Liftable rational varieties

Question

Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?

Answer 1

Two such examples were given by Achinger and Zdanowicz [AZ17], both of which satisfy a whole bunch of other good properties (e.g. their classes in the Grothendieck ring of varieties are polynomials in the Lefschetz motive $\mathbb L = [\mathbb A^1]$). The easiest one to state is probably the following:

Example. (Achinger–Zdanowicz [AZ17, §4]) Let $k = \bar{\mathbb F}_p$, let $P = \mathbb P^3_k$, and construct $X$ from $P$ by first blowing up all $\mathbb F_p$-points of $P$, and then blowing up all strict transforms of lines defined over $\mathbb F_p$. Then $X$ cannot be lifted to any ring $A$ in which $pA \neq 0$.

The proof is relatively elementary, and relies on matroid theory and elementary projective geometry, as well as a little deformation theory.

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References.

[AZ17] Achinger, Piotr; Zdanowicz, Maciej, Some elementary examples of non-liftable varieties. Proc. Amer. Math. Soc. 145 (2017). ZBL06769127.