Given a smooth quasiprojective variety $X$ with a coherent sheaf $E$, if $X$ is not projective, then the sheaf cohomology of $E$ may not be finite-dimensional. However, if we also have the action of a torus $T$ on $X$ so that $E$ is equivariant, then the cohomology groups are naturally representations of $T$, and it could be the case that the weight spaces are finite-dimensional. Under what hypotheses will this be true? For instance, if $X$ has proper fixed locus, is that enough?
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1$\begingroup$ Your hypothesis is not enough. For instance, start with a coherent sheaf $F$ on a quasi-projective scheme $Y$ such that $H^i(Y,F)$ is not finite-dimensional. Now take $X$ to be $Y\times \mathbb{G}_m$ with the torus $\mathbb{G}_m$ acting via the natural action on the second factor. There is a $\mathbb{G}_m$-invariant projection $p:Y\times \mathbb{G}_m\to Y$, and the pullback $E=p^*F$ is a coherent sheaf on $Y\times \mathbb{G}_m$ together with a $\mathbb{G}_m$-linearization. Using the Leray spectral sequence for $p$ (or other methods), the weight spaces for $H^i(X,E)$ are $H^i(Y,F)$. $\endgroup$– Jason StarrCommented Apr 11, 2012 at 12:29
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$\begingroup$ I think if you add the hypothesis that there is some one-parameter subgroup so that every point of $X$ has a well-defined limit in $X^T$, then the statement should be true. $\endgroup$– anonCommented Apr 11, 2012 at 23:00
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$\begingroup$ @anon: I still think this is not enough. Let $\mathbb{G}_m$ act on $\mathbb{P}^2$ by $t\cdot [X_0,X_1,X_2] = [X_0, tX_1, t^2X_2]$. The fixed points are $p_0 = [1,0,0]$, $p_1=[0,1,0]$ and $p_2=[0,0,1]$. If you look at the Bialynicki-Birula stratification, then the open stratum limits to $p_0$, a $1$-dimensional stratum limits to $p_1$, and only $p_2$ limits to $p_2$. So define $X$ to be $\mathbb{P}^2\setminus \{p_2\}$. This scheme satisfies your hypotheses. Yet consider the first cohomology of the structure sheaf. $\endgroup$– Jason StarrCommented Apr 14, 2012 at 16:26
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