Some results about holomorphic vector bundles over general Hopf manifolds (Q983663)

A general Hopf manifold of dimension \(n\geq2\) is a manifold \(M\) of the form \(M:=W/G\), where \(W={\mathbb C}^n\setminus\{0\}\) and \(G=\pi_1(M)\). The fundamental group \(G\) has an infinite cyclic subgroup \({\mathbb Z}\) contained in the centre of \(G\) such that \(G={\mathbb Z}.K\) with \(K\) finite. When \(K\) is trivial, the Hopf manifold \(M\) is called primary. The purpose of this note is to give a brief survey of some results extending the theory of holomorphic bundles over primary Hopf manifolds to the general case. The results are a version of the Douady sequence, an isomorphism and vanishing theorem for the cohomology of \(\Omega^p(E)\) and a theorem stating that, if \(n\geq3\), the pullback of a holomorphic vector bundle of rank \(r\) to \(W\) is trivial if and only if the bundle is filtrable by subbundles.