Journal of the European Mathematical Society

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Published online first: 2021-04-14
DOI: 10.4171/JEMS/1063

Global Frobenius liftability I

Piotr Achinger[1], Jakub Witaszek[2] and Maciej Zdanowicz[3]

(1) Polish Academy of Sciences, Warsaw, Poland
(2) University of Michigan, Ann Arbor, USA
(3) École Polytechnique Fédérale de Lausanne, Switzerland

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 étale cover, admit a toric fibration over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wiśniewski, 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 generalization and specialization results. Furthermore, we prove a positive characteristic analogue of Winkelmann’s theorem on varieties with trivial logarithmic tangent bundle (generalizing 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.

Keywords: Frobenius lifting, toric variety, abelian variety, trivial log tangent bundle

Achinger Piotr, Witaszek Jakub, Zdanowicz Maciej: Global Frobenius liftability I. J. Eur. Math. Soc. Electronically published on April 14, 2021. doi: 10.4171/JEMS/1063 (to appear in print)