Thom-Gysin Sequences and Stratifications

Question

Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ into orbits is sometimes called a stratification of $X$. However, I have encountered many different definitions of "stratification" in the literature. I am looking for a type of stratification sufficient to warrant using Thom-Gysin sequences to inductively compute cohomology (ex. as was done by Kirwan in her thesis). I suspect my situation is sufficiently regular, but I would definitely benefit from references and/or advice.

Answer 1

I haven't read Kirwan's thesis but I think this is what you are after. If $U \subset V$ is an open subvariety (no conditions on the variety $V$) then there is a long exact sequence (Thom-Gysin) $$ H^\bullet_c(U) \to H^\bullet_c(V) \to H^\bullet_c(V \setminus U) \to H^{\bullet+1}_c(U). $$

Let's now declare a stratification of $X$ to be a collection of disjoint locally closed subvarieties $X_\alpha$ ("strata") whose union is $X$, and such that the closure of a stratum is a union of strata. Then for every such stratification the above LES can in principle be used to inductively study the compactly supported cohomology of $X$, one stratum at a time: start with the smallest strata and in each step let $V \setminus U$ be what you have already computed and $V$ the union with one more stratum.

If the stratification is actually a filtration by closed subvarieties, $$ X = T_d \supset T_{d-1} \supset \cdots \supset T_0,$$ then there is a spectral sequence $$ E_1^{pq} = H^{p+q}_c(T_p\setminus T_{p-1})\implies H^{p+q}_c(X)$$ which encodes this procedure.

So one needs almost no conditions at all on the stratification. A slogan is that compactly supported cohomology always behaves better w.r.t. stratified spaces.

But probably you are interested in ordinary cohomology; one should add hypotheses relating compactly supported and usual cohomology. For instance if $X$ is compact then $H^\bullet_c(X) = H^\bullet(X)$, same for all closures of strata, and if all strata are smooth varieties then their cohomology with compact support is determined from the usual cohomology by Poincare duality. This covers many cases of interest.