Related papers: Twistor spaces for supersingular K3 surfaces

We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli…

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…

We show how to construct non-isotrivial families of supersingular K3 surfaces over rational curves using a relative form of the Artin-Tate isomorphism and twisted analogues of Bridgeland's results on moduli spaces of stable sheaves on…

Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of…

We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties.…

When studying mirror symmetry in the context of K3 surfaces, the hyperkaehler structure of K3 makes the notion of exchanging Kaehler and complex moduli ambiguous. On the other hand, the metric is not renormalized due to the higher amount of…

A holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. In this paper, we completely determine its structure as an automorphic function on the moduli space of such K3 surfaces. On every…

We generalise the notion of the Tate-Shafarevich group of an elliptic K3 surface with a section to the Tate-Shafarevich group of a K3 surface endowed with a linear system. The construction, which uses Grothendieck's special Brauer group,…

In this paper, we study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and we have natural…

In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3…

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits…

We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to…

M-theory suggests the study of 11-dimensional space-times compactified on some 7-manifolds. From its intimate relation to superstrings, one possible class of such 7-manifolds are those that have Calabi-Yau threefolds as boundary. In this…

These are notes of my lectures given at the school on intersection theory and moduli at the ICTP, Trieste. We construct moduli spaces of K3 surfaces and higherdimensional hyperkaehler manifolds, including moduli spaces of (2,2)-conformal…

We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces…

Let (S,H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c_1(E),H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether…

We construct explicitly moduli curves of polarized supersingular K3 surfaces in characteristic 2 with Artin invariant 2. As an application, we detect a "jump" phenomenon in a family of automorphism groups of supersingular K3 surfaces with a…

We give construction of singular K3 surfaces with discriminant 3 and 4 as double coverings over the projective plane. Focusing on the similarities in their branching loci, we can generalize this construction, and obtain a three dimensional…

We show that supersingular K3 surfaces in characteristic $p\geq5$ are related sequences of very special correspondences. This is not enough to conclude that they are unirational. As a byproduct, we exhibit a fibration structure on the…

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of…