The Hartogs extension problem for holomorphic parabolic and reductive geometries

Abstract

Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold is the pullback of a unique such geometry on the envelope of holomorphy of the domain. We use this result to classify the Hopf manifolds which admit holomorphic reductive geometries, and to classify the Hopf manifolds which admit holomorphic parabolic geometries. Every Hopf manifold which admits a holomorphic parabolic geometry with a given model admits a flat one. We classify flat holomorphic parabolic geometries on Hopf manifolds. For every generalized flag manifold there is a Hopf manifold with a flat holomorphic parabolic geometry modelled on that generalized flag manifold.