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Subvarieties of homogeneous and almost homogeneous manifolds

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Contributions to Complex Analysis and Analytic Geometry / Analyse Complexe et Géométrie Analytique

Part of the book series: Aspects of Mathematics ((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. Ahiezer,D.N.: Dense orbits with two ends. Math. USSR, Izvestija II (1977), 293–307

    Google Scholar 

  2. Ahiezer,D.N.: Invariant analytic hypersurfaces in complex nilpotent Lie groups. Ann. Glob. Analysis and Geometry 2 (1984), 129–140

    Google Scholar 

  3. Ahiezer,D.N.: Invariant meromorphic functions on complex semi-simple Lie groups. Invent. Math. 65 (1982), 325–329

    Google Scholar 

  4. Abe,Y.: Holomorphic sections of line bundles over (H,C)-groups, Manu. Math., 60 (1988), 379–385

    Google Scholar 

  5. Andreotti,A.; Gherardelli,F.: Estensioni commutative di Varieta’ Abeliane. Quaderno manoscritto del Centro di Analisi Globale del CNR, Firenze (1972), 1–48

    Google Scholar 

  6. Berteloot,F.: Habilitation, Universite de Lille (1993)

    Google Scholar 

  7. Berteloot,F.: Fonctions plurisubharmonique sur S/2(C)-invariant par un sous-groupe monogène. J. d’analyse Math. 48 (1987), 267–276

    Google Scholar 

  8. Berteloot,F.: Existence of a Kahler metric on semi-simple homogeneous manifolds, C.R. de L’Acad. des Sc.,Tome 305 (1987), 809–812

    MathSciNet  MATH  Google Scholar 

  9. Berteloot,F.; Oeljeklaus,K.: Invariant Plurisubharmonic Functions and Hypersurfaces on Semisimple Complex Lie Groups. Math. Ann. 281 (1988), 513–530

    Article  MathSciNet  MATH  Google Scholar 

  10. Borel,A.: Linear algebraic groups. New York, Benjamin (1969)

    Google Scholar 

  11. Borel,A.; Remmert,R.: Über kompakte homogene Kählersche Mannigfaltigkeiten. Math. Ann. 145 (1962), 429–439

    Article  MathSciNet  MATH  Google Scholar 

  12. Capocase,F.; Catanese,F.: Periodic meromorphic functions, Acta Math., 166 (1991), 27–68

    Article  MathSciNet  Google Scholar 

  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. Cousin,P.: Sur les fonctions triplement periodique de deux variables, Acta Math. 10 (1910), 105–232

    Article  MathSciNet  Google Scholar 

  15. Dieudonné,J.: Grundzüge der modernen Analysis Bd. 5. Vieweg-Verlag, Braunschweig Wiesbaden 1979

    Book  Google Scholar 

  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), 293–302

    Article  MathSciNet  MATH  Google Scholar 

  17. Fujiki,A.: On automorphism groups of compact Kahler manifolds. Invent. math. 44 (1978), 225–258

    Article  MathSciNet  MATH  Google Scholar 

  18. Gellhaus,C.: Äquivariante Kompaktifizierungen des C1, MZ 206 (1991), 211–217

    Article  MathSciNet  MATH  Google Scholar 

  19. Goodman,R.: Analytic and entire vectors for representations of Lie groups, Trans. AMS 143 (1969), 55–76

    Article  MATH  Google Scholar 

  20. Green,M.; Griffiths,P: Two Applications of Algebraic Geometry to Entire Holomorphic Mappings. The Chern Symposium 1979, Springer Verlag (1988)

    Google Scholar 

  21. Gilligan,B.; Oeljeklaus,K.; Richthofer,W.: Homogeneous complex manifolds with more than one end. Can. J. Math. Vol. XLI, No. 1 (1989), 163–177

    Article  MathSciNet  Google Scholar 

  22. Grauert,H.; Remmert,R.: Über kompakte homogene komplexe Mannigfaltigkeiten. Arch. Math. 13 (1962), 498–507

    Article  MathSciNet  MATH  Google Scholar 

  23. Guillemin,V.; Sternberg,S.: Symplectic techniques in physics. Cambridge Univ. Press 1984

    Google Scholar 

  24. Harvey,R.: Holomorphic Chains and their Boundaries, Proc. of Symp. in Pure Math. XXX Part 1 (1977), 303–382

    Google Scholar 

  25. Huckleberry,A.T.: Analytic hypersurfaces in homogeneous spaces. Publication Institut Elie Cartan, Journees Complexes Nancy 82 (1983), 134–153

    MathSciNet  Google Scholar 

  26. Huckleberry,A.T.; Margulis,G.A.: Invariant analytic hypersurfaces. Invent. Math. 71 (1983), 235–240

    Article  MathSciNet  MATH  Google Scholar 

  27. Huckleberry,A.T.; Oeljeklaus,E.: Classification Theorems for Almost Homogeneous Spaces. Publication de l’Institut Elie Cartan, Nancy, Janvier 1984, 9. 178 pp.

    MathSciNet  Google Scholar 

  28. Huckleberry,A.T.; Oeljeklaus,E.: A Characterization of Complex Homogeneous Cones Math. Z. 170 (1980), 181–194

    Article  MathSciNet  MATH  Google Scholar 

  29. Huckleberry,A.T.; Oeljeklaus,E.: On holomorphically separable complex solvmanifolds. Annales de l’Institut Fourier (Grenoble), Tome XXXVI–Fascicule 3 (1986), 57–65

    MathSciNet  Google Scholar 

  30. Huckleberry,A.T.; Snow,D.: A Classification of Strictly Pseudoconcave Homogeneous Manifolds. Ann. Scuola Norm. Sup. Pisa 8 (1981), 231–255

    MathSciNet  MATH  Google Scholar 

  31. Humphreys,J.E.: Linear Algebraic Groups. Springer-Verlag, New York 1975

    Book  MATH  Google Scholar 

  32. Huckleberry.A.; Winkelmann,J.: Compact subvarieties in paralleliziable complex manifolds, Math. Ann. 295, 469–483 (1993)

    MathSciNet  Google Scholar 

  33. Kaup,W.: Reelle Transformationsgruppen und invariante Metriken auf komplexen Räumen. Invent. Math. 3 (1967), 43–70

    Article  MathSciNet  MATH  Google Scholar 

  34. Kiselman,C.O.: The partial Legendre transformation for plurisubharmonic functions. Invent. Math. 49 (1978), 137–148

    Article  MathSciNet  MATH  Google Scholar 

  35. Lattes,M.S.: Sur les formes reduites des transformations ponctuelles a deux variables, Comptes Rendue 152 (1911), 1566–1569

    MATH  Google Scholar 

  36. Lescure,F.: Sur les compactifications équivariantes des groupes commutatifs. Annales de l’Institut Fourier 38.4 (1988), 93–120

    Google Scholar 

  37. Lescure,F.: Compactifications équivariantes par des courbes. Bull. Soc. Math. France 115, Mém. 26 (1987)

    Google Scholar 

  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), 59–97

    Article  MathSciNet  Google Scholar 

  39. Matsushima,Y.: On discrete subgroups and homogeneous spaces of nilpotent Lie groups. Nagoya Math. J. 2, 95–110, (1951)

    MathSciNet  MATH  Google Scholar 

  40. Matsushima,Y.: Espaces homogènes de Stein des groupes de Lie complexes I. Nagoya Math. J. 16 (1960), 205–218

    MathSciNet  MATH  Google Scholar 

  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. Morimoto,A.: On the classification of non-compact abelian Lie groups, Trans, AMS123(1966),200–228

    Google Scholar 

  43. Moser,L.: Dissertation, Universite de Lausanne (1989)

    Google Scholar 

  44. Mumford,D.: Abelian Varieties, Studies in Math. 5, London, Oxford Univ. Press (1974)

    Google Scholar 

  45. Oeljeklaus,K.: Hyperfiächen und Geradenbündel auf homogenen komplexen Mannigfaltigkeiten. Schriftenr. Math. Inst. Univ. Münster, Serie 2, Heft 36 (1985)

    Google Scholar 

  46. Oeljeklaus,K.; Richthofer,W.: On the Structure of Complex Solvmanifolds. J. Diff. Geom. 27 (1988), 399–421

    MathSciNet  MATH  Google Scholar 

  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. 1277

    Google Scholar 

  48. Oeljeklaus,K.; Richthofer,W.: Complex Solv-manifolds, Bochum Preprint (1985)

    Google Scholar 

  49. Potters,J.: On almost homogeneous compact complex analytic surfaces. Invent. math. 8 (1969), 244–266

    Article  MathSciNet  MATH  Google Scholar 

  50. Penny,R.: Entire vectors and holomorphic extensions of representations, Trans. AMS 198 (1974), 107–121

    Article  Google Scholar 

  51. Richthofer,W.: Currents in Homogeneous Manifolds. Bochum preprint (1987)

    Google Scholar 

  52. Raghunatan,M.S.: Discrete subgroups of Lie groups. Erg. Math. Grenzgeb. 68, Springer (1972)

    Google Scholar 

  53. Skoda,H.: Fibrés holomorphes à base at à fibre de Stein. Invent. Math. 43 (1977), 97–107

    Article  MathSciNet  MATH  Google Scholar 

  54. Ueno,K.: Classification theory of algebraic varietes and compact complex spaces. Lecture Notes in Mathematics 439, Springer-Verlag Berlin 1975

    Google Scholar 

  55. Varouchas,J.: Kahler spaces and proper open morphisms. Math. Ann. 283 (1989), 13–52

    Article  MathSciNet  MATH  Google Scholar 

  56. Wang,H.C.: Complex parallelisable manifolds. Proc. Am. Math. Soc 5 (1954), 771–776

    Article  MATH  Google Scholar 

  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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Authors
  1. Alan Huckleberry
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Editors and Affiliations

  1. Département de Mathématiques, Université Paris 6, Tour 45-46, 5ème étage, 4, Place Jussieu, F-75252, Paris Cedex 05, France

    Henri Skoda  & Jean-Marie Trépreau  & 

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Dedicated to Professor P. Dolbeault

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Huckleberry, A. (1994). Subvarieties of homogeneous and almost homogeneous manifolds. In: Skoda, H., Trépreau, JM. (eds) Contributions to Complex Analysis and Analytic Geometry / Analyse Complexe et Géométrie Analytique. Aspects of Mathematics, vol E 26. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14196-9_7

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  • DOI: https://doi.org/10.1007/978-3-663-14196-9_7

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-528-06633-8

  • Online ISBN: 978-3-663-14196-9

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