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.
- Josef Dorfmeister, Homogeneous Kähler manifolds admitting a transitive solvable group of automorphisms, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 1, 143–180 (English, with French summary). MR 803198
- Josef Dorfmeister and Kazufumi Nakajima, The fundamental conjecture for homogeneous Kähler manifolds, Acta Math. 161 (1988), no. 1-2, 23–70. MR 962095, DOI https://doi.org/10.1007/BF02392294
- 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
- V. V. Gorbacevič, Decomposition of Lie groups and its applications to the study of homogeneous spaces, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1227–1258 (Russian). MR 567035
- V. V. Gorbatsevich, Plesio-compact homogeneous spaces, Sibirsk. Mat. Zh. 30 (1989), no. 2, 61–72, 226 (Russian); English transl., Siberian Math. J. 30 (1989), no. 2, 217–226. MR 997468, DOI https://doi.org/10.1007/BF00971376
- Daniel Guan, Examples of compact holomorphic symplectic manifolds which admit no Kähler structure, Geometry and analysis on complex manifolds, World Sci. Publ., River Edge, NJ, 1994, pp. 63–74. MR 1463964
- Daniel Guan, A splitting theorem for compact complex homogeneous spaces with a symplectic structure, Geom. Dedicata 63 (1996), no. 2, 217–225. MR 1413633, DOI https://doi.org/10.1007/BF00148221
- Daniel Guan, On compact symplectic manifolds with Lie group symmetries, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3359–3373. MR 2135752, DOI https://doi.org/10.1090/S0002-9947-05-03657-3
[Gu4]Gu4 D. Guan: Classification of Compact Complex Homogeneous Manifolds with Pseudo-Kählerian Structures, Preprint, 2007.
- A. T. Huckleberry, Homogeneous pseudo-Kählerian manifolds: a Hamiltonian viewpoint, Note Mat. 10 (1990), no. suppl. 2, 337–342. MR 1221949
- M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. MR 0507234
- Takumi Yamada, A pseudo-Kähler structure on a nontoral compact complex parallelizable solvmanifold, Geom. Dedicata 112 (2005), 115–122. MR 2163892, DOI https://doi.org/10.1007/s10711-004-3397-4
[Dm]Dm J. Dorfmeister: Homogeneous Kähler Manifolds Admitting a Transitive Solvable Group of Automorphisms, Ann. Scient. Ec. Norm. Sup., 4 Ser., 18 (1985), 143–180.
[DN]DN J. Dorfmeister & K. Nakajima: The Fundamental Conjecture for Homogeneous Kähler Manifolds, Acta Math. 161 (1988), 23–70.
[DG]DG J. Dorfmeister & Z.-D. Guan: Classification of Compact Homogeneous Pseudo-Kähler Manifolds, Comment. Math. Helv. 67 (1992), 499–513.
[Gb1]Gb1 V. V. Gorbatsevich: Splittings of Lie Groups and Their Application to the Study of Homogeneous Spaces, Math. USSR Izv. 15 (1979), 441–467.
[Gb2]Gb2 V. V. Gorbatsevich: Plesiocompact Homogeneous Spaces, Siber. Math. J. 30 (1989), 217–226.
[Gu1]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.
[Gu2]Gu2 D. Guan: A Splitting Theorem for Compact Complex Homogeneous Spaces with a Symplectic Structure, Geom. Dedi. 67 (1996), 217–225.
[Gu3]Gu3 D. Guan: On Compact Symplectic Manifolds with Lie Group Symmetries, Transactions of AMS 357 (2005), 3359–3373.
[Gu4]Gu4 D. Guan: Classification of Compact Complex Homogeneous Manifolds with Pseudo-Kählerian Structures, Preprint, 2007.
[Hk]Hk A. Huckleberry: Homogeneous Pseudo-Kählerian Manifolds: A Hamiltonian Viewpoint, Note Mat. 10 (1990), 337–342.
[Rg]Rg M. S. Raghunathan: Discrete Subgroups of Lie Groups, Springer-Verlag, Berlin, 1972.
[Ym]Ym T. Yamada: A Pseudo-Kähler Structure on a Nontoral Compact Complex Parallelizable Solvmanifold, Geom. Dedicata 112 (2005), 115–122.
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
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