Abstract
Let G be a reductive complex Lie group acting holomorphically on X = ℂn. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂn such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: X → V 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 ℂn, only that X is a Stein manifold.
Similar content being viewed by others
References
H. Derksen, F. Kutzschebauch, Nonlinearizable holomorphic group actions, Math. Ann. 311 (1998), no. 1, 41-53.
P. Heinzner, F. Kutzschebauch, An equivariant version of Grauert’s Oka principle, Invent. Math. 119 (1995), no. 2, 317-346.
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.
M. Jiang, On the Holomorphic Linearization and Equivariant Serre Problem, PhD thesis, Brandeis University, 1992.
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.
F. Kutzschebauch, F. Làrusson, G. W. Schwarz, Homotopy principles for equivariant isomorphisms, preprint, arXiv:1503.00797 (2015).
F. Kutzschebauch, F. Làrusson, G. W. Schwarz, An Oka principle for equivariant isomorphisms, J. reine angew. Math. 706 (2015), 193-214.
M. Koras, P. Russell, Linearization problems, in: Algebraic Group Actions and Quotients, Hindawi Publ. Corp., Cairo, 2004, pp. 91-107.
H. Kraft, P. Russell, Families of group actions, generic isotriviality, and linearization, Transform. Groups 19 (2014), no. 3, 779-792.
H. Kraft, Challenging problems on affine n-space, in: Séminaire Bourbaki, Vol. 1994/95, Astérisque 237 (1996), Exp. No. 802, 5, pp. 295-317.
D. Luna, Slices étales, Bull. Soc. Math. France, Mém. 33 (1973), 81-105.
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.
M. Roberts, A note on coherent G-sheaves, Math. Ann. 275 (1986), no. 4, 573-582.
G. W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. (1980), no. 51, 37-135.
G. W. Schwarz, Exotic algebraic group actions, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 2, 89-94.
G. W. Schwarz, Lifting diffierential operators from orbit spaces, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 3, 253-305.
G. W. Schwarz, Vector fields and Luna strata, J. Pure Appl. Algebra 217 (2013), 54-58.
G. W. Schwarz, Quotients, automorphisms and differential operators, J. Lond. Math. Soc. (2) 89 (2014), no. 1, 169-193.
D. M. Snow, Reductive group actions on Stein spaces, Math. Ann. 259 (1982), no. 1, 79-97.
Author information
Authors and Affiliations
Corresponding author
Additional information
(FRANK KUTZSCHEBAUCH) Partially supported by Schweizerischer Nationalfond grants 140235 and 153120, thanks the University of Adelaide for hospitality and the Australian Research Council for financial support.
(FINNUR LÁRUSSON) Partially supported by Australian Research Council grants DP120104110 and DP150103442, thanks the University of Bern for hospitality and financial support.
(GERALD W. SCHWARZ) Thanks the University of Bern for hospitality and financial support, the University of Adelaide for hospitality and the Australian Research Council for financial support.
Rights and permissions
About this article
Cite this article
KUTZSCHEBAUCH, F., LÁRUSSON, F. & SCHWARZ, G.W. SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION. Transformation Groups 22, 475–485 (2017). https://doi.org/10.1007/s00031-016-9376-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-016-9376-7