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.



Criteria for acyclicity
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.


