Global Frobenius Liftability I

TL;DR

This paper explores Frobenius liftability of smooth projective varieties in positive characteristic, proposing a conjecture linking it to toric fibrations over abelian varieties, and proves several related results including special cases and applications.

Contribution

It formulates a conjecture on Frobenius liftability, proves implications for toric varieties, and verifies the conjecture for homogeneous spaces, advancing understanding in positive characteristic geometry.

Findings

Frobenius liftability implies a toric fibration structure after an étale cover.

Proved a positive characteristic analogue of Winkelmann's theorem.

Verified the conjecture for homogeneous spaces using rational curve deformations.

Abstract

We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite \'etale cover, admit a toric fibration over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wi\'sniewski, which states that in characteristic zero a smooth image of a projective toric variety is a toric variety. To this end we analyse the behaviour of toric varieties in families showing some generization and specialization results. Furthermore, we prove a positive characteristic analogue of Winkelmann's theorem on varieties with trivial logarithmic tangent bundle (generalising a result of Mehta-Srinivas), and thus obtaining an important special case of our conjecture. Finally, using deformations of rational curves we verify our conjecture for homogeneous spaces, solving a problem posed by Buch-Thomsen-Lauritzen-Mehta.