Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the equivariant cohomologies $H_{T}(X;\mathbb{C})$ and $H_{T}(X^T;\mathbb{C})$ using the existing localization theory. Yet, I am finding it difficult to verify some of the hypotheses conventionally imposed in the main theorems (ex. that $H_T(X;\mathbb{C})$ is a free module over $H_{T}(*;\mathbb{C})$) for the examples I am considering. What are some of the most general conclusions that follow in the situation described above, and what sorts of things should I investigate in order to relate the equivariant cohomologies?

Usually one uses a BiałynickiBirula decomposition derived from a general circle in $T$ to find cycles that give a basis of this free module. But this decomposition is less useful when the space is singular. For a basic example, let $X$ be the triangle $\{ [x,y,z] : xyz = 0 \} \subset {\mathbb P}^2$. This space has $H^1$ and $H^1_T$, but its fixed points do not. 


This is a helpful example for you. Consider Torusequivariant cohomology of weighted projective spaces WPS (complex, projective, nonsmooth in general, but with finite singular locus under some easy conditions on the weights). It has been computed by A.Bahri,M.Franz and N.Ray.(Unfortunatly, their computations are toric. I mean they are not in terms of the intrinsic geometry of WPS's (which do not need any fan to exist !). But however their toric calculations allowed them to answer a question on Chern classes (conjectured by AlAmrani) in a particular case.). 

