Journal of the European Mathematical Society
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Global Frobenius liftability IPiotr Achinger, Jakub Witaszek and Maciej Zdanowicz (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)