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.



Information from Moment Polytopes
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.


