Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.


We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with residue field $k$ and a flat scheme $\mathcal X$ over $R$, such that
$$ \mathcal X \otimes\_R k \simeq X.$$


In words: There exists a family over a ring of mixed characteristic, which has our $X$ for a special fibre.


For most classes of surfaces in Kodaira dimension zero, liftability is known: For K3 surfaces, liftability was established by Deligne, and for abelian surfaces, one can use more general theories developed for abelian varieties. Bi-elliptic and quasi-bi-elliptic surfaces can be dealt with explicitly.


As far as I know, there is nothing in the literature about Enriques Surfaces. For this class, the case p = 2 is the most interesting.


The question seems natural, so it would struck me as strange if it were still open. Does anyone around here know anything about this?


Thanks a lot.



Liftability of Enriques Surfaces (from char. p to zero)
This may not be exactly the answer you are looking for: I and Nick
Shepherd-Barron have an unpublished (so far) proof of liftability in
characteristic $2$, the only non-trivial case. To atone for the fact that I
refer to unpublished results I give a quick sketch of proof.


The proof starts by showing that in a family $X/S$ of Enriques surfaces
$\mathrm{Pic}^\tau(X/S)$ is flat (of order $2$) over $S$ and
$\mathrm{Pic}(X/S)/\mathrm{Pic}^\tau(X/S)$ is locally constant. This implies
that the tensor square of any line bundle of an Enriques surface extends along
any formal deformation and hence it is enough to find a formal lifting.


For a surface with $h^2(T\_X)=0$ the deformations are unobstructed so we are
OK. There are two types of surfaces with $h^2(T\_X)=1$ (which is the only other
possibility); surfaces with $\mathrm{Pic}^\tau(X)=\alpha\_2$ and surfaces
$\mathrm{Pic}^\tau(X)=\mathbb Z/2$ having non-trivial vector fields (the latter case
exists). The first case is nicer in that we get a map from deformations of
such surfaces to deformations of $\alpha\_2$ and this map is formally smooth. As
we can lift $\alpha\_2$ (a formal deformation has base $\mathbb
Z\_2[[x,y]]/(xy-2)$) we can also lift the surface, in fact over a base with
absolute ramification of order $2$. The second case we know less
about but as $h^2(T\_X)=1$ the base of the deformation is (at most) a
hypersurface singularity and as one can show that it is a very small family one
can show that a versal deformation is flat over $\mathbb Z\_2$. We know nothing
about the ramification necessary in this case.


