Vanishing theorems for pluri-canonical bundle I would like to know if  Grauer-Riemenschneider vanishing theorem  is still true in the setting of pluri-canonical bundle, i.e. the power of canonical bundle.   
Let me recall
Grauer-Riemenschneider vanishing theroem:  Let $\pi: X\to Y$ be a surjective projective morphism of two varieties with $X$ smooth. Let $K_X$ be the canonical line bundle. 
 Then 
$$R^i\pi_*(K_X)= 0, \qquad  \text{ if }  i> \dim Y-\dim X.$$
In the book by Lazarfeld's book,  positivity in algebraic geometry, this theorem follow from 
Kawamata-Viehweg's vanishing theorem:
  Let $X$ be a projective smooth variety. Then for any nef and big line bundle $L$, 
$$H^i(X,K_X\otimes L)=0, \qquad, \text{ for any } i>0 .$$
My question is: if  we replace $K_X$ by $K_X^{\otimes n}$, $n>0$  is Grauer-Riemenschneider vanishing theorem still holds?
As pointed out by Jason Starr that  Kawamata-Viehweg would fail for pluri-canonical line bundle.  Still it is quite possible to some generalization,  for example take $L^{\otimes r_n}$, where $r_n$ depends on $n$.
 A: First of all the theorem you are citing is not Grauert-Riemenschneider vanishing, but Kollár's vanishing and I doubt that you can prove it as a consequence of Kawamata-Viehweg vanishing. GR vanishing is for the case $\dim Y=\dim X$.
Second, of course, you can do what you are asking  with an $r_n$: Let $r_n$ be such that $\omega_X^{n-1}\otimes L^{r_n}$ is ample and apply the previous version.
As Jason explained, you will always run into trouble with Fano varieties.
Here is the issue: cohomology of the canonical bundle is very different from that of pluricanonical bundles. Some aspects of this is explained in the introduction of Kollár's paper on subadditivity of Kodaira dimension. One simple difference is that the canonical bundle appears in the Hodge decomposition while the pluricanonical ones don't. Even if this does not seem a big difference, it is. 
If you want vanishing of an adjoint bundle, i.e., something like $\omega_X\otimes L$, then you generally need some positivity of $L$. Just think of the dual version of KV vanishing and write $L^{-1}$ as an adjoint bundle. So to have vanishing for pluricanonical bundles you need some positivity of them. 
