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Classification of compact complex homogeneous spaces with invariant volumes


Author: Daniel Guan
Journal: Trans. Amer. Math. Soc. 354 (2002), 4493-4504
MSC (2000): Primary 53C30, 32M10, 32M05, 14M17; Secondary 14M20, 53C10, 53C56
DOI: https://doi.org/10.1090/S0002-9947-02-03102-1
Published electronically: July 2, 2002
MathSciNet review: 1926885
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Abstract | References | Similar Articles | Additional Information

Abstract: We solve the problem of the classification of compact complex homogeneous spaces with invariant volumes (see Matsushima, 1961).


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  • 1. J. F. Adams: Lectures on Lie Groups. W. A. Benjamin, New York, 1969. MR 40:5780
  • 2. D. N. Akhiezer: A Bound for the Dimension of the Automorphism Group of a Compact Complex Homogeneous Space. Soobshch. Akad. Nauk. Grus. SSR (Russian) 110 (1983), 469-472.
  • 3. D. N. Akhiezer: Homogeneous Complex Manifolds, in Several Complex Variables IV, Encyclopaedia of Math. Sci., vol. 10, (1994), 195-244.
  • 4. A. Borel: Kählerian Coset Spaces of Semisimple Lie Groups, Proc. Nat. Acad. Sci. USA, 40 (1954) 1147-1151. MR 17:1108e
  • 5. A. Borel and R. Remmert: Über kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1962), 429-439. MR 26:3088
  • 6. J. Dorfmeister and Z. Guan: Classification of Compact Homogeneous Pseudo-Kähler Manifolds, Comment. Math. Helv. 67 (1992), 499-513. MR 93i:32042
  • 7. J. Dorfmeister and Z. Guan: Fine Structure of Reductive Pseudo-Kählerian Spaces, Geom. Dedicata 39 (1991) 321-338. MR 92h:53081
  • 8. J. Dorfmeister and Z. Guan: Pseudo-Kählerian Homogeneous Spaces Admitting a Reductive Transformation Group of Automorphisms, Math. Z. 209 (1992) 89-100. MR 92k:32058
  • 9. J. Dorfmeister and K. Nakajima: The Fundamental Conjecture for Homogeneous Kähler Manifolds, Acta. Math. 161 (1988) 23-70. MR 89i:32066
  • 10. H. Grauert and R. Remmert: Über kompakte homogene komplexe Mannigfaltigkeiten, Arch. Math. 13 (1962) 498-507. MR 26:3089
  • 11. D. Guan: Examples of Compact Holomorphic Symplectic Manifolds Which Admit no Kähler Structure. In Geometry and Analysis on Complex Manifolds--Festschrift for Professor S. Kobayashi's 60th Birthday, World Scientific, River Edge, NJ, 1994, 63-74. MR 98h:53109
  • 12. D. Guan: A Splitting Theorem for Compact Complex Homogeneous Spaces with a Symplectic Structure, Geom. Dedicata 63 (1996), 217-225. MR 98a:53105
  • 13. D. Guan: Classification of Compact Homogeneous Spaces with Invariant Symplectic Structures, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 52-54. MR 99a:53065
  • 14. D. Guan: Toward a Classification of Complex Homogeneous Spaces, preprint, 1998.
  • 15. D. Guan: Classification of Compact Complex Homogeneous Spaces with Invariant Volumes, Electron. Res. Announc. Amer. Math. Soc. 3 (1997) 90-92. MR 98i:32056
  • 16. J. Hano: Equivariant Projective Immersion of a Complex Coset Space with Non-degenerate Canonical Hermitian Form, Scripta Math. 29 (1973), 125-139. MR 51:3557
  • 17. J. Hano: On Compact Complex Coset Spaces of Reductive Lie Groups, Proc. Amer. Math. Soc. 15 (1964), 159-163. MR 28:1258
  • 18. G. Hochschild: The Structure of Lie Groups, Holden-Day, 1965. MR 34:7696
  • 19. A. T. Huckleberry: Homogeneous Pseudo-Kählerian Manifolds: A Hamiltonian Viewpoint, Note di Matematica 10 (1990) suppl. 2, 337-342. MR 94f:53052
  • 20. J. Hano and S. Kobayashi: A Fibering of a Class of Homogeneous Complex Manifolds, Trans. Amer. Math. Soc. 94 (1960), 233-243. MR 22:5990
  • 21. A. T. Huckleberry and E. Oeljeklaus: Classification Theorems for Almost Homogeneous Spaces, Publ. de l'Institut Élie Cartan, vol. 9, Nancy, Janvier 1984, 178 pages. MR 86g:32050
  • 22. J. E. Humphreys: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, 1972. MR 48:2197
  • 23. T. Iwamoto: Density Properties of Complex Lie Groups, Osaka J. Math. 23 (1986) 859-865. MR 88f:22023
  • 24. T. Iwamoto: Algebraic Groups and Co-compact Subgroups of Complex Linear Groups, Memoirs of the Faculty of Sciences, Kyushu University, Ser. A, 42 (1988) 1-7. MR 89d:22005
  • 25. S. Kobayashi: Differential Geometry of Complex Vector Bundles, Iwanami Shoten, Tokyo and Princeton University Press, 1987. MR 89e:53100
  • 26. R. C. Kirby and L. C. Siebenmann: On the Triangulation of Manifolds and the Hauptvermutung, Bull. Amer. Math. Soc., vol. 75 (1969), 742-749. MR 39:3500
  • 27. J. L. Koszul: Sur la Forme Hermitienne Canonique des Espaces Homogènes Complexes, Canad. J. Math. 7 (1955), 562-576. MR 17:1109a
  • 28. A. I. Malcev: On a class of Homogeneous Spaces. Amer. Math. Soc. Transl., no. 39 (1951). MR 12:589e
  • 29. Y. Matsushima: Sur les Espaces Homogènes Kählériens d'un Groupe de Lie Réductif, Nagoya Math. J. 11 (1957), 53-60. MR 19:315c
  • 30. Y. Matsushima: Sur Certaines Variétés Homogènes Complexes, Nagoya Math. J. 18 (1961) 1-12. MR 25:2147
  • 31. M. S. Raghunathan: Discrete Subgroups of Lie Groups, Ergebnisse der Math. und ihrer Grenzgebiete, Band 68 (1972). MR 58:22394a
  • 32. A. Selberg: On Discontinuous Groups in Higher-Dimensional Symmetric Spaces, In Contributions to Function Theory, Tata Institute of Fundamental Research, Bombay, 1960, 147-164. MR 24:A188
  • 33. J. Tits: Espaces Homogènes Complexes Compacts, Comment. Math. Helv. 37 (1962), 111-120. MR 27:4248
  • 34. Hsien-Chung Wang: Complex Parallisable Manifolds, Proc. Amer. Math. Soc. 5 (1954), 771-776. MR 17:531a
  • 35. Hsien-Chung Wang: Closed Manifolds with Homogeneous Complex Structure, Amer. J. Math. 76 (1954) 1-32. MR 16:518a
  • 36. J. Winkelmann: Complex Analytic Geometry of Complex Parallelizable Manifolds. Mém. Soc. Math. Fr. (N. S.) vol. 72-73 (1998). MR 99g:32058

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Additional Information

Daniel Guan
Affiliation: Department of Mathematics, The University of California at Riverside, Riverside, California 92521
Email: zguan@math.ucr.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03102-1
Keywords: Invariant volume, homogeneous, product, fiber bundles, complex manifolds, parallelizible manifolds, discrete subgroups, classifications
Received by editor(s): September 28, 2001
Received by editor(s) in revised form: April 21, 2002
Published electronically: July 2, 2002
Additional Notes: Supported by NSF Grants DMS-9401755 and DMS-9627434
Article copyright: © Copyright 2002 American Mathematical Society

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