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

TL;DR

This paper demonstrates that symplectic circle actions on non-Kähler Calabi-Yau manifolds can have fixed points, contrasting with the non-existence of fixed points in the Kähler case, by analyzing a specific example with known Betti numbers.

Contribution

It provides an explicit example of a non-Kähler symplectic Calabi-Yau manifold with a symplectic circle action that has fixed points, challenging previous assumptions.

Findings

The constructed manifold has vanishing first Chern class.

It has Betti numbers b1=3, b2=8, b3=12.

It does not admit a Kähler structure.

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.