Nilmanifolds [Elektronische Ressource] : complex structures, geometry and deformations / Sönke Rollenske

Nilmanifolds: complex structures,geometry and deformationsVon der Universit¨at Bayreuthzur Erlangung des Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigte AbhandlungvonS¨onke Rollenskeaus Bonn1. Gutachter: Prof. Dr. Fabrizio Catanese2. Gutachter: Prof. Dr. Jo¨rg Winkelmann3. Gutachter: Prof. Dr. Simon SalamonTag der Einreichung: 31. Mai 2007Tag des Kolloquiums: 13. Juli 2007AbstractWe consider nilmanifolds with left-invariant complex structure and provethat in the generic case small deformations of such structures are againleft-invariant.The relation between nilmanifolds and iterated principal holomorphictorus bundles is clarified and we give criteria under which deformationsin the large are again of such type. As an application we obtain a fairlycomplete picture in dimension three.WeshowbyexamplethattheFr¨olicherspectralsequenceofanilmanifoldmay be arbitrarily non degenerate thereby answering a question mentionedin the book of Griffith and Harris.On our way we prove Serre Duality for Lie algebra Dolbeault cohomol-ogy and classify complex structures on nilpotent Lie algebras with smallcommutator subalgebra.MS Subject classification: 32G05; (32G08, 17B30, 53C30, 32C10)Erklarung¨Hiermit erkl¨are ich, dass ich die vorliegende Dissertation mit dem Titel Nil-manifolds: complex structures, geometry and deformations selbststa¨ndig an-gefertigt habe.