Non-Hamiltonian actions with fewer isolated fixed points

TL;DR

This paper constructs non-Hamiltonian symplectic circle actions on 6-dimensional manifolds with fewer fixed points than previously known, specifically achieving exactly 2k fixed points for any k ≥ 5.

Contribution

It improves existing examples by reducing the fixed points count in non-Hamiltonian symplectic actions on six-dimensional manifolds.

Findings

Constructed non-Hamiltonian symplectic circle actions with 2k fixed points for all k ≥ 5.

Reduced the number of fixed points from 32 to 2k in these examples.

Demonstrated the existence of such actions for an infinite family of fixed point counts.

Abstract

In an earlier paper, the second author resolved a question of McDuff by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points. In this paper, we improve on this example by reducing the number of fixed points. More concretely, we construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly $2k$ fixed points for any $k \geq 5$.