Real multiplication on K3 surfaces and Kuga Satake varieties

The endomorphism algebra of a K3 type Hodge structure is a totally real field or a CM field. In this paper we give a low brow introduction to the case of a totally real field. We give existence results for the Hodge structures, for their polarizations and for certain K3 surfaces. We consider the Kuga Satake variety of these Hodge structures and we discuss some examples. Finally we indicate various open problems related to the Hodge conjecture. A K3 type Hodge structure is a simple, rational, polarized weight two Hodge structure V with dimV 2,0 = 1. Zarhin [Z] proved that the endomorphism algebra of a K3 type Hodge structure is either a totally real field or a CM field. Conversely, K3 type Hodge structures whose endomorphism algebra is a given such field exist under fairly obvious conditions. For the totally real case, see Lemma 3.2. Similar to the case of abelian varieties and their polarized weight one Hodge structures, given a polarization and a totally positive endomorphism, one can define a new polarization (see Lemma 4.2). For a polarized abelian variety, this follows from the well-known relation between the Rosati invariant endomorphisms and the Néron Severi group. In case the K3 type Hodge structure is a Hodge substructure of the H of a smooth surface, it comes with a natural polarization induced by the cupproduct. It is then interesting to consider whether the polarization obtained by means of a totally real element a is also realized as the natural polarization for some other surface Sa. Thus H (Sa) has a Hodge substructure isomorphic to the original one, but the isomorphism does not preserve the natural polarizations. It follows easily from general results on K3 surfaces that, under a condition on the dimension of the Hodge structure, such K3 surfaces do exist (see section 4.7). The isomorphism of Hodge substructures, in combination with the Hodge conjecture, then leads one to wonder whether there is an algebraic cycle realizing the isomorphism. We discuss some aspects of this question in ∗Università di Milano, Dipartimento di Matematica, via Saldini 50, I-20133 Milano, Italia. e-mail: [email protected]