On compact symplectic and Kählerian solvmanifolds which are not completely solvable

Aleksy Tralle

Colloquium Mathematicae (1997)

  • Volume: 73, Issue: 2, page 261-283
  • ISSN: 0010-1354

Abstract

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We are interested in the problem of describing compact solvmanifolds admitting symplectic and Kählerian structures. This was first considered in [3, 4] and [7]. These papers used the Hattori theorem concerning the cohomology of solvmanifolds hence the results obtained covered only the completely solvable case}. Our results do not use the assumption of complete solvability. We apply our methods to construct a new example of a compact symplectic non-Kählerian solvmanifold.

How to cite

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Tralle, Aleksy. "On compact symplectic and Kählerian solvmanifolds which are not completely solvable." Colloquium Mathematicae 73.2 (1997): 261-283. <http://eudml.org/doc/210490>.

@article{Tralle1997,
abstract = {We are interested in the problem of describing compact solvmanifolds admitting symplectic and Kählerian structures. This was first considered in [3, 4] and [7]. These papers used the Hattori theorem concerning the cohomology of solvmanifolds hence the results obtained covered only the completely solvable case\}. Our results do not use the assumption of complete solvability. We apply our methods to construct a new example of a compact symplectic non-Kählerian solvmanifold.},
author = {Tralle, Aleksy},
journal = {Colloquium Mathematicae},
keywords = {Kähler structure; symplectic structure; solvmanifold; compact solvmanifolds; Kählerian structure; cohomology},
language = {eng},
number = {2},
pages = {261-283},
title = {On compact symplectic and Kählerian solvmanifolds which are not completely solvable},
url = {http://eudml.org/doc/210490},
volume = {73},
year = {1997},
}

TY - JOUR
AU - Tralle, Aleksy
TI - On compact symplectic and Kählerian solvmanifolds which are not completely solvable
JO - Colloquium Mathematicae
PY - 1997
VL - 73
IS - 2
SP - 261
EP - 283
AB - We are interested in the problem of describing compact solvmanifolds admitting symplectic and Kählerian structures. This was first considered in [3, 4] and [7]. These papers used the Hattori theorem concerning the cohomology of solvmanifolds hence the results obtained covered only the completely solvable case}. Our results do not use the assumption of complete solvability. We apply our methods to construct a new example of a compact symplectic non-Kählerian solvmanifold.
LA - eng
KW - Kähler structure; symplectic structure; solvmanifold; compact solvmanifolds; Kählerian structure; cohomology
UR - http://eudml.org/doc/210490
ER -

References

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  1. [1] E. Abbena, An example of an almost Kähler manifold which is not Kählerian, Boll. Un. Mat. Ital. (6) 3-A (1984), 383-392. Zbl0559.53023
  2. [2] L. Auslander, An exposition of the structure of solvmanifolds, Bull. Amer. Math. Soc. 79 (1973), 227-285. Zbl0265.22016
  3. [3] C. Benson and C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513-518. Zbl0672.53036
  4. [4] C. Benson and C. S. Gordon, Kähler structures on compact solvmanifolds, Proc. Amer. Math. Soc. 108 (1990), 971-980. Zbl0689.53036
  5. [5] L. A. Cordero, M. Fernandez and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), 375-380. Zbl0596.53030
  6. [6] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274. Zbl0312.55011
  7. [7] M. Fernández, M. de León and M. Saralegui, A six-dimensional compact symplectic solvmanifold without Kähler structures, Osaka J. Math. 33 (1996), 19-35. Zbl0861.53032
  8. [8] R. Gompf, Some new symplectic 4-manifolds, Turkish J. Math. 18 (1994), 7-15. Zbl0863.53025
  9. [9] S. Halperin, Lectures on Minimal Models, Hermann, 1982. 
  10. [10] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), 67-71. Zbl0691.53040
  11. [11] A. Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. 1 8 (1960), 289-331. Zbl0099.18003
  12. [12] K. Hess, Twisted tensor products of DGA's and the Adams-Hilton model for the total space of a fibration, in: London Math. Soc. Lecture Note Ser. 175, Cambridge Univ. Press, 1992, 29-51. Zbl0763.55006
  13. [13] D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449. Zbl0146.19201
  14. [14] D. Lehmann, Théorie homotopique des formes différentielles (d'après D. Sullivan), Astérisque 45 (1977). 
  15. [15] G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), 193-207. Zbl0789.55010
  16. [16] D. McDuff, Examples of symplectic simply connected manifolds with no Kähler structure, J. Differential Geom. 20 (1984), 267-277. Zbl0567.53031
  17. [17] C. McCord and J. Oprea, Rational Lusternik-Schnirelmann category and the Arnold conjecture for nilmanifolds, Topology 32 (1993), 701-717. Zbl0798.58017
  18. [18] G. D. Mostow, Factor spaces of solvable groups, Annals of Math. 60 (1954), 1-27. Zbl0057.26103
  19. [19] M. Raghunathan, Discrete Subgroups of Lie Groups, Springer, 1972. Zbl0254.22005
  20. [20] M. Schlessinger and J. Stasheff, Deformation theory and rational homotopy type, preprint (1992), 44 pp. Zbl0813.55004
  21. [21] J.-P. Serre, Homologie singulière des espaces fibrés, Ann. of Math. 54 (1951), 425-505. Zbl0045.26003
  22. [22] N. Steenrod, The Topology of Fiber Bundles, Princeton Univ. Press, 1951. Zbl0054.07103
  23. [23] D. Tanré, Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan, Springer, 1983. 
  24. [24] J.-C. Thomas, Homotopie rationnelle des fibrés de Serre, Université des Sciences et Techniques de Lille I, 1980. 
  25. [25] J.-C. Thomas, Rational homotopy of Serre fibrations, Ann. Inst. Fourier (Grenoble) 31 (3) (1981), 71-90. Zbl0446.55009
  26. [26] E. Vinberg, V. Gorbatsevich and O. Shvartsman, Discrete Subgroups of Lie Groups, Itogi Nauki i Tekhniki. Sovremennye Problemy Matematiki 21 (1988), 5-115 (in Russian). Zbl0656.22004

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