A tuple of multiplication operators defined by twisted holomorphic proper maps

This paper mainly concerns the von Neumann algebras induced by a tuple of multiplication operators on Bergman spaces which arise essentially from holomorphic proper maps over higher dimensional domains. We study the structures and abelian properties of the related von Neumann algebras, and in interesting cases they turns out to be tightly related to a Riemann manifold. There is a close interplay between operator theory, geometry and complex analysis. Many examples are presented.