Let $X$ be a smooth projective variety (say, over a field of characteristic zero).
Let us say that strong Kodaira vanishing holds for $X$ if 
$$
H^q(X,\Omega^p\otimes L)=0
$$
for every $p\geq 0$, $q>0$ and an ample line bundle $L$ on $X$. 


My questions are now these:


1) Does strong Kodaira vanishing hold for $X={\mathbb P}^N$?


2) Does it hold for partial flag varieties of a semi-simple group $G$?


3) What tools are there for proving that strong Kodaira vanishing holds for 
a given variety $X$? 



Strong Kodaira vanishing
It turns out that questions 1 and 2 are completely answered here <http://arxiv.org/abs/alg-geom/9508009> (and some technique for 3 is there as well). In particular, the statement is true
for ${\mathbb P}^N$ but not true for most flag varieties.


