Cohomology of Structure sheaves

Question

What techniques are out there to calculate the cohomology groups of the structure sheaf $\mathcal{O}_X$ of a smooth quasi-projective variety $X$?

For example can we conclude something from the dimension of the complement $Z = \bar{X} \X$, where $\bar{X}$ is smooth and projective. I know I could use Hodge theory to calculate the cohomology groups $H^i(\bar{X},\mathcal{O}_{\bar X})$.

Is there some way to relate the groups $H^i(\bar{X},\mathcal{O}_{\bar X})$ and $H^i(X,\mathcal{O}_X)$,.

I should mention here that I do know the cohomology groups $H^i(\bar{X},\mathcal{O}_{\bar X})$ and the space $Z$ quite well.

Answer 1

Not knowing anything else about your situation, the approach that comes to mind is to use the long exact sequence: $$\cdots \to H^i_Z(\overline{X},\mathcal O_{\overline{X}}) \to H^i(\overline{X}, \mathcal O_{\overline{X}}) \to H^i(X,\mathcal O_X) \to \cdots $$ (described somewhere in a Hartshorne exercise). This requires knowing the local cohomology $H^i_Z(\overline{X},\mathcal O_{\overline{X}})$, which you can hopefully get a handle on, if you know enough about $Z$ and its embedding in $\overline{X}$.

Answer 2

Your problem basically boils down to computing the groups $H^i_Z(X,\mathcal O)$, the local cohomology with support in $Z$. Once these groups are known the local cohomology exact sequence takes the form

$\cdots \to H^i_Z(\bar{X},\mathcal O) \to H^i(\bar{X},\mathcal O) \to H^i(X, \mathcal O) \to H^{i+1}_Z(X,\mathcal O) \to \cdots$

If you know $\bar{X}$ and $Z$ well there are various methods to compute $H^i_Z(X,\mathcal O)$, e.g., it is by definition the group $\lim_\leftarrow Ext^i(\mathcal{O}_{\bar{X}}/I_Z^n,\mathcal{O}_{\bar{X}})$.