K3 Surfaces Research Articles

On irreducible symplectic varieties of K3[n]-type in positive characteristic

We show that there is a good notion of irreducible symplectic varieties of K3[n]-type over an arbitrary field of characteristic zero or p>n+1. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.

On the period of Lehn, Lehn, Sorger, and van Straten’s symplectic eightfold

For the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane, we show that a natural Abel–Jacobi map from Hprim4(Y) to Hprim2(Z) is a Hodge isometry. We describe the full H2(Z) in terms of the Mukai lattice of the K3 category A of Y. We give numerical conditions for Z to be birational to a moduli space of sheaves on a K3 surface or to Hilb4(K3). We propose a conjecture on how to use Z to produce equivalences from A to the derived category of a K3 surface.

New supersymmetric string moduli spaces from frozen singularities

The current classification of mathcal{N} = 1 string theories in eight and seven dimensions is completely captured by K3 surfaces with F-Theory or M-Theory frozen singularities. In this note we show that there are inequivalent ways of freezing certain collections of singularities which have the same ADE type, and so there are more connected components in the moduli space than previously thought; namely, one more in eight dimensions and three more in seven dimensions. We argue that the new component in eight dimensions decompactifies to a string theory in nine dimensions with rank 1 gauge group, which has been so far unidentified. Constructing and studying the stringy descriptions corresponding to these moduli spaces is the subject of a companion paper.

Deformations of Hyperelliptic and Generalized Hyperelliptic Polarized Varieties

The purpose of this article is twofold. Firstly, we address and completely solve the following question: Let (X, L) be a smooth, hyperelliptic polarized variety and let varphi : X longrightarrow Y subset textbf{P}^N be the morphism induced by |L|; when does varphi deform to a birational map? Secondly, we introduce the notion of “generalized hyperelliptic varieties” and carry out a study of their deformations. Regarding the first topic, we settle the non trivial, open cases of (X, L) being Fano-K3 and of (X, L) having dimension m ge 2, sectional genus g and L^m=2g. This was not addressed by Fujita in his study of hyperelliptic polarized varieties and requires the introduction of new methods and techniques to handle it. In the Fano-K3 case, all deformations of (X, L) are again hyperelliptic except if Y is a hyperquadric. By contrast, in the L^m=2g case, with one exception, a general deformation of varphi is a finite birational morphism. This is especially interesting and unexpected because, in the light of earlier results, varphi rarely deforms to a birational morphism when Y is a rational variety, as is our case. The Fano-K3 case contrasts with canonical morphisms of hyperelliptic curves and with hyperelliptic K3 surfaces of genus g ge 3. Regarding the second topic, we completely answer the question for generalized hyperelliptic polarized Fano and Calabi–Yau varieties. For generalized hyperelliptic varieties of general type we do this in even greater generality, since our result holds for Y toric. Standard methods in deformation theory do not work in the present setting. Thus, to settle these long standing open questions, we bring in new ideas and techniques building on those introduced by the authors concerning deformations of finite morphisms and the existence and smoothings of certain multiple structures. We also prove a new general result on unobstructedness of morphisms that factor through a double cover and apply it to the case of generalized hyperelliptic varieties.

Unirational moduli spaces of some elliptic K3 surfaces

We show that the moduli space of $U\oplus \langle -2k \rangle$-polarized K3 surfaces is unirational for $k \le 50$ and $k \notin \{11,35,42,48\}$, and for other several values of $k$ up to $k=97$. Our proof is based on a systematic study of the projective models of elliptic K3 surfaces in $\mathbb{P}^n$ for $3\le n \le 5$ containing either the union of two rational curves or the union of a rational and an elliptic curve intersecting at one point.

Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a K3 surface to a Hirzebrunch surface such that the branch divisor is that effective divisor. Furthermore, he will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface. Subsequently, he studies the same theme for Enriques surfaces.

Equivariant derived equivalence and rational points on K3 surfaces

Equivariant derived equivalence and rational points on K3 surfaces

SINGULARITIES OF QUOTIENT SURFACES OF K3 SURFACES

We study cyclic quotient singularities given by automorphisms of maximum order on K3 surfaces. In particular, we describe fixed loci of such automorphisms, and provide types of these singularities

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

AbstractWe describe the Galois action on the middle$\ell $-adic cohomology of smooth, projective fourfolds$K_A(v)$that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surfaceAwith Mukai vectorv. We show this action is determined by the action on$H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$and on a subgroup$G_A(v) \leqslant (A\times \hat {A})[3]$, which depends onv. This generalizes the analysis carried out by Hassett and Tschinkel over${\mathbb C}$[21]. As a consequence, over number fields, we give a condition under which$K_2(A)$and$K_2(\hat {A})$are not derived equivalent.The points of$G_A(v)$correspond to involutions of$K_A(v)$. Over${\mathbb C}$, they are known to be symplectic and contained in the kernel of the map$\operatorname {\mathrm {Aut}}(K_A(v))\to \mathrm {O}(H^2(K_A(v),{\mathbb Z}))$. We describe this kernel for all varieties$K_A(v)$of dimension at least$4$.When$K_A(v)$is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed-point locus on fourfolds$K_A(0,l,s)$over${\mathbb C}$whereAis$(1,3)$-polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of$K_A(0,l,s)$.

Extendability of automorphisms of K3 surfaces

Extendability of automorphisms of K3 surfaces

Canonical currents and heights for K3 surfaces

Canonical currents and heights for K3 surfaces

Derived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

Finite subgroups of automorphisms of K3 surfaces

AbstractWe give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields.

Compactifications of moduli of elliptic K3 surfaces: Stable pair and toroidal

We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${\bf A_n}$ ($n\ge1$), ${\bf C_n}$ ($n\ge0$) and ${\bf E_n}$ ($n\ge0$).

Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperk\"{a}hler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.

Contractions of hyper-Kähler fourfolds and the Brauer group

We study the geometry of exceptional loci of birational contractions of hyper-Kähler fourfolds that are of K3[2]-type. These loci are conic bundles over K3 surfaces and we determine their classes in the Brauer group. For this we use the results on twisted sheaves on K3 surfaces, on contractions and on the corresponding Heegner divisors. For a general K3 surface of fixed degree there are three (T-equivalence) classes of order two Brauer group elements. The elements in exactly two of these classes are represented by conic bundles on such fourfolds. We also discuss various examples of such conic bundles.

ENRIQUES INVOLUTIONS AND BRAUER CLASSES

AbstractWe prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${\mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${\mathcal Enr}(X)\to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.

Large Orbits on Markoff-Type K3 Surfaces over Finite Fields

Abstract We study the surface $\mathcal {W}_k: x^2 + y^2 + z^2 + x^2 y^2 z^2 = k x y z$ in $(\mathbb {P}^1)^3$, a tri-involutive K3 (TIK3) surface. We explain a phenomenon noticed by Fuchs, Litman, Silverman, and Tran: over a finite field of order $\equiv 1$ mod $8$, the points of $\mathcal {W}_4$ do not form a single large orbit under the group $\Gamma $ generated by the three involutions fixing two variables and a few other obvious symmetries, but rather admit a partition into two $\Gamma $-invariant subsets of roughly equal size. The phenomenon is traced to an explicit double cover of the surface.

On lattice polarizable cubic fourfolds

We extend non-emptyness and irreducibility of Hassett divisors to the moduli spaces of M-polarizable cubic fourfolds for higher rank lattices M, which in turn provides a systematic approach for describing the irreducible components of intersection of Hassett divisors. We show that Fermat cubic fourfold is contained in every Hassett divisor, which yields a new proof of Hassett’s existence theorem of special cubic fourfolds. We obtain an algorithm to determine the irreducible components of the intersection of any two Hassett divisors and we give new examples of rational cubic fourfolds. Moreover, we derive a numerical criterion for the algebraic cohomology of a cubic fourfold having an associated K3 surface and answer a question of Laza by realizing infinitely many rank 11 lattices as the algebraic cohomologies of cubic fourfolds having no associated K3 surfaces.

Moduli of Stable Sheaves on a K3 Surface of Picard Number 1

Moduli of Stable Sheaves on a K3 Surface of Picard Number 1