A Remark on the Douady Sequence for Non-primary Hopf Manifolds

1. Introduction. To determine cohomology groups of holomorphic vector bundles or more general coherent analytic sheaves on complex manifolds is very important in several complex variables and complex geometry. For example, Cartan-Serre's theorem B and Kodaira's vanishing theorem are fundamental respectively in the studies of two classes of complex manifolds: Stein manifolds and projective algebraic manifolds. Deformation and moduli space of complex structures are also closely related to the determination of the cohomology groups of the bundles according to Kodaira-Spencer's theory. Some concrete calculations of cohomology groups on some well-known manifolds have been given. For example, cohomology groups of holomorphic line bundles (and more general holomorphic vector bundles with trivial pull back) on primary Hopf manifolds were calculated, where the Douady sequence (a short exact sequence of complexes, cf. [1,4,10,11,14]) plays a key role in the calculation. The purpose of the present note is to generalize the Douady sequence to the case of non-primary Hopf manifolds, so that the cohomology groups of a holomorphic line bundles (and more general holomorphic vector bundles with trivial pull back) over arbitrary Hopf manifolds could be calculated. This could be used to obtain a criterion for a continuous complex vector bundle over any Hopf manifold admitting a holomorphic structure and to study the filtrability of the holomorphic vector bundles over any Hopf manifold and the moduli space of any Hopf manifold.