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


$C_w = \{x\in X | \lim_{t\to 0} t\cdot x = w \}$ 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.

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I can't seem to get curly braces to show up in my math markup (the sets describing $C_w$ and $C^w$. If someone more knowledgeable could explain how to fix this I would be extremely grateful! – Reladenine Vakalwe Nov 6 '12 at 20:56
You need to enclose the math in backticks (`) sometimes to get these things to show up. – Dan Petersen Nov 6 '12 at 21:00
@RV: It would probably be helpful to point to one particular paper by B-B which is most relevant to your question, since he wrote a lot of papers involving torus actions on varieties. Do you have his Annals paper especially in mind? – Jim Humphreys Nov 6 '12 at 22:34
They are indeed smooth on the flag variety -- this is essentially Kleiman transversality. Also, I think assuming that the two are transverse would imply your stratification assumption. Unfortunately this is quite rare. – Allen Knutson Nov 6 '12 at 22:58
@Dan and Jim: Thank you for the markup assistance! – Reladenine Vakalwe Nov 7 '12 at 0:20

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