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Oberseminar WiSe 2010/11 Zusammenfassung des Vortrags von Christian Liedtke (Stanford, USA)

On the birational nature of lifting

Given a smooth projective variety over an algebraically closed field k, one would like to know whether it lifts over the ring W2 of Witt vectors of length 2. For example, by a famous result of Deligne and Illusie, the Hodge-de Rham spectral sequence degenerates at E1 if this is true. However, in nature it happens that constructions yield a birational model of the variety one is interested in. Moreover, we should allow canonical singularities. Thus, knowing that a variety with at worst canonical singularities lifts over W2, does that imply that all birational models with at worst canonical singularities lift over W2? Examples of Cynk, Schuett, and van Straten show that this is false from dimension 3 on. However, for surfaces the situation is different, using the fact that surfaces with canonical singularities are coarse spaces of smooth Artin stacks. This project is joint with Matthew Satriano (UC Berkeley/Ann Arbor).