臺灣博碩士論文加值系統

緊緻辛環面流形是一個緊緻連通的2n維辛流形，並且在這個流形上有一個n維環面群的有效哈密頓作用。Delzant多面體是一個在n維向量空間中的簡單有理光滑多面體。 Delzant證明了緊緻辛環面流形與Delzant多面體之間有一個對應關係，在這篇論文裡,我們依照Lerman及Tolman論文中的方法來證明這個對應，我們並給出一些例子，說明流形上幾何性質與多面體組合性質的關係。

A compact symplectic toric manifold is a 2n-dimensional compact connected sym-
plectic manifold with an effective Hamiltonian action of an n-dimensional torus. A
Delzant polytope is a simple smooth rational polytope in an n-dimensional real space.
Delzant [5] proved that there is a one-to-one correspondence between compact symplec-
tic toric manifolds up to equivariant symplectomorphism and Delzant polytopes up to
translation. In this thesis, we follow the paper of Lerman and Tolman [13] for the proof
of this correspondence. We also give examples relating the geometry of the manifolds
with the combinatorics of the polytopes.

Contents
1 Introduction 1
2 Preliminaries 1
2.1 Proper Maps.. . . . 1
2.2 Lie Group Actions.. . . . . 2
2.2.1 Lie Groups.. . . . . 2
2.2.2 The Exponential Map.. . . 4
2.3 Basic Forms.. . . . . 5
2.4 Sheaves.. . . . . 6
3 Moment Maps 10
3.1 Symplectic Manifolds.. . . . . .10
3.2 Hamiltonian Actions.. . . . . . 12
3.3 Convexity Theorem.. . . . . 15
3.4 Symplectic Reduction... . . . 15
4 Delzant Theorem 19
4.1 Uniqueness.. . . . .20
4.2 Existence.. . . . . .29
5 Cohomology Of Toric Manifolds 32
5.1 Morse Theory.. . . . 32
5.2 Cohomology Ring.. . . . . . 34
References 35

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