Information from Moment Polytopes

Question

Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^*$. If $\dim(T)=\frac{1}{2}\dim(X)$, then $X$ is classified up to $T$-equivariant symplectomorphism by its moment polytope $\mu(X)$. If we relax this dimension condition, then what information about $X$ might we hope to obtain from an explicit description of its moment polytope? For instance, is the $T$-equivariant cohomology of $X$ nicely related to $\mu(X)$? My question is somewhat vague, so please feel free to answer in any fashion you deem to be acceptable.

Answer 1

Here's two natural things to ask about any compact group action on a compact manifold: (1) what are the (finitely many) conjugacy classes of stabilizer groups? Assume our group is a torus, so we can omit "conjugacy classes of". (2) For each such stabilizer, what are the (finitely many) components of its fixed point set?

So, we have a poset of such submanifolds, and a torus subgroup for each. Now comes the additional question in the Hamiltonian situation: (3) what's the moment polytope for each of these submanifolds?

In actual examples, (1-3) are easy to figure out, and it's silly to answer (3) only for the whole manifold (or one might say, only for the fixed points).

You ask if $H^*_T(X;\mathbb Q)$ can be computed from this sort of data alone. In the case that the minimal strata in (2) are all $S^2$s, $X$ is called a GKM space (for Goresky-Kottwitz-MacPherson), and indeed it can; it's easy to find references with that keyword.

Answer 2

The difference $\frac12\dim(X)-\dim(T)$ is known as the complexity of the $T$-space (assumed effective), so that's the keyword you want to use. Such results as I've heard of are mainly for complexity one, by Yael Karshon and Sue Tolman:

• Centered complexity one Hamiltonian torus actions (2001);

• Complete invariants for Hamiltonian torus actions with two dimensional quotients (2003);

• Classification of Hamiltonian torus actions with two dimensional quotients (2011).

These papers may perhaps not quite have the focus you're asking for: rather than deducing information from the moment polytope alone, they are about enhancing it with extra data (the Duistermaat-Heckman measure, a genus and a "painting") so that the resulting invariant completely determines the $T$-space.

Still, they should provide you with plenty of examples showing what information one cannot hope to obtain from the moment polytope alone.

The third one also quotes some results on $S^1$-spaces of dimension 6, i.e. complexity 2.