Transformation Groups

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




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.


Closed Orbit Stein Manifold Categorical Quotient Reductive Subgroup Principal Orbit 
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  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

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© Springer Science+Business Media New York 2016

Authors and Affiliations

    • 1
    • 2
    • 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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