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Subvarieties of homogeneous and almost homogeneous manifolds

  • Alan Huckleberry
Chapter
Part of the Aspects of Mathematics book series (ASMA, volume E 26)

Abstract

The general context of this paper is that of compact subvarieties of a complex manifold X which is equipped with the action of a Lie group G of holomorphic transformations. Here we restrict our attention to the case where X is homogeneous or at least almost homogeneous. For example, in Chapter 2 we consider the analytic hypersurfaces H(X), i. e., the lcodimensional complex analytic subsets, in a homogeneous space X = G / H. Although the results are primarily formulated in the language of complex geometry, we are mainly interested in determining the complex analytic objects which define the hypersurfaces, e.g., holomorphic functions coming from associated Stein manifolds, rational functions from related projective varieties, Θ-functions on abelian groups, Fourier-Jacobi series,... Difficulties arise, for example, because the ambient manifolds are in general non-compact, non-Kählerian, and the relevant cohomology groups are infinite-dimensional.

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Bibliography

  1. [1]
    Ahiezer,D.N.: Dense orbits with two ends. Math. USSR, Izvestija II (1977), 293–307Google Scholar
  2. [2]
    Ahiezer,D.N.: Invariant analytic hypersurfaces in complex nilpotent Lie groups. Ann. Glob. Analysis and Geometry 2 (1984), 129–140Google Scholar
  3. [3]
    Ahiezer,D.N.: Invariant meromorphic functions on complex semi-simple Lie groups. Invent. Math. 65 (1982), 325–329Google Scholar
  4. [4]
    Abe,Y.: Holomorphic sections of line bundles over (H,C)-groups, Manu. Math., 60 (1988), 379–385Google Scholar
  5. [5]
    Andreotti,A.; Gherardelli,F.: Estensioni commutative di Varieta’ Abeliane. Quaderno manoscritto del Centro di Analisi Globale del CNR, Firenze (1972), 1–48Google Scholar
  6. [6]
    Berteloot,F.: Habilitation, Universite de Lille (1993)Google Scholar
  7. [7]
    Berteloot,F.: Fonctions plurisubharmonique sur S/2(C)-invariant par un sous-groupe monogène. J. d’analyse Math. 48 (1987), 267–276Google Scholar
  8. [8]
    Berteloot,F.: Existence of a Kahler metric on semi-simple homogeneous manifolds, C.R. de L’Acad. des Sc.,Tome 305 (1987), 809–812MathSciNetzbMATHGoogle Scholar
  9. [9]
    Berteloot,F.; Oeljeklaus,K.: Invariant Plurisubharmonic Functions and Hypersurfaces on Semisimple Complex Lie Groups. Math. Ann. 281 (1988), 513–530MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Borel,A.: Linear algebraic groups. New York, Benjamin (1969)Google Scholar
  11. [11]
    Borel,A.; Remmert,R.: Über kompakte homogene Kählersche Mannigfaltigkeiten. Math. Ann. 145 (1962), 429–439MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Capocase,F.; Catanese,F.: Periodic meromorphic functions, Acta Math., 166 (1991), 27–68MathSciNetCrossRefGoogle Scholar
  13. [13]
    Coeure,F; Loeb,J.-J.: A counterexample to the Serre problem with a bounded domain in C2 as fiber, Ann. of Math.(2)122(1985)Google Scholar
  14. [14]
    Cousin,P.: Sur les fonctions triplement periodique de deux variables, Acta Math. 10 (1910), 105–232MathSciNetCrossRefGoogle Scholar
  15. [15]
    Dieudonné,J.: Grundzüge der modernen Analysis Bd. 5. Vieweg-Verlag, Braunschweig Wiesbaden 1979CrossRefGoogle Scholar
  16. [16]
    Demailly,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), 293–302MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Fujiki,A.: On automorphism groups of compact Kahler manifolds. Invent. math. 44 (1978), 225–258MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Gellhaus,C.: Äquivariante Kompaktifizierungen des C1, MZ 206 (1991), 211–217MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Goodman,R.: Analytic and entire vectors for representations of Lie groups, Trans. AMS 143 (1969), 55–76zbMATHCrossRefGoogle Scholar
  20. [20]
    Green,M.; Griffiths,P: Two Applications of Algebraic Geometry to Entire Holomorphic Mappings. The Chern Symposium 1979, Springer Verlag (1988)Google Scholar
  21. [21]
    Gilligan,B.; Oeljeklaus,K.; Richthofer,W.: Homogeneous complex manifolds with more than one end. Can. J. Math. Vol. XLI, No. 1 (1989), 163–177MathSciNetCrossRefGoogle Scholar
  22. [22]
    Grauert,H.; Remmert,R.: Über kompakte homogene komplexe Mannigfaltigkeiten. Arch. Math. 13 (1962), 498–507MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Guillemin,V.; Sternberg,S.: Symplectic techniques in physics. Cambridge Univ. Press 1984Google Scholar
  24. [24]
    Harvey,R.: Holomorphic Chains and their Boundaries, Proc. of Symp. in Pure Math. XXX Part 1 (1977), 303–382Google Scholar
  25. [25]
    Huckleberry,A.T.: Analytic hypersurfaces in homogeneous spaces. Publication Institut Elie Cartan, Journees Complexes Nancy 82 (1983), 134–153MathSciNetGoogle Scholar
  26. [26]
    Huckleberry,A.T.; Margulis,G.A.: Invariant analytic hypersurfaces. Invent. Math. 71 (1983), 235–240MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Huckleberry,A.T.; Oeljeklaus,E.: Classification Theorems for Almost Homogeneous Spaces. Publication de l’Institut Elie Cartan, Nancy, Janvier 1984, 9. 178 pp.MathSciNetGoogle Scholar
  28. [28]
    Huckleberry,A.T.; Oeljeklaus,E.: A Characterization of Complex Homogeneous Cones Math. Z. 170 (1980), 181–194MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    Huckleberry,A.T.; Oeljeklaus,E.: On holomorphically separable complex solvmanifolds. Annales de l’Institut Fourier (Grenoble), Tome XXXVI–Fascicule 3 (1986), 57–65MathSciNetGoogle Scholar
  30. [30]
    Huckleberry,A.T.; Snow,D.: A Classification of Strictly Pseudoconcave Homogeneous Manifolds. Ann. Scuola Norm. Sup. Pisa 8 (1981), 231–255MathSciNetzbMATHGoogle Scholar
  31. [31]
    Humphreys,J.E.: Linear Algebraic Groups. Springer-Verlag, New York 1975zbMATHCrossRefGoogle Scholar
  32. [32]
    Huckleberry.A.; Winkelmann,J.: Compact subvarieties in paralleliziable complex manifolds, Math. Ann. 295, 469–483 (1993)MathSciNetGoogle Scholar
  33. [33]
    Kaup,W.: Reelle Transformationsgruppen und invariante Metriken auf komplexen Räumen. Invent. Math. 3 (1967), 43–70MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    Kiselman,C.O.: The partial Legendre transformation for plurisubharmonic functions. Invent. Math. 49 (1978), 137–148MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    Lattes,M.S.: Sur les formes reduites des transformations ponctuelles a deux variables, Comptes Rendue 152 (1911), 1566–1569zbMATHGoogle Scholar
  36. [36]
    Lescure,F.: Sur les compactifications équivariantes des groupes commutatifs. Annales de l’Institut Fourier 38.4 (1988), 93–120Google Scholar
  37. [37]
    Lescure,F.: Compactifications équivariantes par des courbes. Bull. Soc. Math. France 115, Mém. 26 (1987)Google Scholar
  38. [38]
    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), 59–97MathSciNetCrossRefGoogle Scholar
  39. [39]
    Matsushima,Y.: On discrete subgroups and homogeneous spaces of nilpotent Lie groups. Nagoya Math. J. 2, 95–110, (1951)MathSciNetzbMATHGoogle Scholar
  40. [40]
    Matsushima,Y.: Espaces homogènes de Stein des groupes de Lie complexes I. Nagoya Math. J. 16 (1960), 205–218MathSciNetzbMATHGoogle Scholar
  41. [41]
    Martin,M.: Über den Körper meromorpher Funktionen auf einer kompakten Sl2((C)fast-homogenen komplex-dreidimensionalen Mannigfaltigkeit, Dissertation, Ruhr Universität Bochum (1992)Google Scholar
  42. [42]
    Morimoto,A.: On the classification of non-compact abelian Lie groups, Trans, AMS123(1966),200–228Google Scholar
  43. [43]
    Moser,L.: Dissertation, Universite de Lausanne (1989)Google Scholar
  44. [44]
    Mumford,D.: Abelian Varieties, Studies in Math. 5, London, Oxford Univ. Press (1974)Google Scholar
  45. [45]
    Oeljeklaus,K.: Hyperfiächen und Geradenbündel auf homogenen komplexen Mannigfaltigkeiten. Schriftenr. Math. Inst. Univ. Münster, Serie 2, Heft 36 (1985)Google Scholar
  46. [46]
    Oeljeklaus,K.; Richthofer,W.: On the Structure of Complex Solvmanifolds. J. Diff. Geom. 27 (1988), 399–421MathSciNetzbMATHGoogle Scholar
  47. [47]
    Oeljeklaus,K.; Richthofer,W.: Recent Results on Homogeneous Complex Manifolds in: Complex Analysis III. Pric. Univ. Maryland 1985–86, 78–119, Lecture Notes in Math. 1277Google Scholar
  48. [48]
    Oeljeklaus,K.; Richthofer,W.: Complex Solv-manifolds, Bochum Preprint (1985)Google Scholar
  49. [49]
    Potters,J.: On almost homogeneous compact complex analytic surfaces. Invent. math. 8 (1969), 244–266MathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    Penny,R.: Entire vectors and holomorphic extensions of representations, Trans. AMS 198 (1974), 107–121CrossRefGoogle Scholar
  51. [51]
    Richthofer,W.: Currents in Homogeneous Manifolds. Bochum preprint (1987)Google Scholar
  52. [52]
    Raghunatan,M.S.: Discrete subgroups of Lie groups. Erg. Math. Grenzgeb. 68, Springer (1972)Google Scholar
  53. [53]
    Skoda,H.: Fibrés holomorphes à base at à fibre de Stein. Invent. Math. 43 (1977), 97–107MathSciNetzbMATHCrossRefGoogle Scholar
  54. [54]
    Ueno,K.: Classification theory of algebraic varietes and compact complex spaces. Lecture Notes in Mathematics 439, Springer-Verlag Berlin 1975Google Scholar
  55. [55]
    Varouchas,J.: Kahler spaces and proper open morphisms. Math. Ann. 283 (1989), 13–52MathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    Wang,H.C.: Complex parallelisable manifolds. Proc. Am. Math. Soc 5 (1954), 771–776zbMATHCrossRefGoogle Scholar
  57. [57]
    Winkelmann,J.: Every compact complex manifold admits a non-trivial holomorphic vector bundle, Revue Romaine de Math. Pure et Appl, v. 38 n. 7 (1993)Google Scholar

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© Springer Fachmedien Wiesbaden 1994

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  • Alan Huckleberry

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