Abstract
Let the circle act in a Hamiltonian fashion on a compact symplectic manifold \((M, \omega )\) of dimension 2n. Then the \(S^1\)-action has at least \(n+1\) fixed points. We study the case when the fixed point set consists of precisely \(n+1\) isolated points. We first show certain equivalence on the first Chern class of M and some particular weight of the \(S^1\)-action at some fixed point. Then we show that the particular weight can completely determine the integral cohomology ring of M, the total Chern class of M, and the sets of weights of the \(S^1\)-action at all the fixed points. We will see that all these data are isomorphic to those of known examples, \({\mathbb{C}\mathbb{P}}^n\), or \(\widetilde{G}_2({\mathbb {R}}^{n+2})\) with \(n\ge 3\) odd, equipped with standard circle actions.
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Acknowledgements
I would like to thank Sue Tolman for some helpful discussion. I thank the referee for a suggestion which helps to improve the exposition of the Introduction. This work is supported by the NSFC grant K110712116.
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Li, H. Hamiltonian circle actions with minimal isolated fixed points. Math. Z. 304, 33 (2023). https://doi.org/10.1007/s00209-023-03288-5
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DOI: https://doi.org/10.1007/s00209-023-03288-5