<em>L</em>²-index theory, the Chern conjecture, and manifolds of special holonomy

Abstract

In this dissertation we study a conjecture attributed to Chern and Hopf, a generalization of Atiyah's <em>L</em>²-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.