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Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties of this lifting.

For exapmle, I read (but without reference, thus I will appreciate some if one knows) that if I have a K3 of finite height, then there exist a lifting $S_W$ such that the restirction map of the Picard groups is an isomorphism. Can I say the same thing for other surfaces?

What about abelian surfaces with a polarization of degree coprime with the characteristic?

For Enriques surfaces in odd characteristic?

For Bielliptic surfaces in characteristic greater than 3?

Thank you very much for any minute you spend reading this! Double thanks if you also post something :D

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    $\begingroup$ If I understand correctly, a reference for the statement about K3s in the second paragraph is Lieblich--Maulik, arXiv:1102.3377 Corollary 4.2. Unfortunately I can't say anything about your actual question. $\endgroup$
    – user5117
    Jul 10, 2014 at 8:51

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For an Enriques surface $S$ in odd characteristic, for any lift $S_W$ over $W$, the restriction map on Picard groups is an isomorphism. Essentially this follows from the fact that $h^2(S,\mathcal{O}_S)$ equals $0$ together with infinitesimal deformation theory of the Picard functor. I believe this is discussed in Cossec-Dolgachev, also cf. p. 7 of the following survey by Dolgachev,

A Brief Introduction to Enriques Surfaces

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You can have a look at Liedtke's survey paper Algebraic Surfaces in Positive Characteristic, arXiv:0912.4291.

There is a section about lifting on the Witt vectors $W(k)$ and more general rings, together with a comprehensive bibliography.

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  • $\begingroup$ Thank you Franccesco. As a matter of fact I had already looked at it and it does not say what I need... $\endgroup$ Jul 10, 2014 at 8:21
  • $\begingroup$ Did you also check the references? $\endgroup$ Jul 10, 2014 at 8:33
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    $\begingroup$ Many of them I am keeping going :D $\endgroup$ Jul 10, 2014 at 8:36

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