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

Abstract

Let $(X,\sigma ,J)$ be a compact Kähler 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-Kähler 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 Kähler structure.

15 pages