Liftability of Enriques Surfaces (from char. p to zero)

Question

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.

Answer 1

This may not be exactly the answer you are looking for: I and Nick Shepherd-Barron have an unpublished (so far) proof of liftability in characteristic $2$, the only non-trivial case. To atone for the fact that I refer to unpublished results I give a quick sketch of proof.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