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L2-index theory, the Chern conjecture, and manifolds of special holonomy

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Hoffmann,  Michael
Max Planck Institute for Mathematics, Max Planck Society;

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Hoffmann, M. (2017). L2-index theory, the Chern conjecture, and manifolds of special holonomy. PhD Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn.


Cite as: https://hdl.handle.net/21.11116/0000-0004-0C94-8
Abstract
In this dissertation we study a conjecture attributed to Chern and Hopf, a generalization of Atiyah's L2-index theorem used by Gromov to prove said conjecture for Kähler manifolds, and an extension to Kähler orbifolds and all known examples of quaternionic Kähler manifolds. The study of a two-form η with values in a bundle G over a quaternionic Kähler manifold motivates us to show vanishing results for the cohomology of so5(C)-modules. We use these results to show that in certain degrees all G-valued d∇-closed forms are d∇-exact.