Lifting of Frobenius on torsors over abelian varieties

Question

This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and $\mathcal{L}$ lifts on the canonical lifting of $A$ on $W(k)$.

Consider $X=(\mathrm{Spec}_A\mathcal{L})\setminus A$ -- the total space of $\mathcal{L}$ without the zero section. $X$ lifts to $W(k)$. When Frobenius morphism of $X$ lifts to $W(k)$? In the previous question I asked about the case $\deg \mathcal{L}=0$.

Clearly, if a lifting of Frobenius maps fibers to fibers then reduction of $\mathcal{L}$ to $k$ has to be trivial which is impossible if $\deg\mathcal{L}\neq 0$. But maybe it is possible to construct a lifting which does not respect projection on $A$?

Answer 1

The canonical lifting $\mathcal{A}$ of $A$ has a canonical lift of the relative Frobenius $F_{\mathcal{A}/W}:\mathcal{A}\to \mathcal{A}'$, where $\mathcal{A}'=F_W^* \mathcal{A}$ (=the canonical lift of the Frobenius twist $A'=F_k^*(A)$). Moreover, any line bundle $L$ on $A$ has a canonical `Teichmueller' lift $\mathcal{L}$ to $\mathcal{A}$ such that $F_{\mathcal{A}/W}^* (\mathcal{L}) \cong \mathcal{L}'^p$, $\mathcal{L}' = w^* \mathcal{L}$ where $w$ is the ($W$-Frobenius-linear) projection $\mathcal{A}'=F_W^*\mathcal{A}\cong \mathcal{A}$.

Let $T(L)$ be the total space of $L$ (treated as a $\mathbb{G}_m$-torsor), in other words, $T(L) = {\rm Spec}_A \bigoplus_{n\in \mathbb{Z}} L^n$. Then $F_{A/k}^* (T(L)) = T(L^p)$, and the relative Frobenius $F_{T(L)/A} : T(L)\to T(L^p)$ is simply ${\rm Spec}_A$ of the inclusion $\bigoplus_{n\in p\mathbb{Z}} L^{n}\subseteq \bigoplus_{n\in\mathbb{Z}} L^n$.

The upshot is that the relative Frobenius $F_{T(L)/A}$ naturally lifts to an $F_{T(\mathcal{L})/\mathcal{A}}: T(\mathcal{L})\to T((\mathcal{L}')^p) = T(F_{\mathcal{A}/W}^* \mathcal{L})$. Then we get a lift of the relative Frobenius $F_{T(L)/k}$ by composition with the natural projection $T(F_{\mathcal{A}/W}^* \mathcal{L})\to T(\mathcal{L}')$.

The above argument works similarly for any torsor under a torus over $A$, or a family of toric varieties.

EDIT. The following commutative diagram with Cartesian squares might be helpful: $\require{AMScd}$ \begin{CD} T(\mathcal{L}) @>F_{T(\mathcal{L})/\mathcal{A}}>> T(\mathcal{L}^p) @>>> T(\mathcal{L}') @>>> T(\mathcal{L}) \\ @. @VVV \square @VVV \square @VVV \\ @. \mathcal{A} @>F_{\mathcal{A}/W}>> \mathcal{A}' @>w>> \mathcal{A} \\ @. @. @VVV \square @VVV\\ @. @. {\rm Spec}(W) @>F_W>> {\rm Spec}(W) \\ \end{CD} The composition of the first two arrows in the top row is the desired lift $F_{T(\mathcal{L})/W}$ of $F_{T(L)/k}$ to $W$.