Complex K3 Surface Research Articles

Constructing rational curves on K3 surfaces

We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.

Chow groups of K3 surfaces and spherical objects

We show that for a K3 surface X the finitely generated subring R (X) ⊂ \mathrm{CH}^* (X) introduced by Beauville and Voisin is preserved under derived equivalences. This is proved by analyzing Chern characters of spherical bundles (and complexes). As for a K3 surface X defined over a number field all spherical bundles on the complex K3 surface X_ℂ are defined over \bar ℚ , this is compatible with the Bloch–Beilinson conjecture. Besides the work of Beauville and Voisin [5], Lazarfeld’s result on Brill–Noether theory for curves in K3 surfaces [15] and the deformation theory developed in [12] are central for the discussion.

K3 surfaces with Picard rank 20

We determine all complex K3 surfaces with Picard rank 20 over ℚ. Here the Neron–Severi group has rank 20 and is generated by divisors which are defined over ℚ. Our proof uses modularity, the Artin–Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell–Weil rank 18 over ℚ is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Neron–Severi group of rank 20, but not necessarily generated by divisors over ℚ.

The modularity of K3 surfaces with non-symplectic group actions

We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type. We will establish the modularity of the Galois representations associated to them. Also we discuss mirror symmetry for these K3 surfaces in the sense of Dolgachev, and show that a mirror K3 surface exists with one exception.

Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic

Kuga and Satake associate with every polarized complex K3 surface (X, ℒ) a complex abelian variety called the Kuga-Satake abelian variety of (X, ℒ). We use this construction to define morphisms between moduli spaces of polarized K3 surfaces with certain level structures and moduli spaces of polarized abelian varieties with level structure over ℂ. In this note we study these morphisms. We prove first that they are defined over finite extensions of ℚ. This is done by proving analogues of the main theorems of complex multiplication for abelian varieties for K3 surfaces. Then we show that they extend in positive characteristic. In this way we give an indirect construction of Kuga-Satake abelian varieties over an arbitrary base. We also give some applications of this construction to canonical lifts of ordinary K3 surfaces.

The Chow Motive of a K3 Surface

In this note we present some results on the Chow motive h(X) of an algebraic surface X and relate them to the conjectures of Bloch, Beilinson and Murre. In particular we illustrate the relations between the finite-dimensionality of h(X) and the geometric properties of the surface. Then we focus on the case, where the conjectures are still open, of a complex K3 surface and prove some results which give some evidence to the finite-dimensionality of h(X).

Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic

We show that Mukai's classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic p under assumptions that (i) the order of the group is coprime to p and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without assumption (ii) we classify all possible new groups which may appear. We prove that assumption (i) on the order of the group is always satisfied if p > 11. For p = 2,3,5,11, we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by p which are not contained in Mukai's list.

K3 surfaces via almost-primes

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually non-isomorphic) projective complex K3 surfaces such that their Picard lattices are not isomorphic but their transcendental lattices are Hodge isometric, or equivalently, their derived categories are mutually equivalent. After reviewing some finiteness result, we also give an explicit formula for the cardinality of the isomorphism classes of projective K3 surfaces having derived categories equivalent to the one of X with Picard number 1 in terms of the degree of X.