For a projective variety $X$, Serre's vanishing theorem says that $H^i(X, \mathcal{F}(n))=0$ for any coherent sheave, $i>1$ and sufficiently large $n$. I am wondering, is there a similar type of vanishing theorem on quasi-projective varieties, namely, let $Y$ be a quasi-projective variety, what can we say about the vanishing of $H^i(Y, \mathcal{F}(n))$ under the same setting as projective case.
Or is there a similar type of theorem for local cohomology, say, when is $H^i_{pt}(X, \mathcal{F}(n))$ vanishing?
Best Answer
Even for quasi-affine variety you don't have a vanishing theorem except Grothendieck's vanishing theorem on a noetherian topological space of finite dimension.
Consider, for example, affine plane without point $\mathbf{A}^2\backslash 0.$ Then the structure sheaf is ample but the first cohomology $H^1(\mathbf{A}^2\backslash 0, \mathcal{O})$ is not trivial and even it is infinite dimensional.