Lifting F-split surfaces to the Witt vectors

Algebraic varieties in positive characteristic are ill behaved compared to characteristic zero ones. Several important tools available over the complex numbers, such as the Hodge decomposition theorem, are either not available or straight-away false. There are two important classes of positive characteristic varieties which have better behavior: Witt liftable varieties and Frobenius split varieties. A folklore conjecture predicts that the latter class is contained in the former. In a joint work with Bernasconi, Kawakami and Witaszek we proved that this is the case for surfaces. I will give an overview of this result as well as some applications thereof.