2017/08/12 by Achinger, Piotr, Witaszek, Jakub, Zdanowicz, Maciej
#14J45 #14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14G17 #Secondary 14M17
paper · doi:10.48550/arxiv.1708.03777
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 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 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.