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arXiv · 1304.0540

An example of circle actions on symplectic Calabi-Yau manifolds with non-empty fixed points

Yunhyung Cho·Min Kyu Kim
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Abstract

Let $(X,\sigma,J)$ be a compact K\"{a}hler Calabi-Yau manifold equipped with a symplectic circle action. By Frankel's theorem \cite{F}, the action on $X$ is non-Hamiltonian and $X$ does not have any fixed point. In this paper, we will show that a symplectic circle action on a compact non-K\"{a}hler symplectic Calabi-Yau manifold may have a fixed point. More precisely, we will show that the symplectic $S^1$-manifold constructed by D. McDuff \cite{McD} has the vanishing first Chern class. This manifold has the Betti numbers $b_1 = 3$, $b_2 = 8$, and $b_3 = 12$. In particular, it does not admit any K\"{a}hler structure.

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Yunhyung Cho, Min Kyu Kim. 2013-04-02. An example of circle actions on symplectic Calabi-Yau manifolds with non-empty fixed points. https://arxiv.org/abs/1304.0540

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