On holomorphic \(L^2\) functions on coverings of strongly pseudoconvex manifolds (Q945584)

Gromov, Henkin, and Shubin proved that if \(M\) is a domain with smooth boundary in a complex manifold and \(M^{\prime}:=M_{G}\) is a regular covering of \(M\) with a transformation group \(G\), then the space of square integrable holomorphic functions on \(M_{G}\) is `big': its von Neumann \(G\)-dimension is infinity, and each point of \(bM_{G}\) is a local peak point (for that space) [see \textit{M.~Gromov, G.~Henkin} and \textit{M.~Shubin}, Geom. Funct. Anal. 8, No.~3, 552--585 (1998; Zbl 0926.32011)]. These authors then asked whether the compact group action is really necessary for the existence of many square integrable holomorphic functions. In the paper under review, the author answers this question for coverings of strictly pseudoconvex domains. He shows in particular that the regularity of \(M^{\prime}\) is irrelevant for the existence of many holomorphic \(L^{2}\)-functions. He also obtains an extension of the result of Gromov-Henkin-Shubin cited above. The author's methods are different from those of GHS; they are based on \(L^{2}\)-cohomology techniques and on geometric properties of \(M\).