Delzant-type classification of near-symplectic toric 4-manifolds
Delzant-type classification of near-symplectic toric 4-manifolds
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近辛复曲面 4 流形的 Delzant 型分类
DOI：
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发表时间：
2005
期刊：
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通讯作者：
Samuel Kaufman
中科院分区：
文献类型：
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作者：
Samuel Kaufman
辛环流形的Delzant定理指出$\mathbb{R}^n$中的某些凸多面体与辛环2n$-流形之间存在一一对应关系，通过矩映射的像来实现。本文回顾了该定理的证明及其所依赖的Atiyah-Guillemin-Sternberg的凸性定理。然后，我描述了Honda关于近辛4-流形局部结构的结果，并受Gay-Symington最近工作的启发，我描述了Delzant定理在近辛环面4-流形上的推广。泛化的一个有趣特征是凸性的失效，我将详细讨论。前三章主要是说明性的，在其他地方可以找到重复的材料，熟悉这些材料的人可能会跳过，但为了完整性而包括在内。
Delzant's theorem for symplectic toric manifolds says that there is a one-to-one correspondence between certain convex polytopes in $\mathbb{R}^n$ and symplectic toric $2n$-manifolds, realized by the image of the moment map. I review proofs of this theorem and the convexity theorem of Atiyah-Guillemin-Sternberg on which it relies. Then, I describe Honda's results on the local structure of near-symplectic 4-manifolds, and inspired by recent work of Gay-Symington, I describe a generalization of Delzant's theorem to near-symplectic toric 4-manifolds. One interesting feature of the generalization is the failure of convexity, which I discuss in detail. The first three chapters are primarily expository, duplicate material found elsewhere, and may be skipped by anyone familiar with the material, but are included for completeness.