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

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Let (X,σ,J) be a compact Kähler Calabi-Yau manifold equipped with a symplectic circle action. By Frankel's theorem 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-Kähler symplectic Calabi-Yau manifold may have a fixed point. More precisely, we will show that the symplectic S¹-manifold constructed by D. McDuff McD has the vanishing first Chern class. This manifold has the Betti numbers b₁ = 3, b₂ = 8, and b₃ = 12. In particular, it does not admit any Kähler structure.