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 $L$?
No. The following counterexample is due to Sommese:
Let $Y$ be the projective bundle $\pi:\mathbb{P}(O\oplus O(1)^{\oplus 3})\to \mathbb{P}^1$. Let $M$ be the tautological bundle on $Y$ and take a general member $X\inM\otimes \pi^*O(1)^{\oplus 4})$. Then $X$ is a normal, projective, Gorenstein 3fold. If $L$ is the line budle $M\otimes \pi^*O(1)$, one can also check that $H^1(X,O(K_X+L))=\mathbb{C}$.
However, it is known that the Kodaira vanishing theorem holds if $X$ has log canonical singularities. There are also weaker versions in the theorem in the paper 'D. Arapura and D. B. Jaffe On Kodaira Vanishing for Singular Varieties Proc. A.M.S, 105, No. 4, pp. 911916, 1989.'

1$\begingroup$ Great, that's exactly what I need! $\endgroup$ – Alexander Braverman Oct 14 '11 at 13:38

$\begingroup$ Can you give me a reference to this result (Kodaira vanishing for log canonical singularities)? $\endgroup$ – Alexander Braverman Oct 14 '11 at 13:39

$\begingroup$ Thanks, I saw your edit. Unfortunately, the results of Arapura and Jaffe are not sufficient for my purposes, but full Kodaira vanishing for log canonical singularities would work. Is this written anywhere, or is it folklore? $\endgroup$ – Alexander Braverman Oct 14 '11 at 13:52

$\begingroup$ Alexander: See Corollary 2.43 in Fujino's paper: math.kyotou.ac.jp/~fujino/MMP21s.pdf. I think this was first proved by Sandor Kovacs, so it is probably also found among his papers. $\endgroup$ – J.C. Ottem Oct 14 '11 at 14:33

$\begingroup$ It is actually also true for semilogcanonical (the nonnormal version of log canonical). Here is a reference for that: Sándor J. Kovács, Karl Schwede, Karen E. Smith, The canonical sheaf of Du Bois singularities, Advances in Mathematics, Volume 224, Issue 4, 10 July 2010, Pages 16181640, ISSN 00018708, 10.1016/j.aim.2010.01.020. (sciencedirect.com/science/article/pii/S0001870810000393) $\endgroup$ – Sándor Kovács Oct 14 '11 at 22:53
Indeed, as JC Ottem points out, Kodaira holds for log canonical (even semilog canonical singularities). There's also a way to quickly deduce that Kodaira vanishing holds for Du Bois singularities (either from the AmbroFujino 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 KawamataViehweg vanishing and Serre duality.

$\begingroup$ Thanks a lot; one comment: I actually need the vanishing of $H^i(\omega_X\otimes L)$ but that is indeed easy to include in your argument. $\endgroup$ – Alexander Braverman Oct 14 '11 at 16:11

$\begingroup$ Sure, as you certainly know that follows from either Serreduality applied directly to $X$ (since $X$ is CohenMacaulay) or from the fact that for rational singularities we also have $R \pi_* \omega_Y \cong \omega_X$. $\endgroup$ – Karl Schwede Oct 14 '11 at 16:20