TITLE: Is there an analog of Kodaira vanishing for singular varieties
QUESTION [12 upvotes]: I would like to know what kind of analogs of Kodaira vanishing theorem are valid for singular varieties. For example, is the following true: let $X$ be a projective Gorenstein variety and let $\omega_X$ be its canonical bundle. Is it true that $H^i(L\otimes \omega_X)=0$
for $i>0$ for an ample line bundle $L$?

REPLY [14 votes]: Indeed, as JC Ottem points out, Kodaira holds for log canonical (even semi-log canonical singularities).  There's also a way to quickly deduce that Kodaira vanishing holds for Du Bois singularities (either from the Ambro-Fujino machinary or mimicking arguments of Kollar, let me know if you want details, perhaps I should put it on mathoverflow since it's not written down anywhere).
However, I should probably point out that it's totally trivial to see that Kodaira vanishing holds for rational singularities.  Here's the proof:
Let $\pi : Y \to X$ be a resolution.  Note $R \pi_* O_Y \cong O_X$ and so $R \pi O_Y(-\pi^* L) \cong O_X(-L)$ for any line bundle $L$.  Fix $L$ to be ample.  By a spectral sequence/composition of derived functors argument:
$$H^i(X, O_X(-L)) = H^i(Y, O_Y(-\pi^* L)).$$
But $\pi^* L$ is nef and big and the vanishing of the right hand side is just Kawamata-Viehweg vanishing and Serre duality.