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.
Beauville, Endomorphisms of hypersurfaces and other manifolds , International Mathematics Research Notices 1 (2001), 53–58
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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.