Smooth versus symplectic circle actions

TL;DR

This paper constructs a 6-manifold with a smooth circle action and a symplectic form, demonstrating that equivalent symplectic forms may not admit symplectic circle actions, highlighting differences between smooth and symplectic symmetries.

Contribution

It provides a counterexample showing that symplectic forms equivalent to a given form can lack symplectic circle actions, revealing subtle distinctions in symplectic geometry.

Findings

Existence of a 6-manifold with a smooth circle action and a symplectic form

Existence of an equivalent symplectic form without a symplectic circle action

Illustration of differences between smooth and symplectic symmetries

Abstract

We construct a 6-manifold M which admits a smooth circle action and a symplectic form w such that if w' is another symplectic form on M equivalent to w, then (M,w') does not admit a symplectic circle action.