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研究生: 蘇栢毅
Su, Bo-Yi
論文名稱: ADE型奇點之米爾諾纖維中的正合拉格朗日子流形
Exact Lagrangian Submanifolds in Milnor Fiber of Type ADE Singularities
指導教授: 江孟蓉
Chiang, Meng-Jung
學位類別: 碩士
Master
系所名稱: 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年: 2024
畢業學年度: 112
語文別: 英文
論文頁數: 43
中文關鍵詞: 辛上同調拉格朗日子流形米爾諾纖維
外文關鍵詞: Symplectic cohomology, Lagrangian submanifold, Milnor fiber
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  • 本篇論文的目標是找出ADE型奇點之米爾諾纖維中的封閉正合拉格朗日子流形。我們將會回顧Kronheimer的研究結果,證明這些米爾諾纖維可以被描述為ALE空間。此外,我們也會回顧Ritter關於辛上同調的工作,證明唯一的封閉正合拉格朗日子流形是球面。

    The goal of this paper is to identify the closed exact Lagrangian submanifolds in the Milnor fibers of ADE singularities. We will recall Kronheimer's results that describe these Milnor fibers as ALE spaces. Additionally, we will review Ritter's work on symplectic cohomologies, demonstrating that the only closed exact Lagrangian submanifolds are spheres.

    中文摘要 i Abstract ii 誌謝 iii Contents iv 1 Introduction 1 2 Symplectic Geometry 4 2.1 Some definitions and basic results 4 2.2 Symplectic cohomology 8 3 Milnor fibers 16 4 Exact Lagrangians in 4-dimensional Milnor fibers 20 4.1 Deformation of symplectic cohomology 21 4.2 Twisted symplectic cohomology 23 4.3 ALE spaces 26 5 Lagrangian spheres in type Am Milnor fibers 34 References 37

    [1] A. Keating. Lagrangian tori in four-dimensional Milnor fibers. Geometric and Functional Analysis, 25 (6) : 1822-1901, 2015.
    [2] A. Ritter, Deformation of symplectic cohomology and exact Lagrangians in ALE spaces, GAFA vol.20 n. 3 (2010), 779-816.
    [3] A. Ritter, Novikov-symplectic cohomology and exact Lagrangian embeddings, Geometry & Topology 13 (2009), 943-978.
    [4] D. McDuff and D. Salamon. Introduction to symplectic topology, Oxford University Press, 1998.
    [5] G. N. Tjurina, The topological properties of isolated singularities of complex spaces of codimension one, Math. USSR-Izvestija 2 (1968), pp. 557-571.
    [6] J. Milnor. Singular points of complex hypersurfaces, Annal of Mathematics Studies, no. 61, Princeton University Press, 1968.
    [7] M. Gromov. Pseudoholomorphic curves in symplectic manifolds. Invent. Math., 82 (2) : 307-347, 1985.
    [8] P.B. Kronheimer, The construction of ALE spaces as hyperkähler quotients, J. Differential Geom. 29 (1989), no. 3, 685-697.
    [9] P. Seidel. Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), no. 1, 103-149.
    [10] P. Seidel. Lagrangian spheres can be symplectically knotted, J. Differential Geom. 52 (1999), no. 1, 145-171.

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