Circle actions on symplectic four-manifolds
Circle actions on symplectic four-manifolds
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DOI：
10.4310/cag.2019.v27.n2.a6
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发表时间：
2015-07
影响因子：
0.7
通讯作者：
T. Holm;Liat Kessler
中科院分区：
文献类型：
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作者：
T. Holm;Liat Kessler
我们完成了辛四维流形上的Hamilton环面和圆作用的分类。继Delzant和Karshon的工作之后，Karshon，Kessler和Pinsonnault刻画了任意固定单连通辛四维流形上的Hamilton圈和2-torus作用。剩下的工作是研究正亏格的黎曼曲面上S^2-丛爆破的哈密顿作用。它们不允许2-torus作用。本文刻画了它们的Hamilton圈作用。然后，我们得出这些行动的存在和计数的组合结果。作为一个副产品，我们提供了一个算法，确定g-约化形式的爆破形式。我们的工作是“软”等变和组合技术的结合，使用的动量映射和相关数据，与“硬”全纯技术，包括Gromov-Witten不变量。
We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of these actions. As a by-product, we provide an algorithm that determines the g-reduced form of a blowup form. Our work is a combination of "soft" equivariant and combinatorial techniques, using the momentum map and related data, with "hard" holomorphic techniques, including Gromov-Witten invariants.