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Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume that $A$ lifts to $W(k)$.

When it is possible to lift the Frobenius endomorphism of $A$ to $W(k)$?

Probably, $B$ has to be ordinary as only in this case Frobenius endomorphism of $B$ lifts. My problem is that I cannot use obstruction theory here as the existence of liftings on finite levels does not yield to a lifting on the whole $W(k)$.

Maybe some small examples can be done. The case $T=\mathbb{G}_m,\dim B=1$ is already very interesting.

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  • $\begingroup$ I assume $k$ is perfect. Let $A'$ be the chosen lift of $A$, giving $B'$ and $T'$ lifting $B$ and $T$ respectively. Using the canonical link between such extensions and line bundles on $B'$ arising from the dual abelian scheme when $T = {\rm{GL}}_1$ (so we can regard $B'$ as primary and $A'$ as secondary), in general $A'$ corresponds to a $W(k)$-point $\xi$ of ${B'}^{\vee} \otimes {\rm{X}}^{\ast}(T)$ (using the dual abelian scheme). So the necessary and sufficient condition is that Frobenius of $B$ lifts to $B'$ and its dual map along with Frobenius for $T$ carries $\xi^{(p)}$ to $\xi$. $\endgroup$
    – nfdc23
    Commented Mar 9, 2016 at 4:24
  • $\begingroup$ What do you mean by `$A$ lifts to $W(k)$'? Do you want the lift to be a scheme or a formal scheme? $\endgroup$
    – Piotr Achinger
    Commented Mar 12, 2016 at 14:10

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My answer to your other question seems to show that if $B$ is ordinary, $A$ has a natural lift over $W$ as a scheme, together with a lift of the Frobenius. I don't see however why the group structure of $A$ should lift, and why it should be compatible with the lift of Frobenius.

EDIT. Regarding the group structure: say $B$ is a $T$-torsor over an abelian $A$, trivialized over $0\in A$. We want to give $B$ a group structure such that $p:B\to A$ is a homomorphism with kernel $T$. This should be a map $b:B\times B\to B\to B$ such that the square $\require{AMScd}$ \begin{CD} B\times B @>b>> B \\ @V{p\times p}VV @VVpV \\ A\times A @>a>> A \\ \end{CD} commutes. This map should factor through a map $b':B\times B\to B' = a^* B$ from a $T\times T$-torsor $B\times B$ over $A\times A$ to the $T$-torsor $B'$ over $A\times A$, equivariant with respect to the addition map $t:T\times T\to T$. It should be easy to write out explicitly in terms of line bundles corresponding to the torsor $B$ what this map is, and what it should satisfy to give $B$ a group structure, and most probably we will see that this structure is preserved by Teichmueller lifts, so that we get a group structure on the canonical lift.

EDIT 2. Some time ago I was pointed in a similar context to look up "bi-extensions" (SGA 7 I, Exp. VII-VIII), but I never did. Perhaps you will find an answer to your question there.

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