The Deligne–Illusie Theorem and exceptional Enriques surfaces

Building on the results of Deligne and Illusie liftings to truncated Witt vectors, we give a criterion for non-liftability that involves only dimension certain cohomology groups vector bundles arising from Frobenius pushforward de Rham complex. Using bundle methods, apply this show exceptional Enriques surfaces, class introduced by Ekedahl Shepherd-Barron, do not lift yet base miniversal formal deformation over vectors is regular. classification Bombieri Mumford, also bielliptic surfaces quotient unipotent group scheme order p ring vectors. These hinge some observations in homological algebra relates splittings derived categories Yoneda extensions diagram completions.