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.

中文翻译：

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全局 Frobenius 提升性 I

我们提出了一个猜想，表征具有正特征的光滑射影簇，其 Frobenius 态射可以模 $p^2$ 提升——我们期望这些簇在有限的étale 覆盖之后，承认普通阿贝尔簇的复曲面纤维化。我们证明这个断言暗示了 Occhetta 和 Wiśniewski 的猜想，它指出在特征零中，投影复曲面簇的平滑图像是复曲面簇。为此，我们分析了家庭中环面变种的行为，显示出一些泛化和专业化的结果。此外，我们证明了 Winkelmann 定理的正特征类比关于具有平凡对数切丛的簇（推广 Mehta-Srinivas 的结果），从而获得我们猜想的一个重要特例。最后，