how to Prove that the set of fixed points of a Hamiltonian action of a torus on a symplectic manifold is a symplectic submanifold.

Question

who can explain how to Prove that the set of fixed points of a Hamiltonian action of a torus on a symplectic manifold is a symplectic submanifold?

Answer 1

I am aware of one way to see this, which I saw in McDuff and Salamons book introduction to symplectic geometry.

We can find a compatible almost complex structure which is invariant by the torus action (This basically follows from Gromov's result that a symplectic manifold always has a compatible almost complex structure plus an averaging argument). Note that together with the symplectic structure we also get an invariant Riemannian metric.

Now consider a point $p$ which is fixed by the action, note that $T_{p}M$ inherits the structure of a complex representation of the torus $\mathbb{T}^n$. The points close to $p$ which are fixed by the action are the image under the exponential map at $p$ of the fixed point set of this representation say $V = Fix(\mathbb{T}^n ) \subset T_p M$. Note that $$ V = \cap_{g \in \mathbb{T}^n} Fix(g)$$ where $Fix(g)$ is the fixed point set of the linear transformation associated to $g$.

Now note that since the representation is complex linear, each $Fix(g)$ is a complex subspace (since it is the eigenspace for the eigenvalue 1), and hence $V$ is a complex subspace. This shows that that the fixed point set is locally an almost complex submanifold and hence symplectic.