Equivariant Stratifications of a Variety

Question

Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are smooth locally closed subvarieties and $$\overline{X_{\beta}}\subseteq\bigcup_{\gamma\leq\beta}X_{\gamma}.$$ For fixed $\beta\in S$, we have the equivariant Thom-Gysin sequences $$\ldots\rightarrow H_T^{i-2d(\beta)}(X_{\beta})\rightarrow H_T^i(\bigcup_{\gamma\geq\beta}X_{\gamma})\rightarrow H_T^i(\bigcup_{\gamma>\beta}X_{\gamma})\rightarrow\ldots,$$ where $d(\beta)$ is the complex codimension of $X_{\beta}$ in $X$. The idea is to inductively compute $H_T^*(X)$ from a knowledge of $H_T^*(X_{\beta})$ for each $\beta\in S$. This seems eminently possible if the partial order on $S$ is a total order. In this case, $X=\bigcup_{\gamma\leq\beta}X_{\gamma}$, where $\beta\in S$ is the maximal element.

However, there are some interesting examples in which $S$ is not totally ordered. Consider the nilpotent cone $\mathcal{N}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$. The nilpotent cone has a stratification into the nilpotent $G$-orbits, where $G$ is the simply-connected group with Lie algebra $\mathfrak{g}$.

$\textbf{Question}:$ Are there some general ways which to inductively compute $H_T^*(X)$ from the $H_T^*(X_{\beta})$ assuming only that $S$ is partially ordered? Are there some nice examples of this in the literature?

Answer 1

This will be a partial answer because in the spirit of Christmas I'm not going to go hunt for references. :)

1. It usually makes no difference whether the set of strata is totally or partially ordered. In the partially ordered case, put $X_d = $ (disjoint) union of all $d$-dimensional strata.
2. You are probably better off working with compactly supported equivariant cohomology. In fact I don't think the short exact sequence you wrote down exists in the generality you claim, but with compact support it's fine. If the strata are smooth and $X$ is compact then you get a sequence also for ordinary cohomology using Poincare duality (whence the funny degree shift in your short exact sequence) and $H^\bullet(X)=H^\bullet_c(X)$.
3. There is a spectral sequence associated to the stratification whose first page is given by the various cohomologies $H^\bullet_{c,T}(X_\beta)$ computing $H^\bullet_{c,T}(X)$ (compactly supported equivariant cohomology). I don't have a reference off hand, but I know that the non-equivariant version is given a complete proof in Arapura's paper on the Leray spectral sequence; maybe it can be adapted. Other places I would start looking: the book of Chriss and Ginzburg and Kirwan's thesis.

Answer 2

One possibility would be to follow the treatment of this subject given in Yang-Mills Equations over Riemann Surfaces by Atiyah and Bott. Call a subset $I\subseteq S$ open if whenever $\beta\in I$ and $\gamma\in S$ satisfy $\beta\leq\gamma$, we have $\gamma\in I$. For open $I$, we have the desirable property that $$X_I:=\bigcup_{\beta\in I}X_{\beta}$$ is open in $X$. Also, if $\beta$ is maximal in the complement of $I$, then $I\cup\{\beta\}$ is also open. So, we have the equivariant Thom-Gysin sequence, $$\cdots\rightarrow H_T^{i-2d(\beta)}(X_{\beta})\rightarrow H_T^*(X_{I\cup\{\beta\}})\rightarrow H_T^*(X_I)\rightarrow\cdots.$$ Assume that these sequences split into short-exact sequences.

We begin by selecting a maximal element $\beta_1\in S$. Set $I_1:=\{\beta_1\}$. Take a maximal element $\beta_2\in S\setminus I_1$ and set $I_2:=I_1\cup\{\beta_2\}$. Write the associated Thom-Gysin sequence to conclude that $$H^*_T(X_{I_2})\cong H_T^{*-2d(\beta_2)}(X_{\beta_2})\oplus H_T^*(X_{I_1})$$ as graded vector spaces. Next, choose a maximal element $\beta_3\in S\setminus I_2$ and set $I_3:=I_2\cup\{\beta_3\}$. Form the associated Thom-Gysin sequence to obtain $H_T^*(X_{I_3})$ as a graded vector space. Continue with this inductive procedure to obtain $H_T^*(X)$.