Student Algebraic Geometry: Cohomologies of K3 surfaces in characteristic p > 0

Continuing from last week’s talk by Saket, the seminar will take a close look at K3 surfaces (but no knowledge from last week’s talk will be assumed). K3 surfaces enjoy some nice cohomological properties and relatively simple deformation theories even in the positive characteristic setting. There are moreover multiple Torelli theorems that are known in this setting, which may be surprising since characteristic 0 Torelli theorems are usually phrased in terms of Hodge theory. In one version, called crystalline Torelli theorem, the use of Hodge structures is replaced by their rough characteristic p analogue--F-crystals which arise from crystalline cohomology. The current plan of the talk is to sketch some general landscape of K3 surfaces and their cohomologies in positive characteristic, without getting into Torelli theorems.