# Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces

Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non algebraically closed field of char $p>0$?

More particularly I have the following situation:

$X$ is a variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Let $x$ be a codimension $2$ point of $X$ and $S=\text{Spec }\mathcal{O}_{X,\ x}$. Then $S$ is a surface over the residue field $k(x)$ ( which is not algebraically closed ). I need to use Kawamata-Viehweg vanishing theorem ( at least some relative version of it ) on this $S$. I know that one version of this vanishing theorem holds for surfaces over an algebraically closed field of char $p>0$ by Hiromu Tanaka ( The $X$-methods for klt surfaces in positive characteristic ).

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Doesn't Kodaira vanishing fail for surfaces in positive characteristic? Raynaud gave counterexamples. –  Angelo Apr 19 '13 at 19:30
Yes Angelo you are right, Kadaira vanishing does fail in char positive even on a surface, but some version of Kawamata-Viehweg vanishing theorem which is sufficient to the run the Minimal Model Program continues to hold even in positive characteristic for surfaces. You can look for a reference in the paper I mentioned above or in a preprint by Kollar and Kovacs, "Birational Geometry of Log Surfaces" –  Omprokash Apr 19 '13 at 19:53
Presumably for surfaces non-algebraically closed fields, depending on exactly what you want, perhaps things are ok by some base change (perhaps as long as your exceptional set is geometrically whatever)? If you are looking for references, I know that some Grauert-Riemenschneider-type statements are even true for excellent two dimensional rings (see for example the work of Lipman on resolution of 2 dimensional singularities and also his work on rational singularities in any characteristic). Can you be more precise in terms of what exactly you need? –  Karl Schwede Apr 20 '13 at 5:18
Since the usual version of the Kawamata-Viehweg vanishing theorem generalized Kodaira's, you'd better write down the exact statement that you need. –  Angelo Apr 20 '13 at 12:03
I don't understand "over $k(x)$". Clearly $S$ is a two-dimensional local scheme with closed point $\mathrm{Spec}\,k(x)$, but in general it is not a $k(x)$-scheme in any natural way. –  Laurent Moret-Bailly Apr 20 '13 at 14:42