Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On compact symplectic manifolds with Lie group symmetries


Author: Daniel Guan
Journal: Trans. Amer. Math. Soc. 357 (2005), 3359-3373
MSC (2000): Primary 53C15, 57S25, 53C30, 22E99, 15A75
DOI: https://doi.org/10.1090/S0002-9947-05-03657-3
Published electronically: March 10, 2005
MathSciNet review: 2135752
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces which leads us to a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessary invariant under the group action) that makes the classification of this kind of manifold possible.


References [Enhancements On Off] (What's this?)

  • 1. F. A. Bogomolov: On Guan's Examples of Simply connected Non-Kähler Compact Complex Manifolds, Amer. J. Math. 118 (1996), 1037-1046. MR 1408498 (97k:32048)
  • 2. A. Borel & R. Remmert: Über Kompakte Homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1962), 429-439. MR 0145557 (26:3088)
  • 3. J. Dorfmeister & Z. Guan: Classifications of Compact Homogeneous Pseudo-Kähler Manifolds, Comm. Math. Helv. 67 (1992), 499-513. MR 1185806 (93i:32042)
  • 4. J. Dorfmeister & Z. Guan: Pseudo-Kählerian Homogeneous Spaces Admitting a Reductive Transitive Group of Automorphisms, Math. Zeischrift 209 (1992), 89-100. MR 1143216 (92k:32058)
  • 5. J. Dorfmeister: Homogeneous Kähler Manifolds Admitting a Transitive Solvable Group of Automorphisms, Ann. Scient. Ec. Norm. Sup., 4 Serie, vol. 18 (1985), 143-180.MR 0803198 (87j:32094)
  • 6. J. Dorfmeister & K. Nakajima: The Fundamental Conjecture for Homogeneous Kähler Manifolds, Acta Math. 161 (1988), 23-70. MR 0962095 (89i:32066)
  • 7. V. V. Gorbatsevich: On the Double Normalizer of the Stationary Subalgebra of a Plesiocompact Homogeneous Spaces, Siberian Math. J. 34 (1993), 451-456.MR 1241168 (94h:53063)
  • 8. V. V. Gorbatsevich: On a Fibration of Compact Homogeneous Spaces, Trans. Moscow Math. Soc. vol. 1 (1983), 129-157.
  • 9. V. V. Gorbatsevich: Splittings of Lie Groups and Their Application to the Study of Homogeneous Spaces, Math. USSR Izvestija. vol. 15 (1980), 441-467.
  • 10. V. V. Gorbatsevich: Plesiocompact Homogeneous Spaces, Siber. Math. J. 30 (1989), 217-226. MR 0997468 (90f:22010)
  • 11. V. Guillemin & S. Sternberg: Symplectic Techniques in Physics, Cambridge Univ. Press. 1984. MR 0770935 (86f:58054)
  • 12. 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, 63-74. MR 1463964 (98h:53109)
  • 13. D. Guan: A Splitting Theorem for Compact Complex Homogeneous Spaces with a Symplectic Structure. Geom. Dedi. 67 (1996), 217-225. MR 1413633 (98a:53105)
  • 14. D. Guan: Classification of Compact Complex Homogeneous Spaces with Invariant Volumes, Transactions of AMS. 254 (2002), 4493-4504. MR 1926885 (2003h:32035)
  • 15. D. Guan: Examples of Compact holomorphic Symplectic Manifolds which are not Kählerian II, Invent. Math. 121 (1995), 135-145. MR 1345287 (97i:32033)
  • 16. D. Guan: Classification of Compact Homogeneous Space with an Invariant Symplectic Structure, preprint 1997, An Announcement appeared in ERA-AMS vol. 3 (1997), 52-54. MR 1464575 (99a:53065)
  • 17. D. Guan: Fine Structure of Compact Homogeneous Space with an Invariant Symplectic Structure, preprint 1997.
  • 18. D. Guan: Toward a Classification of Compact Complex Homogeneous Spaces, J. Algebra vol. 273 (2004), 33-59. MR 2032450
  • 19. A. T. Huckleberry: Homogeneous Pseudo-Kählerian Manifolds: A Hamiltonian Viewpoint, Note di Matematica 10(1990) suppl. 2, 337-342. MR 1221949 (94f:53052)
  • 20. A. T. Huckleberry & E. Oeljeklaus: Classification Theorems for Almost Homogeneous Spaces, Publ. de l'Inst. Elie Cartan, Nancy, Janvier 1984, 9. 178 pages. MR 0782881 (86g:32050)
  • 21. J. L. Koszul: Sur la Form Hermitienne Canonique des Spaces Homogenes Complexes, Canad. J. Math. 7 (1968), 562-576. MR 0077879 (17:1109a)
  • 22. A. L. Onishchik: On Lie Groups Transitive on Compact Manifolds II, Math. USSR Sbornik. vol. 3 (1967), 373-388.
  • 23. J. Tits: Espaces Homogènes Complexes Compacts, Comm. Math. Helv. 37 (1962), 111-120. MR 0154299 (27:4248)
  • 24. Ph. B. Zwart & W. M. Boothby: On Compact, Homogeneous Symplectic Manifolds, Ann. Inst. Fourier Grenoble 30, 1 (1980), 129-157. MR 0576076 (81g:53040)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C15, 57S25, 53C30, 22E99, 15A75

Retrieve articles in all journals with MSC (2000): 53C15, 57S25, 53C30, 22E99, 15A75


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-05-03657-3
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): May 22, 2002
Received by editor(s) in revised form: February 26, 2004
Published electronically: March 10, 2005
Additional Notes: This work was supported by NSF Grant DMS-9627434 and DMS-0103282
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society