# The Deligne-Illusie Theorem and exceptional Enriques surfaces

## TL;DR

This paper establishes criteria for non-liftability of certain algebraic surfaces to Witt vectors, using cohomological and vector bundle methods, and applies these to exceptional Enriques and bielliptic surfaces.

## Contribution

It introduces a new cohomological criterion for non-liftability and demonstrates its application to specific classes of algebraic surfaces, expanding understanding of their deformation properties.

## Key findings

- Exceptional Enriques surfaces do not lift to truncated Witt vectors.
- Bielliptic surfaces from unipotent group quotients do not lift to Witt vectors.
- The base of the miniversal deformation for these surfaces remains regular.

## Abstract

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