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研究生：	胡喬晏 Joanne Hu
論文名稱：	Effective Hamiltonian Circle Actions with Finite Fixed Points on the Complex Projective Plane
指導教授：	姚美琳 Mei-Lin Yau
口試委員:
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系 Department of Mathematics
論文出版年：	2023
畢業學年度：	111
語文別：	英文
論文頁數：	52
中文關鍵詞：	Symplectic geometry 、Hamiltonian action 、Effective Hamiltonian action
外文關鍵詞：	Symplectic geometry, Hamiltonian action, Effective Hamiltonian action
相關次數：	點閱：11 下載：0
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我們根據 Y. Karshon 博士在1998年發表的論文《 Periodic Hamiltonian Flows on Four Dimensional Manifolds 》研究對於在複射影平面上具有有限不動點的一維球作用進行分類。我們的結論是:每個這樣的作用都會辛同構到線性作用。

We classify the effective Hamiltonian $S^1$-actions with finite fixed points on the complex projective plane based on the work of Y. Karshon, ``Periodic Hamiltonian Flows on Four Dimensional Manifolds". Our conclusion is that every such action is symplectomorphic to a standard linear case.

中文摘要 i
Abstract ii
Acknowledgements iii
Contents iv
List of Figures v
List of Tables v
List of Figures v
List of Tables vi
Chapter 1: Introduction 1
Chapter 2: Backgrounds 3
1 Symplectic vector spaces 3
2 Symplectic manifolds 4
3 Lie group and group action 5
3.1 One-parameter groups of diffeomorphisms 5
3.2 Lie groups 6
4 Hamiltonian Vector Fields and Hamiltonian actions 9
5 Compatible almost complex structures 11
5.1 Complex structures on vector spaces 11
5.2 Compatible structures 12
6 The index of a vector field, Morse index and the Euler number 13
6.1 The index of a vector field 13
6.2 Morse theory 14
6.3 The Euler Number 15
7 Gradient flows and stable/unstable submanifolds 15
Chapter 3: Hamiltonian S1-actions on (CP2, ω = ωF S ) 18
1 The relation between the S1-action and the moment map h 18
2 The moment map and Hamiltonian vector field around a fixed point 18
3 The existence of gradient spheres 21
Chapter 4: Linear Hamiltonian S1-actions on CP2 22
1 The types of the linear actions 22
1.1 Case I: gcd(p, q) = 1 22
1.2 Case II: gcd(p, q) = k 6 = 1, say p =  ̃pk, q =  ̃qk, and gcd( ̃p,  ̃q) = 1 23
2 The graph of a linear action 23
2.1 Case (i) p &gt; q &gt; 0 25
2.2 Case (ii) p &gt; 0 &gt; q 25
2.3 Case (iii) q &gt; p &gt; 0 26
2.4 Case (iv) q &gt; 0 &gt; p 27
2.5 Case (v) 0 &gt; p &gt; q 28
2.6 Case (vi) 0 &gt; q &gt; p 29
Chapter 5: General Hamiltonian S1 action on CP2 30
1 The fixed points of an action under our assumptions 30
1.1 The minimum occurs at the point guarantees z1 6 = 0  30
1.2 The minimum occurs at the point guarantees z2 6 = 0  31
1.3 The minimum occurs at the point guarantees z3 6 = 0  31
2 The six cases of such action 32
2.1 Case 1: At minimum z3 6 = 0 and at maximum z1 6 = 0 32
2.2 Case 2: At minimum z2 6 = 0 and at maximum z1 6 = 0 33
2.3 Case 3: At minimum z3 6 = 0 and at maximum z2 6 = 0 34
2.4 Case 4: At minimum z1 6 = 0 and at maximum z2 6 = 0 35
2.5 Case 5: At minimum z2 6 = 0 and at maximum z3 6 = 0 36
2.6 Case 6: At minimum z1 6 = 0 and at maximum z3 6 = 0 37
Conclusion 39
References 41
List of Figures
Case (i) p &gt; q &gt; 0 25
Case (ii) p &gt; 0 &gt; q 26
Case (iii) q &gt; p &gt; 0 27
Case (iv) q &gt; 0 &gt; p 28
Case (v) 0 &gt; p &gt; q 28
Case (vi) 0 &gt; q &gt; p 29
Case 1: At minimum z3 6 = 0 and at maximum z1 6 = 0 33
Case 2: At minimum z2 6 = 0 and at maximum z1 6 = 0 34
Case 3: At minimum z3 6 = 0 and at maximum z2 6 = 0 35
Case 4: At minimum z1 6 = 0 and at maximum z2 6 = 0 36
Case 5: At minimum z2 6 = 0 and at maximum z3 6 = 0 37
Case 6: At minimum z1 6 = 0 and at maximum z3 6 = 0 38
The graph of an effective Hamiltonian circle action on the complex projective plane 39
List of Tables
Table of linear action graphs 40
                                

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