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 $X$?
Perhaps I might add that the "strong Kodaira vanishing" holds more generally for smooth projective toric varieties in any characteristic. This goes back to Danilov. This includes your case 1 of course. I can't remember how he did this, but an argument observed by number of people (Fujino, myself,...) is to use a what I might call a mock Frobenius splitting argument. The idea is to exploit the map $\phi$ given by multiplication by $r$ on the fan. For projective space, this is just $[x_0,\ldots, x_N]\mapsto [x_0^r,\ldots, x_N^r]$. If $r>1$ is prime to the characteristic, then $\phi^*$ can be shown to give an injection $$H^q(X,\Omega_X^p\otimes L)\hookrightarrow H^q(X,\Omega_X^p\otimes L^r)$$ So choosing $r\gg 0$, we get the desired result by Serre 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.
Re question 1: yes, see Deligne, Cohomologie des intersections compl`etes, SGA 7 II, th\'eor`eme 1.1. The proof is by "force brutale" as Deligne himself puts it, so I'm not sure this generalizes.
