A theorem by Kollár asserts that if $X$ and $Y$ are projective varieties with $X$ smooth, and $f : X \to Y$ is a surjective map, then the higher direct images $R^if\_\*\omega\_X$ vanish for $i$ greater than the generic fiber dimension. I'd like to know if one can weaken the assumption that $X$ is smooth, for example, to $X$ having quotient singularities, or something even weaker (whereby the sheaf $\omega\_X$ represents an appropriate dualizing sheaf in such a case). 


Kollár's original proof uses the fact that $X$ is smooth to prove that the higher direct images $R^if\_\*\omega\_X$ are torsion-free which I suspect may fail for $X$ having singularities. 



Extension of Kollár's vanishing theorem to singular varieties?
If $X$ has rational singularities, then this vanishing (and the torsion-freeness as well) follows almost trivially. Let $g:Z\to X$ be a resolution of singularities. Then 
$Rg\_\*\omega\_Z\simeq \omega\_X$ and hence $R^if\_\*\omega\_X\simeq R^i(f\circ g)\_\*\omega\_Z=0$ for $i>\dim X-\dim Y$ by Kollár's theorem (I suppose that's what you meant, not what you wrote, right?! ;-) 


Quotient singularities are rational, so this includes for what you are hoping. Also, since rational singularities are Cohen-Macaulay there is indeed a dualizing sheaf and it plays nice.


p.s.: If I remember correctly, Kollár also mentions this in his paper(s).


