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Register a new userLifts of twisted K3 surfaces to characteristic 0
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Authors
Daniel Bragg
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Deligne showed that every K3 surface over an algebraically closed field of positive characteristic admits a lift to characteristic 0. We show the same is true for a twisted K3 surface. To do this, we study the versal deformation spaces of twisted K3 surfaces, which are particularly interesting when the characteristic divides the order of the Brauer class. We also give an algebraic construction of certain moduli spaces of twisted K3 surfaces over $mathrm{Spec}mathbf{Z}$ and apply our deformation theory to study their geometry. As an application of our results, we show that every derived equivalence between twisted K3 surfaces in positive characteristic is orientation preserving.
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We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of K3 surfaces isogenous to a given K3 surface $X$ in terms of certain integral sublattices of the second rational $ell$-adic and crystalline cohomology groups of $X$. This is a positive characteristic analog of a result of Huybrechts, and extends results of the second author. We give applications to the reduction types of K3 surfaces and to the surjectivity of the period morphism. To prove these results we describe a theory of B-fields and Mukai lattices in positive characteristic, which may be of independent interest. We also prove some results on lifting twisted Fourier--Mukai equivalences to characteristic 0, generalizing results of Lieblich and Olsson.
Let $k$ be a field finitely generated over the finite field $mathbb F_p$ of odd characteristic $p$. For any K3 surface $X$ over $k$ we prove that the prime to $p$ component of the cokernel of the natural map $Br(k)to Br(X)$ is finite.
For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over QQ associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.
We study the derived categories of twisted supersingular K3 surfaces. We prove a derived crystalline Torelli theorem for twisted supersingular K3 surfaces, characterizing Fourier-Mukai equivalences in terms of isomorphisms between their associated K3 crystals. This is a positive characteristic analog of the Hodge-theoretic derived Torelli theorem of Orlov, and its extension to twisted K3 surfaces by Huybrechts and Stellari. We give applications to various questions concerning Fourier-Mukai partners, extending results of Cu{a}ldu{a}raru and Huybrechts and Stellari. We also give an exact formula for the number of twisted Fourier-Mukai partners of a twisted supersingular K3 surface.
We develop a theory of twistor spaces for supersingular K3 surfaces, extending the analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are obtained as relative moduli spaces of twisted sheaves on universal gerbes associated to the Brauer groups of supersingular K3 surfaces. In rank 0, this is a geometric incarnation of the Artin-Tate isomorphism. Twistor spaces give rise to curves in moduli spaces of twisted supersingular K3 surfaces, analogous to the analytic moduli space of marked K3 surfaces. We describe a theory of crystals for twisted supersingular K3 surfaces and a twisted period morphism from the moduli space of twisted supersingular K3 surfaces to this space of crystals. As applications of this theory, we give a new proof of the Ogus-Torelli theorem modeled on Verbitskys proof in the complex analytic setting and a new proof of the result of Rudakov-Shafarevich that supersingular K3 surfaces have potentially good reduction. These proofs work in characteristic 3, filling in the last remaining gaps in the theory. As a further application, we show that each component of the supersingular locus in each moduli space of polarized K3 surfaces is unirational.
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