K3 Surfaces Research Articles

A special configuration of 12 conics and generalized Kummer surfaces

A generalized Kummer surface X obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $$9\mathbf{A}_{2}$$ -configuration of $$(-2)$$ -curves. Such a configuration plays the role of the $$16\mathbf{A}_{1}$$ -configurations for usual Kummer surfaces. In this paper we construct 9 other such $$9\mathbf{A}_{2}$$ -configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $$9\mathbf{A}_{2}$$ -configurations are obtained by taking the pullback of a certain configuration of 12 conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of X and of the generic fiber of its natural elliptic pencil.

Lattice duality for coupling pairs admitting polytope duality with trivial toric contribution

We study a lattice duality among families of K3 surfaces associated to coupling pairs that admit polytope duality with trivial toric contribution.

K3 Surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers

We classify complex K3 surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers. These surfaces are characterized by the fact that they admit a unique elliptic fibration with infinite automorphism group. We furnish an explicit list of 32 Néron-Severi lattices corresponding to such surfaces. Incidentally, we are able to decide which of these 32 classes of surfaces admit a unique elliptic pencil. Finally, we prove that all K3 surfaces with Picard rank ≥19 and infinite automorphism group have positive entropy.

Vertex operator superalgebra/sigma model correspondences: The four-torus case

Abstract We propose a correspondence between vertex operator superalgebras and families of sigma models in which the two structures are related by symmetry properties and a certain reflection procedure. The existence of such a correspondence is motivated by previous work on ${\cal N}=(4,4)$ supersymmetric non-linear sigma models on K3 surfaces, and on a vertex operator superalgebra with Conway group symmetry. Here we present an example of the correspondence for ${\cal N}=(4,4)$ supersymmetric non-linear sigma models on four-tori, and compare it to the K3 case.

Limits and singularities of normal functions

We find a new method to detect the linearly independence of \({\mathbb {R}}\)-regulator indecomposable \(K_1\)-cycles which is based on the singularities and limits of admissible normal functions. We also construct a collection of higher Chow cycles on certain surfaces in \({\mathbb {P}}^3\) of degree \(d \geqslant 4\) which degenerate to an arrangement of d planes in general position. By applying our method, we show that these higher Chow cycles are enough to show the surjectivity of the real regulator map when \(d = 4\). Hence our construction gives a new explicit proof of the Hodge-\({\mathcal {D}}\)-Conjecture for a certain type of K3 surfaces. As an application, we also construct a general semistable degeneration family of degree \(d + 1\) threefolds in \({\mathbb {P}}^3 \,{\times }\, {\mathbb {P}}^1 \,{\times }\, {\mathbb {P}}^1\) such that a codimension 1 stratum is a surface of the above type. The real regulator indecomposability of our higher Chow cycles implies that the Griffiths group of the general fiber of these threefolds is non-trivial.

Rank one sheaves over quaternion algebras on Enriques surfaces

Abstract Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

Gromov\u2013Witten theory of K3 surfaces and a Kaneko\u2013Zagier equation for Jacobi forms

We prove the existence of quasi-Jacobi form solutions for an analogue of the Kaneko–Zagier differential equation for Jacobi forms. The transformation properties of the solutions under the Jacobi group are derived. A special feature of the solutions is the polynomial dependence of the index parameter. The results yield an explicit conjectural description for all double ramification cycle integrals in the Gromov–Witten theory of K3 surfaces.

Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface

We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.

Systolic inequalities for K3 surfaces via stability conditions

We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface X, there exists a constant C depending only on the rank and discriminant of NS(X), such that sys(σ)2≤C·vol(σ)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathrm {sys}(\\sigma )^2\\le C\\cdot \\mathrm {vol}(\\sigma ) \\end{aligned}$$\\end{document}holds for any stability condition on mathcal {D}^bmathrm {Coh}(X). This is an algebro-geometric generalization of a classical systolic inequality on two-tori. We also discuss applications of this inequality in symplectic geometry.

A short note on inadmissible coefficients of weight 2 and 2k+1 newforms

Let f(z)=q+sum _{nge 2}a(n)q^n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for k=2 by ruling out or locating all odd prime values |ell |<100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights kge 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

K3 curves with index k>1

Let mathcal {KC}_g ^k be the moduli stack of pairs (S, C) with S a K3 surface and Csubseteq S a genus g curve with divisibility k in mathrm {Pic}(S). In this article we study the forgetful map c_g^k:(S,C) mapsto C from mathcal {KC}_g ^k to {mathcal {M}}_g for k>1. First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when S is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending C in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether c_g^k dominates the locus in {mathcal {M}}_g of k-spin curves with the appropriate number of independent sections. We are able to do this only when S is a complete intersection, and obtain in these cases some classification results for spin curves.

Higher rank Clifford indices of curves on a K3 surface

Let (X, H) be a polarized K3 surface with mathrm {Pic}(X) = mathbb {Z}H, and let Cin |H| be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when gge r^2ge 4, the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree d(ge 5) smooth plane curve is d-4, which is the same as the Clifford index of the curve.

Geometry and dynamics on Riemann and K3 surfaces

Surfaces are some of the simplest yet geometrically rich manifolds. Geometric structures on surfaces illuminate their topology and are useful for studying dynamical systems on surfaces. We illustrate below how some of these concepts blend together, and relate them to algebraic geometry.

Geometry and dynamics on Riemann and K3 surfaces

Geometry and dynamics on Riemann and K3 surfaces

F-theory models with 3 to 8 U(1) factors on K3 surfaces

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell–Weil ranks of 3 to 8. We utilize the method of quadratic base change to glue pairs of rational elliptic surfaces together to yield the aforementioned types of K3 surfaces. The moduli of elliptic K3 surfaces constructed in the study include Kummer surfaces of specific complex structures. We show that the tadpole cancels in F-theory compactifications with flux when these Kummer surfaces are paired with appropriately selected attractive K3 surfaces. We determine the matter spectra on F-theory on the pairs.

On the Mumford–Tate conjecture for hyperkähler varieties

We study the Mumford–Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O’Grady’s ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford–Tate conjecture in every codimension under the assumption that the Künneth components in even degree of its André motive are abelian. Our results extend a theorem of André.

Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

Complex multiplication and Brauer groups of K3 surfaces

We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its idèles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field K and a CM number field E, returns a finite list of groups which contains Br(X‾)GK for any K3 surface X/K that has CM by the ring of integers of E. We run our algorithm when E is a quadratic imaginary field (a condition that translates into X having maximal Picard rank) generalizing similar computations already appearing in the literature.

AN ALGORITHM OF COMPUTING COHOMOLOGY INTERSECTION NUMBER OF HYPERGEOMETRIC INTEGRALS

AbstractWe show that the cohomology intersection number of a twisted Gauss–Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain family of K3 surfaces.

Moduli Space of Quasi-Polarized K3 Surfaces of Degree 6 and 8

In this paper, the authors 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 they have natural GIT constructions for the corresponding moduli spaces and they show that the K3 surfaces with at worst ADE singularities are GIT stable. They give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion. They compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga’s compactifications from arithmetic perspective. One of the main ingredients is the study of the projective models of K3 surfaces in terms of Noether-Lefschetz divisors.