A descent theorem: a finite étale cover with a quasi-canonical lifting, of degree prime to p, forces the base variety to have a canonical lifting over the Witt vectors.
Varieties in positive characteristic with numerically flat log cotangent bundle
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In this paper, we prove that a smooth projective globally $F$-split variety with numerically flat tangent bundle is an \'etale quotient of an ordinary abelian variety. We also show its logarithmic analog, which contains a characterization of toric varieties. We further prove that, without assumption of global $F$-splitting, a smooth projective separably rationally connected variety of arbitrary characteristic with numerically flat log cotangent bundle is a toric variety.
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Quasi-canonical lifting of projective varieties in positive characteristic
A descent theorem: a finite étale cover with a quasi-canonical lifting, of degree prime to p, forces the base variety to have a canonical lifting over the Witt vectors.