Derived categories and scalar extensions [Elektronische Ressource] / vorgelegt von Pawel Sosna

Derived categories and scalarextensionsDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen Fakult¨atderRheinischen Friedrich-Wilhelms-Universit¨at Bonnvorgelegt vonPawel SosnaausOdessa, UkraineBonn 2010Angefertigt mit Genehmigung der Mathematisch-NaturwissenschaftlichenFakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn1. Gutachter: Prof. Dr. Daniel Huybrechts2. Gutachter: Prof. Dr. Jan Schr¨oerTag der mu¨ndlichen Pru¨fung: 4. November 2010Erscheinungsjahr: 2010SummaryThis thesis consists of three parts all of which deal with questions related toscalar extensions and derived categories.In the first part we consider the question whether the conjugation of a com-plex projective K3 surfaceX by an automorphism of the complex numbers canproduce a non-isomorphic Fourier–Mukai partner of X. The answer is affir-mative. The conjugate surface is thus in particular a moduli space of locallyfree sheaves onX. The proof consists of constructing non-isomorphic conjugatederived equivalent K3 surfaces over an extension field ofQ and then lifting thesituation to the complex numbers. We use our result to give higher-dimensionalexamples of derived equivalent conjugate varieties. We furthermore prove thata similar result holds for abelian surfaces.The topic of the second part is the behaviour of stability conditions underscalar extensions.