Let $X$ be a projective variety endowed with an algebraic $\mathbb{C}^*$-action. Assume the fixed point set $W$ is discrete. Then we have two cell decompositions of $X$:

$X = \bigsqcup_{w\in W} C_w$ and $X = \bigsqcup_{w\in W} C^w$,

where

$C_w = \{x\in X | \lim_{t\to 0} t\cdot x = w \}$ and $C^w=\{x\in X | \lim_{t\to \infty} t\cdot x = w\}$,

the so-called plus and minus Bialynicki-Birula decomposition. Assume that both of these decompositions are in fact stratifications (although this latter bit may be irrelevant to the question).

My (slightly vague) question: Are there any general results about the structure of the intersections $C_w \cap C^v$? When non-empty are they smooth in general? My favorite example of the flag variety, these are smooth (I think). It would make me extremely happy if someone could point me to towards a general description of the cohomology groups of these intersections?

Note: In the special situation of flag varieties I am familiar with Deodhar's and Curtis' results describing these intersections. But, unless I am missing something, their results don't help in describing the cohomology groups.

Annalspaper especially in mind? – Jim Humphreys Nov 6 '12 at 22:34