All Strict Calabi--Yau Threefolds of Height One Lift to Characteristic 0

Abstract

We prove that every strict Calabi--Yau threefold of Artin--Mazur height one over an algebraically closed field k of characteristic p>0 admits a projective lift to W(k). No assumption is made on the torsion in its crystalline cohomology, and p is arbitrary. Using the crystalline period map of Brantner and Taelman, we show that the obstruction to extending a lift over W_n(k) is the Bockstein of a torsion coordinate of its period point, and that this coordinate can always be made to vanish by varying the lift. The mixed-characteristic deformation functor need not be smooth: for a Calabi--Yau threefold of height one constructed by Addington and Bragg in characteristic 3, some lifts over W_2(k) do not extend over W_3(k). This disproves a conjecture of Achinger and Zdanowicz.