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? 



Equivariant Cohomology of a Complex Projective Variety
Usually one uses a Białynicki-Birula‎ 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.


