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 p2-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 Wísniewski, 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.

Original languageEnglish (US)
Pages (from-to)2601-2648
Number of pages48
JournalJournal of the European Mathematical Society
Volume23
Issue number8
DOIs
StatePublished - 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics

Keywords

  • Abelian variety
  • Frobenius lifting
  • Toric variety
  • Trivial log tangent bundl

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