Circle actions on four dimensional almost complex manifolds with discrete fixed point sets
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866910443016028160 |
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| author | Jang, Donghoon |
| author_facet | Jang, Donghoon |
| contents | We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we provide a necessary and sufficient condition for a pair of integers to arise as the Chern numbers of such an action, answering negatively a question by Sabatini whether $c_1^2[M] \leq 3 c_2[M]$ holds for any such manifold $M$. We achieve this by demonstrating that pairs of integers that arise as weights of a circle action, also arise as weights of a restriction of a $\mathbb{T}^2$-action. Furthermore, we discuss applications to circle actions on complex/symplectic 4-manifolds and semi-free circle actions with discrete fixed point sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_04128 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Circle actions on four dimensional almost complex manifolds with discrete fixed point sets Jang, Donghoon Differential Geometry We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we provide a necessary and sufficient condition for a pair of integers to arise as the Chern numbers of such an action, answering negatively a question by Sabatini whether $c_1^2[M] \leq 3 c_2[M]$ holds for any such manifold $M$. We achieve this by demonstrating that pairs of integers that arise as weights of a circle action, also arise as weights of a restriction of a $\mathbb{T}^2$-action. Furthermore, we discuss applications to circle actions on complex/symplectic 4-manifolds and semi-free circle actions with discrete fixed point sets. |
| title | Circle actions on four dimensional almost complex manifolds with discrete fixed point sets |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1912.04128 |