Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762

 
 

 

Modification and the cohomology groups of compact solvmanifolds


Author: Daniel Guan
Journal: Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 74-81
MSC (2000): Primary 53C15, 57S25, 53C30, 22E99, 15A75
DOI: https://doi.org/10.1090/S1079-6762-07-00176-X
Published electronically: December 7, 2007
MathSciNet review: 2358304
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we give a modification theorem for a compact homogeneous solvmanifold such that a certain Mostow type condition will be satisfied. An application of this result is a simpler way to calculate the cohomology groups of compact quotients of real solvable Lie group over a cocompact discrete subgroup. Furthermore, we apply the second result to obtain a splitting theorem for compact complex homogeneous manifolds with symplectic structures. In particular, we are able to classify compact complex homogeneous spaces with pseudo-Kählerian structures.


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

  • [Dm] J. Dorfmeister: Homogeneous Kähler Manifolds Admitting a Transitive Solvable Group of Automorphisms, Ann. Scient. Ec. Norm. Sup., 4 Ser., 18 (1985), 143-180. MR 803198 (87j:32094)
  • [DN] J. Dorfmeister & K. Nakajima: The Fundamental Conjecture for Homogeneous Kähler Manifolds, Acta Math. 161 (1988), 23-70. MR 962095 (89i:32066)
  • [DG] J. Dorfmeister & Z.-D. Guan: Classification of Compact Homogeneous Pseudo-Kähler Manifolds, Comment. Math. Helv. 67 (1992), 499-513. MR 1185806 (93i:32042)
  • [Gb1] V. V. Gorbatsevich: Splittings of Lie Groups and Their Application to the Study of Homogeneous Spaces, Math. USSR Izv. 15 (1979), 441-467. MR 0567035 (82e:22019)
  • [Gb2] V. V. Gorbatsevich: Plesiocompact Homogeneous Spaces, Siber. Math. J. 30 (1989), 217-226. MR 997468 (90f:22010)
  • [Gu1] 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. MR 1463964 (98h:53109)
  • [Gu2] D. Guan: A Splitting Theorem for Compact Complex Homogeneous Spaces with a Symplectic Structure, Geom. Dedi. 67 (1996), 217-225. MR 1413633 (98a:53105)
  • [Gu3] D. Guan: On Compact Symplectic Manifolds with Lie Group Symmetries, Transactions of AMS 357 (2005), 3359-3373. MR 2135752 (2006c:53094)
  • [Gu4] D. Guan: Classification of Compact Complex Homogeneous Manifolds with Pseudo-Kählerian Structures, Preprint, 2007.
  • [Hk] A. Huckleberry: Homogeneous Pseudo-Kählerian Manifolds: A Hamiltonian Viewpoint, Note Mat. 10 (1990), 337-342. MR 1221949 (94f:53052)
  • [Rg] M. S. Raghunathan: Discrete Subgroups of Lie Groups, Springer-Verlag, Berlin, 1972. MR 0507234 (58:22394a)
  • [Ym] T. Yamada: A Pseudo-Kähler Structure on a Nontoral Compact Complex Parallelizable Solvmanifold, Geom. Dedicata 112 (2005), 115-122. MR 2163892 (2006e:53061)

Similar Articles

Retrieve articles in Electronic Research Announcements 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 at Riverside, Riverside, CA 92507
Email: zguan@math.ucr.edu

DOI: https://doi.org/10.1090/S1079-6762-07-00176-X
Keywords: Solvmanifolds, cohomology, invariant structure, homogeneous space, product, fiber bundles, symplectic manifolds, splittings, prealgebraic group, decompositions, modification, Lie group, compact manifolds, uniform discrete subgroups, locally flat parallelizable manifolds
Received by editor(s): August 10, 2006
Published electronically: December 7, 2007
Communicated by: Keith Burns
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society