This is related to my [previous question](https://mathoverflow.net/questions/233152/lifting-of-frobenius-on-semi-abelian-varieties) 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 $A$?



Lifting of Frobenius on torsors over abelian varieties
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 $W$.


