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Let $X$ be an algebraic variety with a $\mathbb C^*$ action such that the fixpoints set is finite. By theorem 4.3 in the paper of Bialynicki-Birula "Some theorems on actions of algebraic groups", there exist the ($+$) and the ($-$)-decomposition if $X$ is complete.

Is there such a decomposition in the case the variety is not complete? Or more precisely, having an algebraic variety $X$ (not necessarily complete) with a $\mathbb C^*$ action such that the fixpoints set is finite, is $X$ known to have a Bialynicki-Birula cell decomposition as well? If not, is there any other cell decomposition that is known for this case? Any suggestions for references? Thanks.

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You should try looking at some examples. –  Angelo Apr 5 '12 at 16:55
    
To amplify Angelo's comment, can you first point to some interesting action on (say) an affine variety for which the fixed point set is finite? In other words, what specific problem motivates the question here? –  Jim Humphreys Apr 5 '12 at 17:54

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One useful case that has arisen in my own work is when $lim_{t\to 0} t\cdot x$ exists for all $x\in X$. This sort of action is "circle compact" and then the BB decomposition goes through. I think the terminology is due to Tamas Hausel -- in the case where $X$ is smooth and quasi-projective, this condition is the same as saying that the moment map for the $S^1$ action is proper and bounded below.

(Proper isn't enough, as the example of $S^1$ acting on {$(x,y) : xy=1$} shows.)

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This happens a bunch. One sufficient condition is that the induced action on $X_{aff}$ (the $Spec$ of the ring of global functions on $X$) makes $X_{aff}$ into a cone, with one ${\mathbb C}^\times$-fixed point that everybody falls into, plus the assumption that $X_{aff} \to X$ is proper. For example, let $X_{aff}$ be the Chow variety of $n$ points in the plane, let ${\mathbb C}^\times$ act on the plane with two weights $a,b>0$ (say much larger than $n$ and coprime), and $X$ itself be the Hilbert scheme of $n$ points in the plane. –  Allen Knutson Apr 6 '12 at 4:30

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