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The Hartogs extension problem for holomorphic parabolic and reductive geometries

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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.

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Authors and Affiliations

  1. School of Mathematical Sciences, University College Cork, Cork, Ireland

    Benjamin McKay

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  1. Benjamin McKay
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Correspondence to Benjamin McKay.

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Communicated by A. Constantin.

It is a pleasure to thank Sorin Dumitrescu and Filippo Bracci for helpful conversations on the problems solved in this paper, and for inviting me to the Laboratoire de Mathématiques J.A. Dieudonné at the University of Nice Sophia–Antipolis and to the University of Rome Tor Vergata where part of this work was carried out. This publication has emanated from activity conducted with the financial support of Science Foundation Ireland under the International Strategic Cooperation Award Grant Number SFI/13/ISCA/2844.

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McKay, B. The Hartogs extension problem for holomorphic parabolic and reductive geometries. Monatsh Math 181, 689–713 (2016). https://doi.org/10.1007/s00605-016-0955-4

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