Abstract
We generalize the notion of the density property for complex manifolds to holomorphic fibrations, and introduce the notion of the fibred density property. We prove that the natural fibration of the spectral ball over the symmetrized polydisc enjoys the fibred density property and describe the automorphism group of the spectral ball.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, R., Marsden, J.E., Raţiu, T.: Manifolds, Tensor Analysis, and Applications, Global Analysis Pure and Applied: Series B. Addison-Wesley Publishing Co., Reading (1983)
Andersén, E.: Volume-preserving automorphisms of \({ C}^n\). Complex Var. Theory Appl. 14(1–4), 223–235 (1990)
Andersén, E., Lempert, L.: On the group of holomorphic automorphisms of \({ C}^n\). Invent. Math. 110(2), 371–388 (1992). doi:10.1007/BF01231337
Baribeau, L., Ransford, T.: Non-linear spectrum-preserving maps. Bull. Lond. Math. Soc. 32(1), 8–14 (2000). doi:10.1112/S0024609399006426
Bercovici, H., Foias, C., Tannenbaum, A.: Spectral radius interpolation and robust control. In: Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1–3. IEEE, New York (1989), pp. 916–917
Bercovici, H., Foias, C., Tannenbaum, A.: A spectral commutant lifting theorem. Trans. Am. Math. Soc. 325(2), 741–763 (1991). doi:10.2307/2001646
Dixmier, J.: Champs de vecteurs adjoints sur les groupes et algèbres de Lie semi-simples. J. Rein. Angew. Math. 309, 183–190 (1979). doi:10.1515/crll.1979.309.183
Donzelli, F., Dvorsky, A., Kaliman, S.: Algebraic density property of homogeneous spaces. Transform. Groups 15(3), 551–576 (2010). doi:10.1007/s00031-010-9091-8
Forstnerič, F.: Stein Manifolds and Holomorphic Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Vol. 56. Springer, Heidelberg (2011)
Forstneric, F., Rosay, J.-P.: Erratum: Approximation of biholomorphic mappings by automorphisms of C\(^{n}\). Invent. Math. 118(3), 573–574 (1994). doi:10.1007/BF01231544
Forstneric, F., Rosay, J.-P.: Approximation of biholomorphic mappings by automorphisms of C\(^{n}\). Invent. Math. 112(2), 323–349 (1993). doi:10.1007/BF01232438
Kaliman, S., Kutzschebauch, F.: On the present state of the Andersén–Lempert theory. In: Affine Algebraic Geometry, CRM Proc. Lecture Notes, Vol. 54, Am. Math. Soc., Providence (2011), pp. 85–122
Kaliman, S., Kutzschebauch, F.: On algebraic volume density property. Transform. Groups 21(2), 451–478 (2016). doi:10.1007/s00031-015-9360-7
Kutzschebauch, F.: Andersén–Lempert-theory with parameters: a representation theoretic point of view. J. Algebra Appl. 4(3), 325–340 (2005). doi:10.1142/S0219498805001216
Kosiński, L.: The group of automorphisms of the spectral ball. Proc. Am. Math. Soc. 140(6), 2029–2031 (2012). doi:10.1090/S0002-9939-2011-11064-5
Kosiński, L.: Structure of the group of automorphisms of the spectral \(2\)-ball. Collect. Math. 64(2), 175–184 (2013). doi:10.1007/s13348-012-0079-7
Kostant, B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)
Kraft, H.: Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig (1984)
Pietsch, A.: Nuclear Locally Convex Spaces. Springer, New York (1972)
Ransford, T.J., White, M.C.: Holomorphic self-maps of the spectral unit ball. Bull. Lond. Math. Soc. 23(3), 256–262 (1991). doi:10.1112/blms/23.3.256
Rostand, J.: On the automorphisms of the spectral unit ball. Stud. Math. 155(3), 207–230 (2003). doi:10.4064/sm155-3-2
Thomas, P.J.: A local form for the automorphisms of the spectral unit ball. Collect. Math. 59(3), 321–324 (2008). doi:10.1007/BF03191190
Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Dover Publications, Inc., Mineola (2006)
Varolin, D.: A general notion of shears, and applications. Mich. Math. J. 46(3), 533–553 (1999). doi:10.1307/mmj/1030132478
Varolin, D.: The density property for complex manifolds and geometric structures. II. Int. J. Math. 11(6), 837–847 (2000). doi:10.1142/S0129167X00000404
Varolin, D.: The density property for complex manifolds and geometric structures. J. Geom. Anal. 11(1), 135–160 (2001). doi:10.1007/BF02921959
Acknowledgements
The authors would like to thank the referee for recommendations to improve the presentation and for pointing out some problems in the first version of this article. Moreover they would like to thank A. Liendo, P.-M. Poloni, S. Maubach, A. van den Essen, H. Derksen and A. Nowicki for discussions about determining the dimension growth of the kernel of a homogeneous derivation and finally again the referee for the contribution of the Proposition 5.2 and its proof which simplified the calculations.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Andrist, R.B., Kutzschebauch, F. The fibred density property and the automorphism group of the spectral ball. Math. Ann. 370, 917–936 (2018). https://doi.org/10.1007/s00208-017-1520-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-017-1520-8