Effective Hamiltonian Circle Actions with Finite Fixed Points on the Complex Projective Plane

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胡喬晏(Joanne Hu) 查詢紙本館藏

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論文名稱

(Effective Hamiltonian Circle Actions with Finite Fixed Points on the Complex Projective Plane)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ Symplectic geometry
★ Hamiltonian action
★ Effective Hamiltonian action

關鍵字(英)

★ Symplectic geometry
★ Hamiltonian action
★ Effective Hamiltonian action

論文目次

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
2.1 Symplectic vector spaces 3
2.2 Symplectic manifolds 4
2.3 Lie group and group action 5
2.3.1 One-parameter groups of diffeomorphisms 5
2.3.2 Lie groups 6
2.4 Hamiltonian Vector Fields and Hamiltonian actions 9
2.5 Compatible almost complex structures 11
2.5.1 Complex structures on vector spaces 11
2.5.2 Compatible structures 12
2.6 The index of a vector field, Morse index and the Euler number 13
2.6.1 The index of a vector field 13
2.6.2 Morse theory 14
2.6.3 The Euler Number 15
2.7 Gradient flows and stable/unstable submanifolds 15
Chapter 3: Hamiltonian S1-actions on (CP2, ω = ωF S ) 18
3.1 The relation between the S1-action and the moment map h 18
3.2 The moment map and Hamiltonian vector field around a fixed point 18
3.3 The existence of gradient spheres 21
Chapter 4: Linear Hamiltonian S1-actions on CP2 22
4.1 The types of the linear actions 22
4.1.1 Case I: gcd(p, q) = 1 22
4.1.2 Case II: gcd(p, q) = k 6 = 1, say p = ̃pk, q = ̃qk, and gcd( ̃p, ̃q) = 1 23
4.2 The graph of a linear action 23
4.2.1 Case (i) p > q > 0 25
4.2.2 Case (ii) p > 0 > q 25
4.2.3 Case (iii) q > p > 0 26
4.2.4 Case (iv) q > 0 > p 27
4.2.5 Case (v) 0 > p > q 28
4.2.6 Case (vi) 0 > q > p 29
Chapter 5: General Hamiltonian S1 action on CP2 30
5.1 The fixed points of an action under our assumptions 30
5.1.1 The minimum occurs at the point guarantees z1 6 = 0 30
5.1.2 The minimum occurs at the point guarantees z2 6 = 0 31
5.1.3 The minimum occurs at the point guarantees z3 6 = 0 31
5.2 The six cases of such action 32
5.2.1 Case 1: At minimum z3 6 = 0 and at maximum z1 6 = 0 32
5.2.2 Case 2: At minimum z2 6 = 0 and at maximum z1 6 = 0 33
5.2.3 Case 3: At minimum z3 6 = 0 and at maximum z2 6 = 0 34
5.2.4 Case 4: At minimum z1 6 = 0 and at maximum z2 6 = 0 35
5.2.5 Case 5: At minimum z2 6 = 0 and at maximum z3 6 = 0 36
5.2.6 Case 6: At minimum z1 6 = 0 and at maximum z3 6 = 0 37
Conclusion 39
References 41
List of Figures
1 Case (i) p > q > 0 25
2 Case (ii) p > 0 > q 26
3 Case (iii) q > p > 0 27
4 Case (iv) q > 0 > p 28
5 Case (v) 0 > p > q 28
6 Case (vi) 0 > q > p 29
7 Case 1: At minimum z3 6 = 0 and at maximum z1 6 = 0 33
8 Case 2: At minimum z2 6 = 0 and at maximum z1 6 = 0 34
9 Case 3: At minimum z3 6 = 0 and at maximum z2 6 = 0 35
10 Case 4: At minimum z1 6 = 0 and at maximum z2 6 = 0 36
11 Case 5: At minimum z2 6 = 0 and at maximum z3 6 = 0 37
12 Case 6: At minimum z1 6 = 0 and at maximum z3 6 = 0 38
13 The graph of an effective Hamiltonian circle action on the complex projective plane 39
List of Tables
1 Table of linear action graphs 40

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指導教授

姚美琳(Mei-Lin Yau)

審核日期

2023-7-24

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