Classification of compact homogeneous spaces with invariant symplectic structures
Author:
Daniel Guan
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 52-54
MSC (1991):
Primary 53C15, 57S25, 53C30; Secondary 22E99, 15A75
DOI:
https://doi.org/10.1090/S1079-6762-97-00023-1
Published electronically:
July 29, 1997
MathSciNet review:
1464575
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Abstract: We solve a longstanding problem of classification of compact homogeneous spaces with invariant symplectic structures. We also give a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessarily invariant under the group action) that makes the classification of this kind of manifolds possible.
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- Josef Dorfmeister and Zhuang Dan Guan, Classification of compact homogeneous pseudo-Kähler manifolds, Comment. Math. Helv. 67 (1992), no. 4, 499–513. MR 1185806, DOI https://doi.org/10.1007/BF02566516
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- D. Guan, Classification of compact complex homogeneous spaces with invariant volumes, preprint 1996.
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- A. T. Huckleberry, Homogeneous pseudo-Kählerian manifolds: a Hamiltonian viewpoint, preprint, 1990.
- Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. MR 909698
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- F. A. Bogomolov, On Guan’s examples of simply connected non-Kähler compact complex manifolds, Amer. J. Math. 118 (1996), 1037–1046.
- A. Borel and R. Remmert, Über kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1962), 429–439.
- J. Dorfmeister and Z. Guan, Classifications of compact homogeneous pseudo-Kähler manifolds, Comm. Math. Helv. 67 (1992), 499–513.
- Z. 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, 1994, pp. 63–74.
- D. Guan, A splitting theorem for compact complex homogeneous spaces with a symplectic structure, Geom. Dedi. 67 (1996), 217–225.
- D. Guan, Classification of compact complex homogeneous spaces with invariant volumes, preprint 1996.
- D. Guan, Examples of compact holomorphic symplectic manifolds which are not Kählerian II, Invent. Math. 121 (1995), 135–145.
- A. T. Huckleberry, Homogeneous pseudo-Kählerian manifolds: a Hamiltonian viewpoint, preprint, 1990.
- S. Kobayashi, Differential geometry of complex vector bundles, Iwanami Shoten Publishers and Princeton University Press, 1987.
- Y. Matsushima, Sur les espaces homogènes kählériens d‘un groupe de Lie réductif, Nagoya Math. J. 11 (1957), 53–60.
- J. Tits, Espaces homogènes complexes compacts, Comm. Math. Helv. 37(1962), 111–120.
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Additional Information
Daniel Guan
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544
Email:
zguan@math.princeton.edu
Keywords:
Invariant structure,
homogeneous space,
product,
fiber bundles,
symplectic manifolds,
splittings,
prealgebraic group,
decompositions,
modification,
Lie group,
symplectic algebra,
compact manifolds,
uniform discrete subgroups,
classifications,
locally flat parallelizable manifolds
Received by editor(s):
February 21, 1997
Published electronically:
July 29, 1997
Additional Notes:
Supported by NSF Grant DMS-9401755 and DMS-9627434.
Communicated by:
Gregory Margulis
Article copyright:
© Copyright 1997
American Mathematical Society
