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Gromov’s Oka Principle for Equivariant Maps

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Abstract

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group G acts on a Stein manifold X and another manifold Y in such a way that Y is G-Oka, then every G-equivariant continuous map \(X\rightarrow Y\) can be deformed, through such maps, to a G-equivariant holomorphic map. Approximation on a G-invariant holomorphically convex compact subset of X and jet interpolation along a G-invariant subvariety of X can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.

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Notes

  1. A G-module is a finite-dimensional vector space with an action of G by linear maps.

  2. A set is saturated if it is the union of fibres of the categorical quotient map \(Z\rightarrow Z/\!\!/G\).

  3. For subsets of a set with a group action, we use the terms invariant and stable interchangeably.

  4. We use the terms isotropy group and stabiliser interchangeably.

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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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Correspondence to Finnur Lárusson.

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F. Kutzschebauch was supported by Schweizerischer Nationalfonds Grant 200021-178730. F. Lárusson was supported by Australian Research Council Grant DP150103442.

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Kutzschebauch, F., Lárusson, F. & Schwarz, G.W. Gromov’s Oka Principle for Equivariant Maps. J Geom Anal 31, 6102–6127 (2021). https://doi.org/10.1007/s12220-020-00520-0

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