Strong Kodaira vanishing

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 $X$?

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For flag varieties you can try to apply Borel--Bott--Weil. –  Sasha Feb 16 '12 at 16:24
Sure, but I was not able to say that carefully for all $L$ (only for sufficiently positive ones) –  Alexander Braverman Feb 16 '12 at 17:13

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.

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Thank you. There is actually a different argument in arxiv.org/abs/alg-geom/9508009 (using the lift of Frobenius mod $p^2$). –  Alexander Braverman Feb 16 '12 at 19:37

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.

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For flag varieties one has a much weaker "diagonal" vanishing of these cohomology groups, cf section 5.2 in Brion & Kumar's Frobenius splitting book (in particular Theorem 5.2.9). One interesting thing is that this diagonal vanishing actually holds for some non-ample line bundles $L$ as well as for all ample ones. –  Chuck Hague Feb 17 '12 at 18:38

Re question 1: yes, see Deligne, Cohomologie des intersections completes, SGA 7 II, th\'eoreme 1.1. The proof is by "force brutale" as Deligne himself puts it, so I'm not sure this generalizes.

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Thank you. I wonder whether Frobenius splitting can be helpful here - it does prove everything for $q=0$ (and $q=\dim X$ is the usual Kodaira vanishing). –  Alexander Braverman Feb 16 '12 at 17:14