Holomorphic \(L^2\) functions on coverings of pseudoconvex manifolds
Let \(\widetilde M\) be a complex manifold, \(\overline {M}= M\cup bM\subset\widetilde M\) be a closed submanifold of \(\widetilde M\) with smooth boundary, \(\dim_{\mathbb C}\widetilde M = \dim_{\mathbb C}M\). Let \(\Gamma\) be a discrete group acting freely and holomorphically on \(\widetilde M\) such that \(\Gamma\) invariates \(\overline {M}\) and \(\overline {M}/\Gamma\) is compact. For a Hilbert \(\Gamma-\) module \(L\) one defines in the theory of von Neumann algebras the \(\Gamma-\) dimension \(\dim_{\Gamma}L\in [0,\infty]\). Choosing a hermitian \(\Gamma-\) invariant metric on \(\overline M\) one endows for any \(p,q\) the reduced \(L^2\) Dolbeault cohomology group \(L^2{\overline H}^{p,q}(M)\) with a structure of Hilbert \(\Gamma-\) module. The authors prove: (1) If \(M\) is strongly pseudoconvex, \(\dim_{\Gamma}L^2{\overline H}^{p,q}(M) < \infty\) for any \(p,q\) with \(q>0\). (2) If \(bM\neq \emptyset\) then \(\dim_{\Gamma}L^2{\mathcal O}(M)=\infty\) and any point in \(bM\) is a local peak point for \(L^2{\mathcal O}(M)\). (3) If \(bM\neq \emptyset\) then for any \(m\in{\mathbb N}, m>0\) there exists a \(\Gamma-\) invariant subspace \(L\subset L^2{\mathcal O}(M)\cap{\mathcal C}^{\infty}({\overline M})\) such that \(\dim_{\Gamma}{\overline L}=m\) where \(\overline L\) is the closure of \(L\) in \(L^2(M)\). An example of \(M\) when the results apply is obtained as follows: take \(X\) be a real analytic manifold with infinite \(\Gamma = \pi_1(X)\). Embed \(X\) into a complexification \(Y\) and choose on \(Y\) a riemannian metric. Let \(X_{\varepsilon}\) be an \(\varepsilon-\) neighbourhood of \(X\) in \(Y\) with \(\varepsilon\) small enough. Finally take \(M\) to be the universal covering of \(X_{\varepsilon}\). The authors study analogues of (1), (2), (3) above when \(M\) is weakly pseudoconvex or \(M\) is strongly pseudoconvex but \(bM\) is not necessarily smooth and also analogues of (1)--(3) for regular coverings of compact strongly pseudoconvex CR-manifolds. The proofs of all these results are based on extensions of \(L^2\) techniques, in particular of Kohn-Morrey estimates, also on refined versions of Hörmander and Andreotti-Vesentini weighted \(L^2\)-estimates.
scientific article; zbMATH DE number 1197117