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An equivariant parametric Oka principle for bundles of homogeneous spaces

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Abstract

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle E with a structure group bundle \({{\mathscr {G}}}\) on a reduced Stein space X, such that the fibre of E is a homogeneous space of the fibre of \({{\mathscr {G}}}\), with the complexification \(K^{{\mathbb {C}}}\) of a compact real Lie group K acting on X, \({{\mathscr {G}}}\), and E. Our main result is that the inclusion of the space of \(K^{{\mathbb {C}}} \hbox {-equivariant}\) holomorphic sections of E over X into the space of \(K\hbox {-equivariant}\) continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch’s classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.

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Acknowledgement

We thank Michael Murray for help with the theory of generalised principal bundles.

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Authors and Affiliations

  1. Institute of Mathematics, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Frank Kutzschebauch

  2. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, 5005, Australia

    Finnur Lárusson

  3. Department of Mathematics, Brandeis University, Waltham, MA, 02454-9110, USA

    Gerald W. Schwarz

Authors
  1. Frank Kutzschebauch
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  2. Finnur Lárusson
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  3. Gerald W. Schwarz
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Corresponding author

Correspondence to Finnur Lárusson.

Additional information

Communicated by Ngaiming Mok.

Dedicated to Peter Heinzner on the occasion of his sixtieth birthday.

F. Kutzschebauch was partially supported by Schweizerischer Nationalfond Grant 200021-140235/1. F. Lárusson was partially supported by Australian Research Council Grant DP150103442. Much of the work on this paper was done at the Centre for Advanced Study at the Norwegian Academy of Science and Letters. The authors would like to warmly thank the Centre for hospitality and financial support. F. Kutzschebauch and G. W. Schwarz would also like to thank the University of Adelaide for hospitality and the Australian Research Council for financial support.

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Kutzschebauch, F., Lárusson, F. & Schwarz, G.W. An equivariant parametric Oka principle for bundles of homogeneous spaces. Math. Ann. 370, 819–839 (2018). https://doi.org/10.1007/s00208-017-1588-1

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