Real Lagrangians in symplectic toric manifolds
- 주제(키워드) real Lagrangian , antisymplectic involution , toric symplectic manifold , Delzant construction , Gromov foliation
- 주제(DDC) 510
- 발행기관 아주대학교
- 지도교수 최수영
- 발행년도 2021
- 학위수여년월 2021. 8
- 학위명 박사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/ajou/000000031222
- 본문언어 영어
- 저작권 아주대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
We study two problems of real aspects in symplectic topology. The first problem concerns the topology of real Lagrangian submanifolds in a toric symplectic manifold. Real Lagrangians we consider come from involutive symmetries on the moment polytope of a toric symplectic manifold. We establish a real analogue of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data of the polytope. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces. In the second problem, we deal with a real analogue of the symplectic mapping class group of a monotone $Q:=S^2\times S^2$, the set $\pi_{0}\mathcal{I}(Q,\ow ,S^2)$ of the isotopy classes of antisymplectic involutions of $Q$ having a Lagrangian sphere as the fixed point set. It is shown that $\pi_{0}\mathcal{I}(Q,\ow ,S^2)$ has a single element. This follows from a stronger result, namely that any two anti-symplectic involutions in that space are Hamiltonian isotopic.
more목차
chapter 1. Introduction 1
section 1.1 On the topology of real Lagrangians 2
section 1.2 On the space of antisymplectic involutions 6
chapter 2. Preliminaries 9
section 2.1 Symplectic manifold ans its Lagrangian submanifolds 9
section 2.2 Hamiltonian vector fields and Hamiltonian torus actions 13
section 2.3 Properties of Hamiltonian T-space 16
section 2.4 Morphisms compatible with a Hamiltonian torus action 21
section 2.5 Symplectic toric maniflds 24
chapter 3. Delzant construction 27
section 3.1 Proof of Delzant theorem 28
chapter 4. Lifted antisymplectic involutions from moment polytopes 35
section 4.1 Symmetries of polygons 35
section 4.2 Lifting of symmetries 36
chapter 5. Real Delzant construction 41
section 5.1 Real Delzant construction 41
chapter 6. Applications of real Delzant construction 51
section 6.1 Convexity and tightness 51
section 6.2 Real Lagrangians in toric symplectic del Pezzo surfaces 55
chapter 7. Antisymplectic involutions of $S^2\times S^2$ 69
section 7.1 J-holomorphic curves and Moduli spaces 72
section 7.2 Gromov's foliations on $S^2\timesS^2$ 76
section 7.3 Real analogue of the work of Hind 78
section 7.4 Diffeomorphism of Q induced by transversal foliations 81
section 7.5 Equivariant Moser trick 83
