Moduli of K3 Surfaces and Abelian Varieties

Abstract

In order to study a class of algebraic varieties one constructs moduli spaces. A point of a moduli space corresponds to an isomorphism class of the varieties we want to study. In this thesis we consider K3 surfaces. Over C moduli spaces of primitively polarized K3 surfaces are constructed as open subspaces of the Shimura variety associated with SO(2, 19). In Chapter 1 we construct these spaces over Z using techniques developed by Grothendieck, Mumford, Artin and others.

Chapter 2 is devoted to K3 surfaces in positive characteristic. We consider the following problem: For a given natural number d and a prime number p determine all Newton polygons of K3 surfaces with a polarization of degree 2d over a field of characteristic p. This is an analogue of the Manin problem for Newton polygons of abelian varieties. Constructing ample line bundles of appropriate degree on Kummer surfaces we show that for any large enough d and any p, prime to 2d, the Manin problem for <K3 surfaces with a polarization of degree 2d has an affirmative answer. The proof is constructive and gives a bound for d.

Let A be an abelian variety with complex multiplication over C. The main theorem of complex multiplication of Shimura and Taniyama describes the action on the torsion points of A of the automorphisms in Gal(Qab/Q) fixing the reflex field of A. P. Deligne uses this description to define canonical models of Shimura varieties and proves thatAg,1,n,Q is the canonical model of Sh(CSp2g,H±)C. In Chapter 3 we prove a similar result for moduli spaces of primitively polarized K3 surfaces. More precisely, for a certain class of compact open subgroups K of SO(2, 19)(Aƒ) we define a period morphism jd,K,C from the moduli space complex of <K3 surfaces with a primitive polarization of degree 2d and a level K-structure to the Shimura variety associated with SO(V2d,ψ2d). Our main result is that jd,K,C is defined over Q. As a corollary we obtain an analogue of the main theorem of Shimura and Taniyama for <K3 surfaces with complex multiplication.

M. Kuga and I. Satake associate to every complex polarized K3 surface (X,L) and abelian variety A. The variety A is called the Kuga-Satake abelian variety of (X,L). The construction uses Hodge theory. In Chapter 4 we use this construction to define, for a d ε N and n ε N, n > 3, a Kuga-Satake morphism from the moduli space of primitively polarized K3 surfaces of degree 2d with a "spin level n-structure" to the moduli space abelian varieties with a polarization of degree d'2 and a level n-structure. This morphism is defined over a finite abelian extension En of Q and maps every complex K3 surface with a polarization and a spin level n-structure to its Kuga-Satake abelian variety with extra structure (a polarization and a level structure). Our main result is that the Kuga-Satake morphism extends over an open part Spec(OEn[1/N]) of Spec(OEn), where OEn is the ring of integers in En and N is a natural number explicitly depending on d, d', n and En. In this way we define indirectly Kuga-Satake abelian varieties in positive characteristic.