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Let $X$ be a smooth projective variety. Let $E$ be a line bundle (or, more generally, a vetor bundle) on $X$. Are there any nice criteria for acyclicity of $E$ (that is, for the property $H^i(X,E)=0$ for all $i>0$)? Here $E$ is considered as a coherent sheaf.

In my case the structure sheaf $O_X$ is acyclic and $E$ is ample.

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Search for the topic vanishing theorems. There are many such criteria, and perhaps this question should be community wiki?

Anyways, I'll highlight the one of the most common situations, for adjoint line bundles. Set $\omega_X = \Omega_X^{\dim X}$. Suppose that:

$E = \omega_X \otimes \text{ample}$

In this case you get the vanishing you want, which is called the Kodaira vanishing theorem.

For example, if $X$ is Calabi-Yau or Fano, then $\omega_X$ is trivial or anti-ample respectively (by definition). In either case, for any ample $E$, the tensor product $E \otimes \omega_X^{-1}$ is also ample and so $E$ is of the form above, and so the vanishing you want holds.

Of course, many common classes of varieties are these types (there are varieties called log-Calabi-Yau and log-Fano which are also close enough). For example, toric varieties need not be Fano, but they are always ``close enough'' to give you the vanishing, they are log Fano.

If $X$ is not Calabi-Yau or Fano, then this sort of argument will fail, and you don't always get vanishing for arbitrary ample $E$. For example, if $X$ is a curve of genus $\geq 2$, then if you set $E = \omega_X$, then $E$ is ample. However, $H^1(X, E) = H^1(X, \omega_X) = H^0(X, O_X)$ which has dimension 1.

For vector bundles, again there are a lot of options when $E$ is various twists of $\Omega_X^j$. Look up Kodaira-Nakano-Akizuki vanishing.

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Thank you very much. This was very helpful. I'll make it community wiki if there will be more answers. –  Rami Feb 28 '12 at 21:21
    
The counter example that you gave seems not to satisfy $H^i(X,O_X)=0$ for $i>0$. Do you have an example that satisfy this and with very ample $E$? If yes it will also give a counter example for my question: mathoverflow.net/questions/90638/… Thank you again, Rami –  Rami Mar 10 '12 at 4:40

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