[I found in the literature](https://arxiv.org/abs/1501.00741) that, in characteristic 0, Kodaira vanishing holds for log-canonical pairs. On the other hand, the usual statement for Kawamata-Viehweg vanishing talks about a klt pair $(X,\Delta)$.


**Question:** Are there (good) examples of failure of Kawamata-Viehweg vanishing for log-canonical pairs (even better, for $\Delta=0$)?



Log canonical counterexample to Kawamata-Viehweg vanishing
In Prop. 3.13 of <http://arxiv.org/pdf/1212.5105.pdf> there is an example of a generically finite map $\lambda : T\to A$ where $A$ is an abelian variety and $T$ is a Gorenstein variety with log canonical singularites, and $R^1\lambda \_\* \omega \_T\ne 0$ is supported at a closed point of $A$ (everything is defined over a field of characteristic 0). But then if $H$ is sufficiently ample on $A$, $\lambda ^\*H$ is nef and big on $T$ and $H^j(R^i\lambda \_\* \omega \_T)=0$ for $j>0$ and $i\geq 0$. Thus $H^1(T,\omega \_T\otimes \lambda ^\*H)\cong H^0(R^1\lambda \_\* \omega \_T \otimes H)\ne 0$.


There are surely more straightforward examples (eg. there is no need to go to the trouble of guaranteeing that there is a generically finite map to an abelian variety).


