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    Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/55095


    Title: K 穩定性與熱帶幾何之研究
    On K Stability and Tropical Geometry
    Authors: 李威德
    Li, Wei De
    Contributors: 蔡炎龍
    Tsai, Yen Lung
    李威德
    Li, Wei De
    Keywords: K穩定性
    熱帶幾何
    法諾超平面
    K stability
    tropical geometry
    Fano hypersurface
    Date: 2011
    Issue Date: 2012-10-30 16:27:29 (UTC+8)
    Abstract: 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.
    Reference: [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.
    Description: 碩士
    國立政治大學
    應用數學研究所
    99751004
    100
    Source URI: http://thesis.lib.nccu.edu.tw/record/#G0997510041
    Data Type: thesis
    Appears in Collections:[Department of Mathematical Sciences] Theses

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