Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic zero ring $R$ with a surjective morphism $R \to k$.

In fact, we can lift $A$ to the Witt ring $W(k)$ if $A$ is ordinary (Norman-Oort) or if $A$ is an elliptic curve (lift the cubic equation defining it).

1). In general, we don't expect to lift $A$ to $W(k)$. But can we always lift $A$ to some ring of integers $\mathcal{O}_K$, where $K/W(k)$ is a finite extension? (Such an extension must be totally ramified to ensure a surjection $K \to k$).

2). If not, is there data internal to $A$ that tells us about when such lifts do exist, analogous to $$\text{Ordinary } \implies \text{lifts to $W(k)$}?$$

serious, even in the ordinary case (where it is affirmative, by an argument of Serre-Tate at the end of Messing's thesis). Beyond dimension 1 there are many non-algebraizable formal deformations over $W(k)$. The Norman-Oort paper assumes alg. closed $k$ (due Dieudonne modules), but this can be bypassed via a deformation ring argument (giving a lift over a $p$-adicorderwith residue field $k$, not that its normalization has that residue field!). Later work of Norman & Ogus gaveobstructions, and showed a lift always exists over $W(k)[\sqrt{p}]$. $\endgroup$2more comments