Hodge classes on self-products of K3 surfaces

Abstract

This thesis consists of four parts which investigate different aspects of Hodge classes on self-products of K3 surfaces.<br /> In the first three parts we present three different strategies to tackle the Hodge conjecture for self-products of K3 surfaces. The first approach is of deformation theoretic nature. We prove that Grothendieck's invariant cycle conjecture would imply the Hodge conjecture for self-products of K3 surfaces. The second part is devoted to the study of the Kuga-Satake variety associated with a K3 surface with real multiplication. Building on work of van Geemen, we calculate the endomorphism algebra of this Abelian variety. This is used to prove the Hodge conjecture for self-products of K3 surfaces which are double covers of the projective plane ramified along six lines. In the third part we show that the Hodge conjecture for the self-product of a K3 surface is equivalent to the Hodge conjecture for the second Hilbert square. Motivated by this, we calculate some algebraic classes on the Hilbert scheme and on its deformations.<br /> The fourth part includes two additional results related with Hodge classes on self-products of K3 surfaces. The first one concerns K3 surfaces with complex multiplication. We prove that if a K3 surface S has complex multiplication by a CM field E and if the dimension of the transcendental lattice of S over E is one, then S is defined over an algebraic number field. This result was obtained previously by Piatetski-Shapiro and Shafarevich but our method is different. The second additional result says that the Andre motive of an irreducible symplectic variety which is deformation equivalent to the Hilbert scheme of a K3 surface is an object of the category generated by its motive in degree 2.