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