The Hartogs extension phenomenon for holomorphic parabolic and reductive geometries

TL;DR

This paper proves that holomorphic parabolic and reductive geometries on domains over Stein manifolds extend uniquely to their envelopes of holomorphy, and classifies Hopf manifolds admitting such geometries.

Contribution

It completes the open problem of extension of holomorphic geometric structures and classifies Hopf manifolds with holomorphic parabolic or reductive geometries.

Findings

Holomorphic geometries extend uniquely to envelopes of holomorphy.

Hopf manifolds admitting these geometries are classified.

Flat geometries are sufficient for manifolds admitting parabolic geometries.

Abstract

Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold extends uniquely to the envelope of holomorphy of the domain. This result completes the open problems of my earlier paper on extension of holomorphic geometric structures on complex manifolds. 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.