Theory Of K3 Surfaces Research Articles

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.

Moduli spaces and Modular forms

The roots of both moduli spaces and modular forms go back to the theory of elliptic curves in the 19th century. Both topics have seen an enormous growth in the second half of the 20th century, but the interaction between the two remained limited. Recently there have been new developments that led to new points of contact between the two topics. One is the theory of K3 surfaces that is rapidly gaining a lot of new interest. Here the link with modular forms on orthogonal groups has led to progress on the Kodaira dimension of the moduli spaces of K3 surfaces. Another new development has been the use of moduli spaces of curves to gather new information about Siegel modular forms. The workshop intended to bring representatives from both the theory of moduli and the theory of modular forms together to further the interaction between the two topics as the time seemed ripe to do this.

Discrminantal groups and Zariski pairs of sextic curves

A series of Zariski pairs and three Zariski triplets were found by using lattice theory of K3 surfaces. Explicit equations for typical ones were calculated.

K3 surfaces with non-symplectic involution and compact irreducibleG2-manifolds

AbstractWe consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomyG2developed by the first named author. The method requires pairs of projective complex threefolds endowed with anticanonicalK3 divisors and the latterK3 surfaces should satisfy a certain ‘matching condition’ intertwining on their periods and Kähler classes. Suitable examples of threefolds were previously obtained by blowing up curves in Fano threefolds.In this paper, we give a large new class of suitable algebraic threefolds using theory ofK3 surfaces with non-symplectic involution due to Nikulin. These threefolds arenotobtainable from Fano threefolds as above, and admit matching pairs leading to topologically new examples of compact irreducibleG2-manifolds. ‘Geography’ of the values of Betti numbersb2,b3for the new (and previously known) examples of irreducibleG2manifolds is also discussed.

The Kähler cone of a compact hyperkähler manifold

This paper is a sequel to [11]. We study a number of questions only touched upon in [11] in more detail. In particular: What is the relation between two birational compact hyperkähler manifolds? What is the shape of the cone of all Kähler classes on such a manifold? How can the birational Kähler cone be described? Most of the results are motivated by either the well-established two-dimensional theory, i.e. the theory of K3 surfaces, or the theory of Calabi-Yau threefolds and string theory.

Every algebraic Kummer surface is the K3-cover of an Enriques surface

A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T. T is an abelian surface if and only if its associated Kummer surface is algebraic. Kummer surfaces are among classical examples of K3-surfaces (which are simply-connected smooth surfaces with a nowhere-vanishing holomorphic 2-form), and play a crucial role in the theory of K3-surfaces. In a sense, all Kummer surfaces (resp. algebraic Kummer surfaces) form a 4 (resp. 3)-dimensional subset in the 20 (resp. 19)-dimensional family of K3-surfaces (resp. algebraic K3 surfaces).

The cohomology of moduli spaces of K3 surfaces of degree 2 (I)

1.1. ONE of the most striking results in algebraic geometry in recent years was the global Torelli theorem for K3 surfaces. A K3 surface S is by definition a nonsingular complex compact analytic variety of dimension 2 whose first Betti number and first Chern class are both zero. When S is projective, there is a hyperplane section class L in the second cohomology H2(S, Z) of S such that the evaluation of the square L2 of this class against the fundamental cycle [SJ is always a positive even integer, known as the degree of the surface S. Since the work of Grothendieck and Mumford (see [6], [14]) it had been known for some time that K3 surfaces of a given degree d form a coarse moduli space K,. In 1971, PiatetskiShapiro and Shafarevich proved that the periods of K3 surfaces give rise to an injection of this moduli space K, into an arithmetic quotient space D/T,, where D is the bounded hermitian domain associated to SO(2, 19; [w) and F,, is an arithmetic subgroup in SO(2, 19; [w) (see [17]). Subsequent work by Kulikov and many others refined this theorem in various ways (see [2], [4], [I 11, [20], [21]). In particular, it was proven that the period map is an isomorphism K, z D/T, and hence there is a link between the theory of K3 surfaces and arithmetic subgroups Fd of SO(2, 19; @--analogous to the classical situation of the moduli space of elliptic curves and SL(2; Iw).