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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Homotopy principles for equivariant isomorphisms
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by Frank Kutzschebauch, Finnur Lárusson and Gerald W. Schwarz PDF
Trans. Amer. Math. Soc. 369 (2017), 7251-7300 Request permission

Abstract:

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $\varphi$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $\varphi$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $\Phi \colon X\to Y$, inducing $\varphi$, which is $G$-biholomorphic on the reduced fibres of the quotient mappings, then $\Phi$ is homotopic, through $G$-diffeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. The second result roughly says that if we have a $G$-homeomorphism $\Phi \colon X\to Y$ which induces a continuous family of $G$-equivariant biholomorphisms of the fibres $p_X{^{-1}}(q)$ and $p_Y{^{-1}}(\varphi (q))$ for $q\in Q_X$ and if $X$ satisfies an auxiliary property (which holds for most $X$), then $\Phi$ is homotopic, through $G$-homeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. Our results improve upon those of our earlier paper [J. Reine Angew. Math. 706 (2015), 193–214] and use new ideas and techniques.
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Additional Information
  • Frank Kutzschebauch
  • Affiliation: Institute of Mathematics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
  • MR Author ID: 330461
  • Email: frank.kutzschebauch@math.unibe.ch
  • Finnur Lárusson
  • Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
  • MR Author ID: 347171
  • Email: finnur.larusson@adelaide.edu.au
  • Gerald W. Schwarz
  • Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110
  • MR Author ID: 157450
  • Email: schwarz@brandeis.edu
  • Received by editor(s): April 23, 2015
  • Received by editor(s) in revised form: January 13, 2016
  • Published electronically: May 5, 2017
  • Additional Notes: The first author was partially supported by Schweizerischer Nationalfond grant 200021-140235/1
    The first and third authors thank the University of Adelaide for hospitality and the Australian Research Council for financial support
    The second author was partially supported by Australian Research Council grants DP120104110 and DP150103442
    The second and third authors thank the University of Bern for hospitality and financial support
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 7251-7300
  • MSC (2010): Primary 32M05; Secondary 14L24, 14L30, 32E10, 32M17, 32Q28
  • DOI: https://doi.org/10.1090/tran/6797
  • MathSciNet review: 3683109