Euler characteristics and geometric properties of quiver Grassmannians [Elektronische Ressource] / Nicolas Haupt. Mathematisch-Naturwissenschaftliche Fakultät

Euler characteristicsand geometric propertiesof quiver GrassmanniansDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen Fakult atderRheinischen Friedrich-Wilhelms-Universit at Bonnvorgelegt vonNicolas HauptausGeorgsmarienhutte, DeutschlandBonn, Mai 2011Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult at derRheinischen Friedrich-Wilhelms-Universit at BonnErstgutachter: Prof. Dr. Jan Schr oerZweitgutachter: Priv.-Doz. Dr. Igor BurbanTag der Promotion: 27. September 2011Erscheinungsjahr: 201134AbstractLet k be an algebraically closed eld, Q a nite quiver and M a nite-dimensionalQ-representation. The quiver Grassmannian Gr (M) is the projective variety of sub-drepresentations of M with dimension vector d.Quiver Grassmannians occur naturally in di erent contexts. Fomin and Zelevinsky in-troduced cluster algebras in 2000. Caldero and Keller used Euler characteristics of quiverGrassmannians for the categori cation of acyclic cluster algebras. This was generalizedto arbitrary antisymmetric cluster algebras by Derksen, Weyman and Zelevinsky. Thequiver Grassmannians play a crucial role in the construction of Ringel-Hall algebras.Moreover, they arise in the study of general representations of quivers by Scho eld andin the theory of local models of Shimura varieties.