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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?

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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.

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  • $\begingroup$ Can it be done locally though? $\endgroup$
    – Anonymous
    Commented Feb 22, 2011 at 23:50
  • $\begingroup$ Yes. Smooth affine $X,Y$ can be lifted. Then obstructions to lifting the graph of $f$ lie in $H^1$ of the normal bundle, which vanishes by the affine condition. $\endgroup$
    – inkspot
    Commented Feb 23, 2011 at 9:28

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