Question

Let X and Y be smooth varieties over a finite field F. Let R be a complete DVR of unequal characteristic with residue field F. I have the following question:

If f is a morphism from X to Y, is it possible to choose formally smooth R-formal schemes, $\mathfrak{X}$ and $\mathfrak{Y}$, whose special fibres are X and Y respectively, together with a morphism g from $\mathfrak{X}$ to $\mathfrak{Y}$ which lifts f under the natural specialization maps?

Answer 1

There are surfaces $X$ over $F$ that cannot be lifted to any CDVR $R$ of the kind you mention and whose dualizing sheaf is ample. So $X$ cannot be lifted as a formal scheme over $R$, since any such formal scheme would be the completion of a scheme over $R$. (For example, there exist $X$ that violate the Bogomolov-Miyaoka-Yau inequality $c_1^2\le 3c_2$.) Then the identity $X\to X$ will be a counterexample for you.