Liftability of Frobenius and images of toric varieties

The celebrated proof of the Hartshorne conjecture by Shigefumi Mori
allowed for the study of the geometry of higher dimensional varieties
through the analysis of deformations of rational curves. One of the
many applications of Mori's results was Lazarsfeld's positive answer
to the conjecture of Remmert and Van de Ven which states that the only
smooth variety that the projective space can map surjectively onto is
the projective space itself. Motivated by this result, a similar
problem has been considered for other kinds of varieties such as
abelian varieties (Demailly-Hwang-Mok-Peternell) or toric varieties
(Occhetta-Wiśniewski). In my talk, I would like to present a
completely new perspective on the problem coming from the study of
Frobenius lifts in positive characteristic. This is based on a joint
project with Jakub Witaszek and Maciej Zdanowicz.