Hodge classes on self-products of K3 surfaces [Elektronische Ressource] / vorgelegt von Ulrich Schlickewei

Hodge classes on self-products ofK3 surfacesDissertationzur Erlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen Fakult atderRheinischen Friedrich-Wilhelms-Universit at Bonnvorgelegt vonUlrich Schlickeweiaus Freiburg im BreisgauBonn 2009Angefertigt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult at der Rheinischen Friedrich-Wilhelms-Universit at Bonn1. Referent: Prof. Dr. D. Huybrechts2.nt: Prof. Dr. B. van GeemenTag der mundlic hen Prufung: 26. Juni 2009Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn unterhttp://hss.ulb.uni-bonn.de/diss online elektronisch publiziert.Erscheinungsjahr: 2009.SummaryThis thesis consists of four parts all of which deal with di erent aspects ofHodge classes on self-products of K3 surfaces.In the rst three parts we present three di erent strategies to tackle theHodge conjecture for self-products of K3 surfaces. The rst approach isof deformation theoretic nature. We prove that Grothendieck’s invariantcycle conjecture would imply the Hodge conjecture for self-products of K3surfaces. The second part is devoted to the study of the Kuga{Satake varietyassociated with a K3 surface with real multiplication. Building on work ofvan Geemen, we calculate the endomorphism algebra of this Abelian variety.This is used to prove the Hodge conjecture for self-products of K3 surfaces2which are double covers of P rami ed along six lines.