TITLE: Lifting varieties to characteristic zero.
QUESTION [16 upvotes]: If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ of $k$. If that succeeds, compute de Rham cohomology of the lift over $W_k$ instead, which in general will be much easier to do. Neglecting torsion, this de Rham cohomology is the same as the crystalline cohomology of $X$.
I would like to have an example at hand where this approach fails: Can you give an example for

A smooth proper variety $X$ over the finite field with $p$ elements, such that there is no smooth proper scheme of finite type over $\mathbb Z_p$ whose special fibre is $X$.

The reason why such examples have to exist is metamathematical: If there werent any, the pain one undergoes constructing crystalline cohomology would be unnecessary.

REPLY [3 votes]: In general the obstruction to lift a scheme $X$ in characteristic zero is in $H^2(X,T_X)$. For examples of $3$-folds in positive characteristic that connot be lifted in characteristic zero you may look at Theorem 22.4 in Hartshorne's "Deformation Theory".