Let $X$ be a smooth complex projective variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite $T$-invariant stratification of $X$ into smooth locally closed subvarieties, so that $$\overline{X_{\beta}}\subseteq\bigcup_{\gamma\leq\beta}X_{\gamma}$$ for all $\beta\in B$. Assume that for each $\beta\in B$, the $T$-equivariant Euler class of the normal bundle of $X_{\beta}$ in $X$ is not a zero-divisor in $H_T^*(X_{\beta})$. The relevant Thom-Gysin sequences then split into the short-exact sequences $$0\rightarrow H_T^{i-2d(\beta)}(X_{\beta})\rightarrow H_T^i(\bigcup_{\gamma\geq\beta}X_{\gamma})\rightarrow H_T^i(\bigcup_{\gamma>\beta}X_{\gamma})\rightarrow 0,$$ where $d(\beta)$ is the complex codimension of $X_{\beta}$ in $X$. By induction, we have the identity $$P_T(X)=\sum_{\beta\in B}t^{2d(\beta)}P_T(X_{\beta})$$ of equivariant Poincare series. This was the subject of an earlier post: Equivariant Stratifications of a Variety.
So, one can obtain the graded vector space structure of $H_T^*(X)$ from a knowledge of the $H_T^*(X_{\beta})$ as graded vector spaces. Yet, this approach does not seem particularly conducive to "computing" the product structure of $H_T^*(X)$. Certainly, if one has a basis of each $H_T^i(X_{\beta})$, then one has generators of $H_T^*(X)$. Still, it is unclear to me how one might find relations.
I would appreciate a reference outlining the computation of a product structure in the context above. Also, I would appreciate any and all pieces of advice.
