Kodaira vanishing for Du Bois singularities

Let $X$ be a projective variety with Du Bois singularities, which is additionally assumed to be Cohen-Macaulay. Then $H^i(X, \mathscr L^{-1}) = 0$ for any ample line bundle $\mathscr L$ and $i < \dim X$, by Thm. 10.42 of Kollár, Singularities of the MMP.

Question: Does this still hold if $\mathscr L$ is only big and nef?

• Hi Patrick. I think this is an open question. Fujino has results in this direction but he usually assumes that $\mathcal{L}$ is ample after being restricted to various strata. Mar 10, 2014 at 17:26
• Hi Karl. Probably you're right, at least I found a gap in my proof attempt ... Mar 17, 2014 at 18:43