TITLE: Examples of manifolds with effective circle actions?
QUESTION [6 upvotes]: I would like to know examples of smooth compact connected manifolds, on which there exists an effective smooth circle action preserving a positive smooth volume, besides the simple example: $[0,1]^d \times \mathbb{S}^1$. Is there a classification of these manifolds?

REPLY [2 votes]: This is more of a comment than an answer.  Any simplectic manifold with an effective circle action preserving the form is an example.  So  any smooth projective toric variety is an example of a manifold with an effective circle action preserving a volume form.
Edit: I forgot to say what the volume is: it's the top power of the symplectic form.