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 $L$?



Is there an analog of Kodaira vanishing for singular varieties
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 $X\in|M\otimes \pi^\*O(-1)^{\oplus 4})|$. Then $X$ is a normal, projective, Gorenstein 3-fold. 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](http://www.jstor.org/pss/2047052) Proc. A.M.S, 105, No. 4, pp. 911-916, 1989.'


