# Points with finite stabilizer in Hamiltonian torus actions

Atiyah-Guillemin-Sternberg theorem asserts that the image of the moment map $\mu$ for a Hamiltonian $(S^1)^m$-action on a smooth compact symplectic manifold $(M^{2n},\omega)$ is a convex polytope of $\mathbb{R}^m$.

Smooth compact toric manifolds are special examples of this ($m=n$) where the moment polytope $\Delta=image(\mu)$ determines the geometry of $M$.
In this case, there is no point with finite stabilizer.

Are there non-toric examples with finite stabilizers, if we assume that the action of $(S^1)^m$ is effective?

• I think Spherical varieties
– user21574
Feb 15, 2016 at 18:52

If I understand the question correctly, what you are after is an effective Hamiltonian action of $\mathbb{T}^m$ on a (closed) symplectic manifold $(M,\omega)$ such that there exists a point $p \in M$ with finite stabilizer. If this is the case, there are plenty of examples. Consider a closed symplectic toric manifold $\mu: (M^4,\omega) \to \mathfrak{t}^*$ with Delzant (= moment) polygon $\Delta = \mu(M)$ with the property that the primitive integral tangent vector to the edge $e$ of $\Delta$ is of the form $(k,a) \in \mathbb{Z}^2$ for $|k| > 1$. Writing $\mathbb{T}^2 = S^1 \times S^1$, consider the restriction of this action to the first $S^1$, thus obtaining what is known as a Hamiltonian $S^1$-space $S^1 \curvearrowright (M^4,\omega)$ (these have been classified by Karshon in this paper). The moment map of this action is the composition $\mathrm{pr}_1 \circ \mu : M \to \mathrm{Lie}(S^1)^* \cong \mathbb{R}$, where $\mathrm{pr}_1 : \mathrm{t}^* \to \mathrm{Lie}(S^1)^*$ is the projection induced by the inclusion $S^1 \hookrightarrow \mathbb{T}^2$. Any point $p \in \mu^{-1}(\mathring{e})$ has finite stabilizer with respect to this $S^1$-action; in fact, the stabilizer is precisely the cyclic group of order $k$!
There are examples of Hamiltonian $S^1$-spaces which do not arise as above (i.e. they cannot be extended to a toric action) which also have finite stabilizers. If you take a look at Karshon's classification, these are precisely the points lying on what she calls $\mathbb{Z}_k$-spheres (except for the "north" and "south" poles of the spheres which are fixed points).
In all of the above examples, the set of points with finite stabilizer is "small", i.e. it has empty interior. This is always the case for effective Hamiltonian actions by a torus $\mathbb{T}^m$ on a connected symplectic manifold $(M,\omega)$. The fact that $\mathbb{T}^m$ is abelian is crucial in the above statement. If you are interested in the case of this set of points with finite stabiliser being "large", say being open and dense, then a good place to look at is multiplicity-free Hamiltonian spaces, which are the non-abelian analogue of closed symplectic toric manifolds. A good reference is this paper by Knop.
• Sounds reasonable, the local example inspired from your solution is the $S^1$-action on $\mathbb{C}^2$, $e^{i\theta}(x,y)=(e^{im\theta}x, e^{in\theta} y)$, where $(m,n)=1$. It is free away intersections with $x=0$ and $y=0$. Feb 16, 2016 at 0:31