, 2007.
[16] G. Tian. The K- energy on hypersurfaces and stability. Communications in Analysis and Geometry. 2(2): 239–265, 1994.
[17] G. Tian. Kähler-Einstein metrics with positive scalar curvature. Inventions Mathematicae. 137: 1–37, 1997.
[18] S. T. Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge- Ampére equation, I. Communications on Pure and Applied Mathematics. 31: 339–441, 1978.
描述 碩士
國立政治大學
應用數學研究所
99751004
100
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0997510041
資料類型 thesis
dc.contributor.advisor 蔡炎龍zh_TW
dc.contributor.advisor Tsai, Yen Lungen_US
dc.contributor.author (Authors) 李威德zh_TW
dc.contributor.author (Authors) Li, Wei Deen_US
dc.creator (作者) 李威德zh_TW
dc.creator (作者) Li, Wei Deen_US
dc.date (日期) 2011en_US
dc.date.accessioned 30-Oct-2012 16:27:29 (UTC+8)-
dc.date.available 30-Oct-2012 16:27:29 (UTC+8)-
dc.date.issued (上傳時間) 30-Oct-2012 16:27:29 (UTC+8)-
dc.identifier (Other Identifiers) G0997510041en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/55095-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 99751004zh_TW
dc.description (描述) 100zh_TW
dc.description.abstract (摘要) 在這篇論文中，我們從K energy的角度探討緊緻法諾超平面上的K穩定性。首先，我們給K energy一個較明確的型式，接著再透過分析的手法求解其導函數。後續，我們引進熱帶幾何的結構來重新分析主要的結果，最後給一些法諾超平面的實例，驗證我們所得到的公式。zh_TW
dc.description.abstract (摘要) In this thesis, we analyze K stability on compact Fano hypersurfaces from K energy. We first represent the K energy into an explicitly formula. Then we compute the derivative by using some analytic techniques. Furthermore, we introduce some structures of tropical geometry to analyze the main result. Finally, we give some examples of compact Fano hypersurface to test and verify the formula we get.en_US
dc.description.tableofcontents 謝辭­ .......................... i
Abstract ­ .......................... iii
中文摘要 ­ .......................... iv
Content ­ .......................... v
1 Introduction ­ .......................... 1
2 Tropical Geometry ­ .......................... 8
3 An explicit formula for the K energy ­ .......................... 16
4 The limit of the derivative of the K energy ­ .......................... 27
5 Some Examples ­ .......................... 43
References ­ .......................... 52		zh_TW
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0997510041en_US
dc.subject (關鍵詞) K穩定性zh_TW
dc.subject (關鍵詞) 熱帶幾何zh_TW
dc.subject (關鍵詞) 法諾超平面zh_TW
dc.subject (關鍵詞) K stabilityen_US
dc.subject (關鍵詞) tropical geometryen_US
dc.subject (關鍵詞) Fano hypersurfaceen_US
dc.title (題名) K 穩定性與熱帶幾何之研究zh_TW
dc.title (題名) On K Stability and Tropical Geometryen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] T. Aubin. Equations du type de Monge-Ampére sur les variétés Kähleriennes compactes. C. R. Acad. Sci. Paris. 283: 119-121, 1976.
[2] D. Burns and P. De Bartolomeis. Stability of vector bundles amd extremal metrics. Inventions Mathematicae. 92(2):403–407, 1988.
[3] W. Y. Ding and G. Tian. Kähler-Einstein metrics and the generalized Futaki invariant. Inventions Mathematicae. 110: 315–335, 1992.
[4] M. Einsiedler, M. Kapranov and D. Lind. Non-Archimedean amoebas and tropical varieties. ArXiv preprint:math.AG/0408311, 2004.
[5] S. K. Donaldson. Scalar curvature and stability of toric varieties. Journal of Differential Geometry. 62(2): 289–349, 2002.
[6] A. Futaki. An obstruction to the existence of Einstein- Kähler metrics. Inventions Mathematicae. 73: 437–443, 1983.
[7] A. Gathmann. Tropical algebraic geometry. Jahresbericht der Deutschen Mathematiker-Vereinigung. 108(1): 3–32, 2006.
[8] Y. J. Hong. Gauge-fixing constant scalar curvature equations on ruled manifolds and the Futaki invariants. Journal of Differential Geometry. 60(3): 389–453, 2002.
[9] M. Kapranov. Amoebas over non-archimedean fields. Preprint. 2000.
[10] Z. Lu. On the Futaki invariants of complete intersections. Duke Mathematical Journal. 100(2): 359–372, 1999.
[11] Z. Lu. K energy and K stability on hypersurfaces. Communications in Analysis and Geometry. 12(3): 599-628, 2004.
[12] T. Mabuchi. K energy maps integrating Futaki invariants. Tohoku Mathematical Journal. 38: 245–257, 1986.
[13] Y. Matsushima. Sur la structure du group d`homeomorphismes analytiques d`une certaine varietie Kahleriennes. Nagoya Mathematical Journal. 11: 145–150, 1957.
[14] D. H. Phong and J. Sturm. Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Annals of Mathematics II. 152(1): 277–329, 2000.
[15] J. Ross and R. Thomas. A study of the Hilbert-Mumford criterion for the stability of projective varieties, Journal of Differential Geometry. 16(2): 201–255, 2007.
[16] G. Tian. The K- energy on hypersurfaces and stability. Communications in Analysis and Geometry. 2(2): 239–265, 1994.
[17] G. Tian. Kähler-Einstein metrics with positive scalar curvature. Inventions Mathematicae. 137: 1–37, 1997.
[18] S. T. Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge- Ampére equation, I. Communications on Pure and Applied Mathematics. 31: 339–441, 1978.		zh_TW