Mirror symmetry and stability conditions on K3 surfaces [Elektronische Ressource] / Heinrich Hartmann

Mirror symmetry andstability conditionson K3 surfacesDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen Fakult¨atderRheinischen Friedrich-Wilhelms-Universit¨at Bonnvorgelegt vonHeinrich HartmannausMainz, DeutschlandBonn 2011Angefertigt mit Genehmigung der Mathematisch-NaturwissenschaftlichenFakult¨ at der Rheinischen Friedrich-Wilhelms-Universit¨ at Bonn1. Gutachter: Prof. Dr. Daniel Huybrechts2.hter: Prof. Dr. Emanuele Macr`ıTag der mundlic¨ hen Prufung:¨ 17. Mai 2011Erscheinungsjahr: 2011SummaryThis thesis is concerned with questions arising in the realm of mirror symmetryfor K3 surfaces. It is divided into two parts.In the first part, we study the bijection between Fourier–Mukai partners ofa K3 surface X and cusps of the K¨ ahler moduli space which was establishedby Shouhei Ma in [Ma09]. The K¨ ahler moduli space can be described as aquotient of Bridgeland’s manifold of stability conditions on the derived categorybof coherent sheavesD (X). We relate stability conditions σ near to a cusp andthe associated Fourier–Mukai partner Y in the following ways.1. We construct a special path of stability conditionsσ(t) such that the heartsconverge to the heart Coh(Y ) of coherent sheaves on Y .2. We study a class of geodesic degenerations of stability conditionsσ(t) andshow that the hearts of σ(t) are related to Coh(Y ) by a tilt.3. We construct Y as moduli space of σ-stable objects.