In this talk, we will discuss the following conjecture.

Conjecture. Let $(M, \omega)$ be a compact symplectic manifold and $S^1$ be the circle group which acts on $(M, \omega)$ symplectically. If the fixed point set is non-empty and isolated, then the action is Hamiltonian.

The conjecture above originated from T. Frankel’s work (Ann. Math, 1959). He proved that any holomorphic circle action on a compact Kähler manifold preserving Kähler structure is Hamiltonian if and only if the fixed point is non-empty. A symplectic analogue of Frankel’s theorem was studied by K. Ono (1987) and D. McDuff (1988). In particular, McDuff proved that Frankel’s theorem does not hold in the symplectic category in general. But it remains still open when the fixed points are all isolated. The main aim of this talk is to describe several techniques to approach this conjecture (localization, Duistermaat-Heckman measure,...).

Also, if time permits, we will discuss the question of existence of symplectic circle action with only two fixed points, which is posed by Tolman and Weitsman (2000).

This is joint work with L. Godinho.