Global Frobenius liftability. I. (English) Zbl 1471.14055

Achinger, Piotr; Witaszek, Jakub; Zdanowicz, Maciej

J. Eur. Math. Soc. (JEMS) 23, No. 8, 2601-2648 (2021).

The paper addresses the problem of characterizing \(F\)-liftable smooth projective varieties in positive characteristic. Here, a variety \(X\) over a perfect field \(k\) of positive characteristic is \(F\)-liftable if it admits a model \(\widetilde{X}\) over the ring of truncated Witt vectors \(W_2(k)\) together with a model \(\widetilde{F}_X:\widetilde{X}\to\widetilde{X}\) of the Frobenius morphism \(F_X\) of \(X\).
The paper puts forward the following conjecture.

Conjecture: Every \(F\)-liftable smooth projective variety over an algebraically closed field of positive characteristic has a finite étale cover whose Albanese morphism is a toric fibration.

The first main result relates the conjecture to a conjecture by G. Occhetta and J. A. Wiśniewski [Math. Z. 241, No. 1, 35–44 (2002; Zbl 1022.14016)]:

If every smooth projective simply connected \(F\)-liftable variety over an algebraically closed field of positive characteristic is a toric variety, then every smooth projective variety \(X\) in characteristic zero with a surjective morphism \(Y\to X\) from a complete toric variety \(Y\) is a toric variety.

The proof is based on a descent result for \(F\)-liftability along morphisms of finite type, and an investigation of toric varieties in families:

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Given a smooth projective family \(X\to S\) such that the fiber \(X_s\) is a toric variety for a dense set of \(s\in S\), if the generic fiber has characteristic 0 then it is a toric variety.

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Given a relative normal crossing pair over a DVR, if the generic fiber is a toric pair, then so is the special fiber.

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A relative normal crossing pair is a toric fibration if and only if any fiber over a geometric point is a toric pair.

As a byproduct of the proof, every smooth projective variety \(X\) in characteristic zero with a surjective morphism \(Y\to X\) from a complete toric variety \(Y\) satisfies Bott vanishing, as \(F\)-liftable varieties satisfy Bott vanishing by A. Buch et al. [Tohoku Math. J. (2) 49, No. 3, 355–366 (1997; Zbl 0899.14026)].
The second main result characterizes projective \(F\)-liftable normal crossing pairs \((X,D)\) with numerically trivial log canonical bundle \(\omega_X(D)\) by showing that they are exactly the varieties \(X\) that admit a finite étale cover whose Albanese map is a toric fibration over an abelian variety. This generalizes the result of V. B. Mehta and V. Srinivas [Compos. Math. 64, 191–212 (1987; Zbl 0639.14024)] for \(F\)-liftable varieties with numerically trivial canonical bundle to the logarithmic setting.
Finally, the paper shows that the Conjecture holds for homogeneous spaces, thus proving a conjectural characterization of \(F\)-liftable rational homogeneous spaces by A. Buch et al. [Tohoku Math. J. (2) 49, No. 3, 355–366 (1997; Zbl 0899.14026)]. More precisely, the paper shows that over an algebraically closed field of positive characteristic:

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The only \(F\)-liftable smooth Fano varieties with nef tangent bundle and Picard rank 1 are projective spaces.

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Every \(F\)-liftable homogeneous space is isomorphic to a product of projective spaces and an ordinary abelian variety.

The first result is deduced from S. Mori’s characterization of the projective space [Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)] via a careful analysis of the restriction of differential forms invariant under \(\frac{d\widetilde{F}_X}{p}:F_X^*\Omega_X^1\to\Omega_X^1\) to rational curves in a special covering family. The second result relies on the classification of homogeneous spaces and their characterization in terms of marked Dynkin diagrams.
The authors illustrate their results and highlight the nuances with plenty of examples. They also announce that \(F\)-liftable varieties of small dimension are investigated in the paper Global Frobenius liftability II: surfaces and Fano threefolds by the same authors.

Reviewer: Marta Pieropan (Utrecht)

MSC:

14G17	Positive characteristic ground fields in algebraic geometry
14M17	Homogeneous spaces and generalizations
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
14J45	Fano varieties

Keywords:

Frobenius lifting; toric variety; abelian variety; trivial log tangent bundle

Citations:

Zbl 1022.14016; Zbl 0899.14026; Zbl 0639.14024; Zbl 0423.14006

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