Twistor Spaces for Supersingular K3 Surfaces

Abstract

We develop a theory of twistor spaces for supersingular K3 surfaces, extending Artin's analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are families of twisted supersingular K3 surfaces over the affine line, and are obtained as relative moduli spaces of twisted sheaves on universal gerbes associated to supersingular K3 surfaces. To study these families, we develop a theory of crystals for twisted supersingular K3 surfaces, and study the resulting period morphism from the moduli space of twisted supersingular K3 surfaces to the space of crystals. As applications of this theory, we give a new proof of the Ogus's crystalline Torelli theorem, inspired by Verbitsky's proof in the complex analytic setting. We also obtain a new proof of the result of Rudakov-Shafarevich that supersingular K3 surfaces have potentially good reduction. Finally, we apply our twistor spaces to study elliptic fibrations. Using results of Max Lieblich, we show that every elliptic fibration on a supersingular K3 surface admits a purely inseparable multisection. As a consequence of this result, we give a new proof of the unirationality of supersingular K3 surfaces. Our techniques work uniformly in odd characteristic, and in particular we are able to extend all of these results to characteristic 3, where they were not previously known.