Global Frobenius Liftability I

Piotr Achinger; Jakub Witaszek; Maciej Zdanowicz

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arXiv · 1708.03777

Global Frobenius Liftability I

Piotr Achinger·Jakub Witaszek·Maciej Zdanowicz

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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.

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BibTeX RIS

Piotr Achinger, Jakub Witaszek, Maciej Zdanowicz. 2017-08-12. Global Frobenius Liftability I. https://arxiv.org/abs/1708.03777

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