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Transformation Groups

, Volume 22, Issue 2, pp 475–485 | Cite as

SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION

  • FRANK KUTZSCHEBAUCH
  • FINNUR LÁRUSSON
  • GERALD W. SCHWARZ
Article

Abstract

Let G be a reductive complex Lie group acting holomorphically on X = ℂ n . The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂ n such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: XV where V is a G-module? There is an intrinsic stratification of the categorical quotient Q X , called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: Q X Q V which is stratified, i.e., the stratum of Q X with a given label is sent isomorphically to the stratum of Q V with the same label.

The counterexamples to the Linearisation Problem construct an action of G such that Q X is not stratified biholomorphic to any Q V .Our main theorem shows that, for most X, a stratified biholomorphism of Q X to some Q V is sufficient for linearisation. In fact, we do not have to assume that X is biholomorphic to ℂ n , only that X is a Stein manifold.

Keywords

Closed Orbit Stein Manifold Categorical Quotient Reductive Subgroup Principal Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [DK98]
    H. Derksen, F. Kutzschebauch, Nonlinearizable holomorphic group actions, Math. Ann. 311 (1998), no. 1, 41-53.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [HK95]
    P. Heinzner, F. Kutzschebauch, An equivariant version of Grauert’s Oka principle, Invent. Math. 119 (1995), no. 2, 317-346.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Huc90]
    A. T. Huckleberry, Actions of groups of holomorphic transformations, in: Several complex variables VI, Encyclopaedia Math. Sci., Vol. 69, Springer, Berlin, 1990, pp. 143-196.Google Scholar
  4. [Jia92]
    M. Jiang, On the Holomorphic Linearization and Equivariant Serre Problem, PhD thesis, Brandeis University, 1992.Google Scholar
  5. [KLM03]
    A. Kriegl, M. Losik, P. W. Michor, Tensor fields and connections on holomorphic orbit spaces of finite groups, J. Lie Theory 13 (2003), no. 2, 519-534.MathSciNetzbMATHGoogle Scholar
  6. [KLS]
    F. Kutzschebauch, F. Làrusson, G. W. Schwarz, Homotopy principles for equivariant isomorphisms, preprint, arXiv:1503.00797 (2015).Google Scholar
  7. [KLS15]
    F. Kutzschebauch, F. Làrusson, G. W. Schwarz, An Oka principle for equivariant isomorphisms, J. reine angew. Math. 706 (2015), 193-214.MathSciNetzbMATHGoogle Scholar
  8. [KR04]
    M. Koras, P. Russell, Linearization problems, in: Algebraic Group Actions and Quotients, Hindawi Publ. Corp., Cairo, 2004, pp. 91-107.zbMATHGoogle Scholar
  9. [KR14]
    H. Kraft, P. Russell, Families of group actions, generic isotriviality, and linearization, Transform. Groups 19 (2014), no. 3, 779-792.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Kra96]
    H. Kraft, Challenging problems on affine n-space, in: Séminaire Bourbaki, Vol. 1994/95, Astérisque 237 (1996), Exp. No. 802, 5, pp. 295-317.Google Scholar
  11. [Lun73]
    D. Luna, Slices étales, Bull. Soc. Math. France, Mém. 33 (1973), 81-105.Google Scholar
  12. [Lya83]
    О. В. Ляшко, Геометрия бифуркационных диаграмм, Итоги науки и техники, Современные проблемы математики, т. 22, ВИНИТИ, М. 1983, стр. 94-129. Engl. transl.: O. V. Lyashko, Geometry of bifurcation diagrams, J. Soviet Math. 27 (1984), no. 3, 2736-2759.Google Scholar
  13. [Rob86]
    M. Roberts, A note on coherent G-sheaves, Math. Ann. 275 (1986), no. 4, 573-582.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Sch80]
    G. W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. (1980), no. 51, 37-135.Google Scholar
  15. [Sch89]
    G. W. Schwarz, Exotic algebraic group actions, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 2, 89-94.MathSciNetzbMATHGoogle Scholar
  16. [Sch95]
    G. W. Schwarz, Lifting diffierential operators from orbit spaces, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 3, 253-305.Google Scholar
  17. [Sch13]
    G. W. Schwarz, Vector fields and Luna strata, J. Pure Appl. Algebra 217 (2013), 54-58.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [Sch14]
    G. W. Schwarz, Quotients, automorphisms and differential operators, J. Lond. Math. Soc. (2) 89 (2014), no. 1, 169-193.Google Scholar
  19. [Sno82]
    D. M. Snow, Reductive group actions on Stein spaces, Math. Ann. 259 (1982), no. 1, 79-97.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • FRANK KUTZSCHEBAUCH
    • 1
  • FINNUR LÁRUSSON
    • 2
  • GERALD W. SCHWARZ
    • 3
  1. 1.Institute of MathematicsUniversity of BernBernSwitzerland
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  3. 3.Department of MathematicsBrandeis UniversityWalthamUSA

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