Advertisement

Actions of Groups of Holomorphic Transformations

  • Alan T. Huckleberry
Chapter

Abstract

This paper primarily deals with three topics: Classification results for homogeneous and almost homogeneous spaces, complex analytic questions on homogeneous spaces, and certain types of actions, e.g. of compact Lie groups on complex spaces. Our goal here is to indicate our own current view of these areas, as opposed to presenting a comprehensive survey.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ch]
    Chevalley, C.: Théorie des groupes de Lie II: Groupes algébriques. Hermann, Paris 1951zbMATHGoogle Scholar
  2. [deM]
    de Mailly, J.-P.: Un example de fibré holomorphe non de Stein à fibre C ayant pour base le disque ou le plan. Invent. math. 48 (1978) 293–302CrossRefMathSciNetGoogle Scholar
  3. [DN]
    Dorfmeister, J., Nakajima, K.: The fundamental conjecture for homogeneous Kähler manifolds. Acta math. 161 (1988) 23–70CrossRefzbMATHMathSciNetGoogle Scholar
  4. [ES]
    Erdmann-Snow, J.: On the classification of solv-manifolds in dimension 2 and 3. 10. Revue de l’Institut Elie Cartan, Nancy, 57–103 (1986)Google Scholar
  5. [Feld]
    Feldmüller, D.: Two-orbit varieties with smaller orbit of codimension two. Arch. Math. 54, no. 6 (1990) 582–593CrossRefzbMATHMathSciNetGoogle Scholar
  6. [Fu]
    Fujiki, A.: On automorphism groups of compact Kähler manifolds. Invent. math. 44 (1978) 225–258CrossRefzbMATHMathSciNetGoogle Scholar
  7. [G]
    Goodman, R.: Analytic and entire vectors for representations of Lie groups. Trans. A.M.S 143 (1969) 55–76CrossRefzbMATHGoogle Scholar
  8. [Gal]
    Galemann, B.: Beschränkte Gebiete über C2 mit holomorphen S1-Wirkungen. Diplomarbeit, Ruhr-Universität Bochum 1989Google Scholar
  9. [Gell]
    Gellhaus, C.: Äquivariante Kompaktifizierungen des Cn. Math. Z. 206, no. 2 (1991) 211–217CrossRefzbMATHMathSciNetGoogle Scholar
  10. [GH1]
    Gilligan, B., Huckleberry, A.T.: Complex homogeneous manifolds with two ends. Mich. J. Math. 28 (1981) 183–196CrossRefzbMATHMathSciNetGoogle Scholar
  11. [GH 2]
    Gilligan, B., Huckleberry, AT.: On non-compact complex nilmanifolds. Math. Ann. 238 (1978) 39–49CrossRefzbMATHMathSciNetGoogle Scholar
  12. [Gi 1]
    Gilligan, B.: On bounded holomorphic reductions of homogeneous spaces. C.R. Math. Rep. Acad. Sci. Canada, vol. VI (1984) 175–178MathSciNetGoogle Scholar
  13. [Gi 2]
    Gilligan, B.: Ends of complex homogeneous manifolds having non-constant holomorphic functions. Arch. Math. 37 (1981) 544–555CrossRefzbMATHMathSciNetGoogle Scholar
  14. [GOR]
    Gilligan, B., Oeljeklaus, K., Richthofer, W.: Homogeneous complex manifolds with more than one end. Can. J. Math. vol. XLI, No. 1 (1989) 163–177CrossRefMathSciNetGoogle Scholar
  15. [GR1]
    Grauert, H., Remmert, R.: Über kompakte homogene komplexe Mannigfaltigkeiten. Arch. Math. 13 (1962) 498–507CrossRefzbMATHMathSciNetGoogle Scholar
  16. [GR 2]
    Grauert, H., Remmert, R.: Komplexe Räume. Math. Ann. 136 (1958) 245–318CrossRefzbMATHMathSciNetGoogle Scholar
  17. [GS]
    Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge Univ. Press 1984zbMATHGoogle Scholar
  18. [Hl]
    Huckleberry, AT.: Hypersurfaces in homogeneous spaces. Journées Complexes, Nancy, Institut Elie Cartan (1982)Google Scholar
  19. [Har]
    Harish-Chandra: Discrete series for semi-simple Lie groups II. Acta Math. 116 (1966) 1–111CrossRefzbMATHMathSciNetGoogle Scholar
  20. [Hei 1]
    Heinzner, P.: Fixpunktmengen kompakter Gruppen in Teilgebieten Steinscher Mannigfaltigkeiten. J. reine angew. Math. 402 (1989) 128–137zbMATHMathSciNetGoogle Scholar
  21. [Hei 2]
    Heinzner, P.: Kompakte Transformationsgruppen Steinscher Räume. Math. Ann. 285 (1989) 13–28CrossRefzbMATHMathSciNetGoogle Scholar
  22. [Hei 3]
    Heinzner, P.: Invariantentheorie in der komplexen Analysis. Habilitationsarbeit, Ruhr-Universität Bochum 1990Google Scholar
  23. [Hei 4]
    Heinzner, P.: Linear äquivariante Einbettungen Steinscher Räume. Math. Ann. 280 (1988) 147–160CrossRefzbMATHMathSciNetGoogle Scholar
  24. [HL]
    Huckleberry, A.T., Livorni, L.: A classification of complex homogeneous surfaces. Can. J. Math. 33 (1981) 1096–1109CrossRefMathSciNetGoogle Scholar
  25. [HM 1]
    Hochschild, G., Mostow, G.D.: Affine embeddings of complex analytic homogeneous spaces. Am. J. Math. 87 (1965) 807–839CrossRefzbMATHMathSciNetGoogle Scholar
  26. [HM 2]
    Hochschild, G., Mostow, G.D.: On the algebra of representative functions of an analytic group, II. Am. J. Math. 86 (1964) 869–887CrossRefzbMATHMathSciNetGoogle Scholar
  27. [HMar]
    Huckleberry, A.T., Margulis, G.A.: Invariant analytic hypersurfaces. Invent. math. 71 (1983) 235–240CrossRefzbMATHMathSciNetGoogle Scholar
  28. [Ho]
    Hochschild, G.: The structure of Lie groups. Holden-Day 1965Google Scholar
  29. [HO 1]
    Huckleberry, A.T., Oeljeklaus, E.: Classification theorems for almost homogeneous spaces. Revue De l’Institut Elie Cartan, Numéro 9, Janvier 1984Google Scholar
  30. [HO 2]
    Huckleberry, A.T., Oeljeklaus, E.: A characterization of complex homogeneous cones. Math. Z. 170 (1980) 181–194CrossRefzbMATHMathSciNetGoogle Scholar
  31. [HO 3]
    Huckleberry, A.T., Oeljeklaus, E.: On holomorphically separable complex solvmanifolds. Annales de l’Institut Fourier, Tome XXXVI – Fascicule 3, 57–65 (1986)MathSciNetGoogle Scholar
  32. [HR]
    Huckleberry, A.T., Richthofer, R.: Recent developments in homogeneous CR-hypersurfaces, Contributions to several complex variables. Aspects of Math. E9, Vieweg, Braunschweig, 149–177 (1986)Google Scholar
  33. [HS 1]
    Huckleberry, AT., Snow, D.: Pseudoconcave homogeneous manifolds. Ann. Scuola Norm. Sup. Pisa, Série IV, vol. VII 29–54 (1980)MathSciNetGoogle Scholar
  34. [HS 2]
    Huckleberry, AT., Snow, D.: A classification of strictly pseudoconcave homogeneous manifolds. Ann. Scuola Norm. Sup. Pisa, Série IV, vol. VIII, 231–255 (1981)MathSciNetGoogle Scholar
  35. [HS 3]
    Huckleberry, A.T., Snow, D.: Almost-homogeneous Kähler-manifolds with hypersurface orbits. Osaka J. Math. 19 (1982) 763–786zbMATHMathSciNetGoogle Scholar
  36. [Hum]
    Humphreys, J.E.: Linear algebraic groups. (Graduate Texts in Mathematics, vol. 21). Springer, New York 1975Google Scholar
  37. [HW]
    Huckleberry, A.T., Wurzbacher, T.: Multiplicity-free complex manifolds. Math. Ann. 286, no. 1–3 (1990) 261–280CrossRefzbMATHMathSciNetGoogle Scholar
  38. [K]
    Kraft, H.: Geometrische Methoden in der Invariantentheorie. Vieweg-Verlag, Braunschweig-Wiesbaden 1985CrossRefzbMATHGoogle Scholar
  39. [Ka 1]
    Kaup, W.: Reelle Transformationsgruppen und invariante Metriken auf komplexen Räumen. Invent, math. 3 (1967) 43–70CrossRefzbMATHMathSciNetGoogle Scholar
  40. [Ki]
    Kiselman, C.O.: The partial Legendre transformation for plurisubharmonic functions. Invent, math. 49 (1978) 137–148CrossRefzbMATHMathSciNetGoogle Scholar
  41. [Kod]
    Kodaira, K.: On compact analytic surfaces II. Ann. Math. 77 (1963) 563–626CrossRefzbMATHGoogle Scholar
  42. [KV]
    Kimelfeld, B.N., Vinberg, E.B.: Homogeneous domains on flag manifolds and spherical subgroups of semi-simple Lie groups. Funct. Analysis and its Appl. 12.3 (1978) 12–19Google Scholar
  43. [L]
    Luna, D.: Slices etales. Bull. Soc. Math. Fr. Mem. 33 (1973) 81–105zbMATHGoogle Scholar
  44. [Le 1]
    Lehmann, R.: Complex-symmetric spaces. Dissertation, Ruhr-Universität Bochum 1988zbMATHGoogle Scholar
  45. [Le 2]
    Lehmann, R.: Complex-symmetric spaces. Ann. Inst. Fourier 39.2 (1989) 373–416CrossRefGoogle Scholar
  46. [Le 3]
    Lehmann, R.: Singular complex-symmetric torus embeddings. (To appear)Google Scholar
  47. [Les 1]
    Lescure, F.: Sur les compactifications équivariantes des groupes commutatifs. Annales de l’Institut Fourier 38.4 (1988) 93–120CrossRefMathSciNetGoogle Scholar
  48. [Les 2]
    Lescure, F.: Compactifications équivariantes par des courbes. Bull. Soc. Math. France 115 (1987) Mém. 26Google Scholar
  49. [Loe]
    Loeb, J.: Actions d’une forme de Lie réelle d’un groupe de Lie complexe sur les fonctions plurisousharmoniques. Annales de l’Institut Fourier 35–4 (1985) 59–97CrossRefMathSciNetGoogle Scholar
  50. [M]
    Mumford, D., Fogarty, J.: Geometrie Invariant Theory. Ergebnisse der Mathematik, vol. 34 (Second Enlarged Edition). Springer, Berlin Heidelberg 1982Google Scholar
  51. [Mal]
    Matsushima, Y.: Sur les espaces homogènes kählériens d’un groupe de Lie réductif. Nagoya Math. J. 11 (1957) 53–60zbMATHMathSciNetGoogle Scholar
  52. [Ma 2]
    Matsushima, Y.: On discrete subgroups and homogeneous spaces of nilpotent Lie groups. Nagoya Math. J. 2 (1951) 95–110zbMATHMathSciNetGoogle Scholar
  53. [Ma 3]
    Matsushima, Y.: Fibres holomorphes sur un tore complexe. Nagoya Math. J. 14 (1959) 1–24zbMATHMathSciNetGoogle Scholar
  54. [Ma4]
    Matsushima, Y: Espaces homogènes de Stein des groupes de Lie complexes I. Nagoya Math. J. 16 (1960) 205–218zbMATHMathSciNetGoogle Scholar
  55. [MJ]
    Moser-Jauslin, L.: The Chow rings of smooth complete SL(2)-embeddings. Compos. Math. 82, no. 1 (1992) 67–106zbMATHMathSciNetGoogle Scholar
  56. [MM]
    Matsushima, Y., Morimoto, A.: Sur certains espaces fibres holomorphes sur une variété de Stein. Bull. Soc. Math. France 88 (1960) 137–155zbMATHMathSciNetGoogle Scholar
  57. [MN]
    Morimoto, Y; Nagano, T.: On pseudo-conformal transformations of hyper-surfaces. J. Math. Soc. Japan 14 (1963) 289–300CrossRefMathSciNetGoogle Scholar
  58. [Mo l]
    Mostow, G.D.: Some applications of representative functions to solv-mani-folds. Am. J. Math. 93 (1971) 11–32CrossRefzbMATHMathSciNetGoogle Scholar
  59. [Mo 2]
    Mostow, G.D.: Factor spaces of solvable groups. Ann. Math. 60 (1954) 1–27CrossRefzbMATHMathSciNetGoogle Scholar
  60. [Oe l]
    Oeljeklaus, E.: Fast homogene Kählermannigfaltigkeiten mit verschwindender erster Bettizahl. Manuskr. Math. 7 (1972) 175–183CrossRefzbMATHMathSciNetGoogle Scholar
  61. [Oe 2]
    Oeljeklaus, E.: Ein Hebbarkeitssatz für Automorphismengruppen kompakter Mannigfaltigkeiten. Math. Ann. 190 (1970) 154–166CrossRefzbMATHMathSciNetGoogle Scholar
  62. [OeK]
    Oeljeklaus, K.: Hyperflächen und Geradenbündel auf homogenen komplexen Mannigfaltigkeiten. Schriftenreihe des Mathematischen Instituts der Universität Münster, Ser. 2, Heft 36, Münster 1985Google Scholar
  63. [OR1]
    Oeljeklaus, K., Richthofer, W.: On the structure of complex solvmanifolds. J. Diff. Geom. 27 (1988) 399–421zbMATHMathSciNetGoogle Scholar
  64. [OR 2]
    Oeljeklaus, K., Richthofer, W.: Homogeneous complex surfaces. Math. Ann. 268 (1984) 273–292CrossRefzbMATHMathSciNetGoogle Scholar
  65. [P]
    Penny, R: Entire vectors and holomorphic extension of representations. Trans. AMS. 198 (1974) 107–121CrossRefGoogle Scholar
  66. [R]
    Rossi, H.: Homogeneous strongly pseudoconvex hypersurfaces. Rice Studies 59 (3) (1973) 131–145Google Scholar
  67. [R1]
    Richthofer, W.: Currents in homogeneous manifolds. (To appear)Google Scholar
  68. [R2]
    Richthofer, W.: Homogene CR-Mannigfaltigkeiten. Dissertation, Ruhr-Universität Bochum 1985zbMATHGoogle Scholar
  69. [Sko]
    Skoda, H.: Fibrés holomorphes à base at à fibre de Stein. Invent, math. 43 (1977) 97–107CrossRefzbMATHMathSciNetGoogle Scholar
  70. [Sn 1]
    Snow, D.M.: Reductive group action on Stein spaces. Math. Ann. 259 (1982) 79–97CrossRefzbMATHMathSciNetGoogle Scholar
  71. [Sn 2]
    Snow, D.M.: Stein quotients of connected complex lie groups. Manuscripta math. 49/50 (1985) 185–214CrossRefMathSciNetGoogle Scholar
  72. [So]
    Sommese, A.J.: Extension theorems for reductive group actions on compact Kähler manifolds. Math. Ann. 218 (1975) 107–116CrossRefzbMATHMathSciNetGoogle Scholar
  73. [SZ]
    Stout, E.L., Zame, W.R.: A Stein manifold topologically but not holomor-phically equivalent to a domain in Cn. Adv. in Math. 60 (1986) 154–160CrossRefzbMATHMathSciNetGoogle Scholar
  74. [T]
    Tits, J.: Espaces homogènes complexes compacts. Comm. Math. Helv. 37 (1962) 111–120CrossRefzbMATHMathSciNetGoogle Scholar
  75. [Wa 1]
    Wang, H.C.: Closed manifolds with homogeneous complex structure. Am. J. Math. 76 (1954) 1–32CrossRefzbMATHGoogle Scholar
  76. [Wa 2]
    Wang, H.C.: Complex parallelisable manifolds. Proc. Am. Math. Soc 5 (1954) 771–776CrossRefzbMATHGoogle Scholar
  77. [Wi 1]
    Winkelmann, J.: Classification des espaces complexes homogènes de dimension 3 I. C.R.A.S. Paris 306 (1988) Série I, 231–234Google Scholar
  78. [Wi2]
    Winkelmann, J.: Classification des espaces complexes homogènes de dimension 3 II. C.R.A.S. Paris 306 (1988) Série I, 405–408Google Scholar
  79. [Wi3]
    Winkelmann, J.: The Kobayashi pseudometric on homogeneous complex manifolds. (To appear)Google Scholar
  80. [Wi4]
    Winkelmann, J.: On Stein homogeneous manifolds and free holomorphic C-actions on Cn. Math. Ann. (To appear in Math. Ann.)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Alan T. Huckleberry

There are no affiliations available

Personalised recommendations