Universität zu Köln

Gareis, Stephan (2015). L^2-Cohomology of Coverings of q-convex Manifolds and Stein Spaces. PhD thesis, Universität zu Köln.

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Abstract

We are interested in the $L^2$ cohomology groups of coverings of different geometric objects. \\ In the first part we are dealing with a non-compact manifold with boundary $X$ that admits a free, holomorphic and properly discontinuous group action of a discrete group $\Gamma$ such that the orbit space $\widetilde{X} = \quotient{X}{\Gamma}$ is a compact $q$-convex manifold with boundary. Assume furthermore that there is a holomorphic $\Gamma$-invariant holomorphic line bundle $E$. We show that the $\Gamma$-dimension of the $L^2$-cohomology groups $H^{0,j}_{(2)}(X,E)$ is finite if $j>q$. \\ In the second part we are dealing with infinite coverings of a relatively compact pseudoconvex domain $X$ in a normal Stein space with isolated singularities that are generated by a group action of a discrete group $\Gamma$. We assume that the group action is again free, holomorphic and properly discontinuous. We show that the space of $L^2$ holomorphic functions on $X$ has infinite $\Gamma$-dimension.