Formality and the Lefschetz property in symplectic and cosymplectic geometry

Giovanni Bazzoni; Marisa Fernández; Vicente Muñoz

Complex Manifolds (2015)

  • Volume: 2, Issue: 1, page 53-77, electronic only
  • ISSN: 2300-7443

Abstract

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We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b1 = 1 case (in the coKähler/cosymplectic situation).

How to cite

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Giovanni Bazzoni, Marisa Fernández, and Vicente Muñoz. "Formality and the Lefschetz property in symplectic and cosymplectic geometry." Complex Manifolds 2.1 (2015): 53-77, electronic only. <http://eudml.org/doc/275956>.

@article{GiovanniBazzoni2015,
abstract = {We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b1 = 1 case (in the coKähler/cosymplectic situation).},
author = {Giovanni Bazzoni, Marisa Fernández, Vicente Muñoz},
journal = {Complex Manifolds},
keywords = {Kähler and coKähler manifolds; symplectic and cosymplectic manifolds; K-cosymplectic manifolds; almost coKähler manifolds; Lefschetz property; Betti numbers; formal spaces},
language = {eng},
number = {1},
pages = {53-77, electronic only},
title = {Formality and the Lefschetz property in symplectic and cosymplectic geometry},
url = {http://eudml.org/doc/275956},
volume = {2},
year = {2015},
}

TY - JOUR
AU - Giovanni Bazzoni
AU - Marisa Fernández
AU - Vicente Muñoz
TI - Formality and the Lefschetz property in symplectic and cosymplectic geometry
JO - Complex Manifolds
PY - 2015
VL - 2
IS - 1
SP - 53
EP - 77, electronic only
AB - We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b1 = 1 case (in the coKähler/cosymplectic situation).
LA - eng
KW - Kähler and coKähler manifolds; symplectic and cosymplectic manifolds; K-cosymplectic manifolds; almost coKähler manifolds; Lefschetz property; Betti numbers; formal spaces
UR - http://eudml.org/doc/275956
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. A (6) 3 (1984), no. 3, 383–392. Zbl0559.53023
  2. [2] J. Amorós, M. Burger, K. Corlette, D. Kotschick and D. Toledo, Fundamental groups of compact Kähler manifolds, Math. Surveys and Monographs 44, Amer. Math. Soc., 1996. Zbl0849.32006
  3. [3] D. Angella, Cohomological Aspects in Complex Non-Kähler Geometry, Lecture Notes inMathematics, 2095, Springer-Verlag, Berlin, 2014. Zbl1290.32001
  4. [4] D. Angella, A. Tomassini and W. Zhang, On cohomological decomposability of almost-Kähler structures, Proc. Amer. Math. Soc. 142 (2014), no. 10, 3615–3630. Zbl1298.53073
  5. [5] V. I. Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Graduate Texts in Mathematics 60, Springer, 1997. 
  6. [6] M. Audin, Exemples de variétés presque complexes, Einseign. Math. (2) 37 (1991), no. 1–2, 175–190. Zbl0736.53036
  7. [7] M. Audin, Torus Actions on Symplectic Manifolds (Second revised edition) Progress in Mathematics 93, Birkhäuser, 2004. 
  8. [8] L. Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227–261. [Crossref] Zbl0265.22016
  9. [9] D. Auroux, Asymptotically holomorphic families of symplectic submanifolds, Geom. Funct. Anal. 7 (1997), no. 6, 971–995. [Crossref] Zbl0912.53020
  10. [10] I. K. Babenko and I. A. Taˇimanov, On nonformal simply-connected symplecticmanifolds, SiberianMath. Journal 41 (2) (2000), 204–217. [Crossref] 
  11. [11] W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 4. Springer-Verlag, Berlin, 2004. 
  12. [12] O. Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology 43 (2004), 903–924. [Crossref] Zbl1059.57022
  13. [13] G. Bazzoni, M. Fernández and V. Muñoz, Non-formal co-symplectic manifolds, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4459–4481. Zbl1317.53040
  14. [14] G. Bazzoni, M. Fernández and V.Muñoz, A 6-dimensional simply connected complex and symplectic manifold with no Kähler metric, preprint http://arxiv.org/abs/1410.6045. 
  15. [15] G. Bazzoni and O. Goertsches, K-cosymplectic manifolds, Ann. Global Anal. Geom. 47 (2015), no. 3, 239–270. [Crossref] 
  16. [16] G. Bazzoni, G. Lupton and J. Oprea, Hereditary properties of co-Kähler manifolds, preprint http://arxiv.org/abs/1311.5675. Zbl1295.53016
  17. [17] G. Bazzoni and V. Muñoz, Classification of minimal algebras over any field up to dimension 6, Trans. Amer. Math. Soc. 364 (2012), no. 2, 1007–1028. Zbl1239.55003
  18. [18] G. Bazzoni and J. Oprea, On the structure of co-Kähler manifolds, Geom. Dedicata 170 (1) (2014), 71–85. Zbl1295.53016
  19. [19] C. Benson and C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518. [Crossref] Zbl0672.53036
  20. [20] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Mathematical Notes (2nd edition), London, 1981. Zbl0357.20027
  21. [21] D. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203, Birkhäuser, 2002. Zbl1011.53001
  22. [22] J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114. Zbl0634.58029
  23. [23] C. Bock, On low-dimensional solvmanifolds, preprint, http://arxiv.org/abs/0903.2926. Zbl06579256
  24. [24] A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2008. 
  25. [25] B. Cappelletti-Montano, A. de Nicola and I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys. 25 (10), 1343002 (2013). [Crossref] Zbl1292.53053
  26. [26] G. R. Cavalcanti, The Lefschetz property, formality and blowing up in symplectic geometry, Trans. Amer. Math. Soc. 359 (2007), no. 1, 333–348. Zbl1115.53060
  27. [27] G. R. Cavalcanti, M. Fernández and V. Muñoz, Symplectic resolutions, Lefschetz property and formality, Adv. Math. 218 (2008), no. 2, 576–599. [Crossref] Zbl1142.53070
  28. [28] D. Chinea, M. de León and J. C. Marrero, Topology of cosymplectic manifolds, J. Math. Pures Appl. 72 (1993), no. 6, 567–591. Zbl0845.53025
  29. [29] S. Console and A. Fino, On the de Rham cohomology of solvmanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 4, 801–818. Zbl1242.53055
  30. [30] S. Console and M.Macrì, Lattices, cohomology and models of six-dimensional almost abelian solvmanifolds, preprint, http: //arxiv.org/abs/1206.5977. 
  31. [31] L. A. Cordero, M. Fernández and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), no. 3, 375–380. [Crossref] Zbl0596.53030
  32. [32] P. Deligne, P. Griflths, J. Morgan and D. Sullivan, Real Homotopy Theory of Kähler Manifolds, Invent. Math. 29 (1975), no. 3, 245–274. [Crossref] Zbl0312.55011
  33. [33] S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996), no. 4, 666–705. Zbl0883.53032
  34. [34] S. K. Donaldson, Two-forms on four-manifolds and elliptic equations, Inspired by S. S. Chern, 153–172, Nankai Tracts.Math. 11, World Sci. Publ. Hackensack, NJ, 2006. [Crossref] Zbl1140.58018
  35. [35] Y. Félix, S. Halperin and J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer, 2001. 
  36. [36] Y. Félix, J. Oprea and D. Tanré, Algebraic Models in Geometry, Oxford Graduate Texts in Mathematics, 17. Oxford University Press, 2008. Zbl1149.53002
  37. [37] M. Fernández, M. de León and M. Saralegui, A six-dimensional compact symplectic solvmanifold without Kähler structures, Osaka J. Math. 33 (1996), no. 1, 19–35. Zbl0861.53032
  38. [38] M. Fernández, M. Gotay and A. Gray, Four-dimensional parallelizable symplectic and complex manifolds, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1209–1212. [Crossref] Zbl0656.53034
  39. [39] M. Fernández and A. Gray, The Iwasawa manifold, Differential geometry, Peñíscola 1985, 157–159, Lecture Notes in Math., 1209, Springer, Berlin, 1986. 
  40. [40] M. Fernández and A. Gray, Compact symplectic four dimensional solvmanifolds not admitting complex structures, Geom. Dedicata 34 (1990), no. 4, 295–299. Zbl0703.53030
  41. [41] M. Fernández and V.Muñoz, Homotopy properties of symplectic blow-ups, Proceedings of the XII Fall Workshop on Geometry and Physics, 95–109, Publ. R. Soc. Mat. Esp., 7, 2004. Zbl1066.57029
  42. [42] M. Fernández and V. Muñoz, Formality of Donaldson submanifolds, Math. Zeit. 250 (2005), no. 1, 149–175. [Crossref] Zbl1071.57024
  43. [43] M. Fernández and V.Muñoz, Non-formal compact manifolds with small Betti numbers, Proceedings of the Conference “Contemporary Geometry and Related Topics”. N. Bokan, M. Djoric, A. T. Fomenko, Z. Rakic, B. Wegner and J. Wess (editors), 231–246, 2006. Zbl1199.55009
  44. [44] M. Fernández and V. Muñoz, An 8-dimensional non-formal simply connected symplectic manifold, Ann. of Math. (2) 167 (2008), no. 3, 1045–1054. Zbl1173.57012
  45. [45] M. Fernández, V. Muñoz and J. Santisteban, Cohomologically Kähler manifolds with no Kähler metrics, IJMMS. 52 (2003), 3315–3325. Zbl1049.53059
  46. [46] R. E. Gompf, A new construction fo symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. Zbl0849.53027
  47. [47] P. Griflths and J. Morgan, Rational Homotopy Theory and Differential Forms (Second edition) Progress in Mathematics 16, Birkhäuser, 2013. 
  48. [48] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. [Crossref] Zbl0592.53025
  49. [49] M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9. Springer-Verlag, Berlin, 1986. 
  50. [50] Z.-D. Guan, Modification and the cohomology groups of compact solvmanifolds, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 74–81. [Crossref] Zbl1134.53024
  51. [51] Z.-D. Guan, Toward a Classification of Compact Nilmanifolds with Symplectic Structures, Int. Math. Res. Not. IMRN (2010), no. 22, 4377–4384. Zbl1236.53045
  52. [52] V. Guillemin, E. Miranda and A. R. Pires, Codimension one symplectic foliations and regular Poisson structures, Bull. Braz. Math. Soc. (N. S.) 42 (2011), no. 4, 607–623. [Crossref] Zbl1244.53093
  53. [53] V. Guillemin, E. Miranda and A. R. Pires, Symplectic and Poisson geometry of b-manifolds, Adv.Math. 264 (2014), 864–896. Zbl1296.53159
  54. [54] K. Hasegawa, Minimal Models of Nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71. [Crossref] Zbl0691.53040
  55. [55] K. Hasegawa, A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), no. 1, 131–135. Zbl1105.32017
  56. [56] A. Hattori, Spectral sequences in the de Rhamcohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331. Zbl0099.18003
  57. [57] D. Huybrechts, Complex geometry. An introduction, Universitext. Springer-Verlag, Berlin, 2005. 
  58. [58] R. Ibáñez, Y. Rudyak, A. Tralle and L. Ugarte, On certain geometric and homotopy properties of closed symplectic manifolds, Top. and its Appl. 127 (2003), no. 1-2, 33–45. [Crossref] Zbl1033.53077
  59. [59] H. Kasuya, Cohomologically symplectic solvmanifolds are symplectic, J. Symplectic Geom. 9 (2011), no. 4, 429–434. Zbl1338.53118
  60. [60] H. Kasuya, Formality and hard Lefschetz property of aspherical manifolds, Osaka J. Math. 50 (2013), no. 2, 439–455. Zbl1283.53068
  61. [61] J. Kędra, Y. Rudyak and A. Tralle, Symplectically aspherical manifolds, J. Fixed Point Theory Appl. 3 (2008), no. 1, 1–21. Zbl1149.53320
  62. [62] K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751–798. [Crossref] Zbl0137.17501
  63. [63] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in “Élie Cartan et les Math. d’Aujourd’Hui”, Astérisque horssérie, 1985, 251–271. 
  64. [64] D. Kotschick, On products of harmonic forms, Duke Math. J. 107 (2001), no. 3, 521–531. Zbl1036.53030
  65. [65] D. Kotschick and S. Terzić, On formality of generalized symmetric spaces, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 3, 491–505. Zbl1042.53035
  66. [66] H. Li, Topology of co-symplectic/co-Kähler manifolds, Asian J. Math., 12 (2008), no. 4, 527–543. Zbl1170.53014
  67. [67] T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Commun. Anal. Geom. 17 (2009), no. 4, 651–683. [Crossref] Zbl1225.53066
  68. [68] P. Libermann, Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact, Colloque Géom. Diff. Globale (Bruxelles 1958), 37–59, 1959. 
  69. [69] P. Libermann and C. Marle, Symplectic Geometry and Analytical Mechanics, Kluwer, Dordrecht, 1987. 
  70. [70] G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), no. 1-3, 193–207. [Crossref] Zbl0789.55010
  71. [71] G. Lupton and J. Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995), no. 1, 261–288. Zbl0836.57019
  72. [72] M.Macrì, Cohomological properties of unimodular six dimensional solvable Lie algebras, Differential Geom. Appl. 31 (2013), no. 1, 112–129. [Crossref] Zbl1263.53045
  73. [73] A. Mal’čev, On a class of homogeneous spaces, Izv. Akad. Nauk. Armyan. SSSR Ser. Mat. 13 (1949), 201–212. 
  74. [74] J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. Zbl0327.58005
  75. [75] D.Martínez Torres, Codimension-one foliations calibrated by nondegenerate closed 2-forms, Pacific J.Math. 261 (2013), no. 1, 165–217. Zbl1276.53032
  76. [76] O. Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Hel. 70 (1995), no. 1, 1–9. [Crossref] Zbl0831.58004
  77. [77] D. McDuff, Examples of symplectic simply connected manifolds with no Kähler structure, J. Diff. Geom. 20 (1984), no. 1, 267–277. 
  78. [78] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Second edition. Oxford Mathematical Monographs, 1998. Zbl0844.58029
  79. [79] D. McDuff and D. Salamon, J-holomophic curves and symplectic topology, Colloquium Publications Volume 52, American Mathematical Society, 2004. Zbl1064.53051
  80. [80] S. A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices (1998), no. 14, 727–733. [Crossref] Zbl0931.58002
  81. [81] J. T. Miller, On the formality of (k − 1)-connected compact manifolds of dimension less than or equal to (4k − 2), Illinois J. Math. 23 (1979), 253–258. Zbl0412.57014
  82. [82] V.Muñoz, F. Presas and I. Sols, Almost holomorphic embeddings in Grassmannians with applications to singular symplectic submanifolds, J. Reine Angew. Math. 547 (2002), 149–189. Zbl1004.53042
  83. [83] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), no. 2, 531–538. [Crossref] Zbl0058.02202
  84. [84] J. Oprea and A. Tralle, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, 1661, Springer-Verlag, Berlin, 1997. Zbl0891.53001
  85. [85] L. Ornea and M. Pilca, Remarks on the product of harmonic forms, Pacific J. Math. 250 (2011), no. 2, 353–363. Zbl1232.53033
  86. [86] S. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2–3, 311–333. Zbl1020.17006
  87. [87] H. Sawai and T. Yamada, Lattices on Benson-Gordon type solvable Lie groups, Topology Appl. 149 (2005), no. 1–3, 85–95. Zbl1069.53059
  88. [88] P. Seidel, Fukaya categories and Picard-Lefschetz theory, European Mathematical Society, Zürich, 2008. Zbl1159.53001
  89. [89] D. Sullivan, Infinitesimal Computations in Topology, Publications Mathématiques de l’I. H. É. S. 47 (1977), 269–331. 
  90. [90] W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468. Zbl0324.53031
  91. [91] D. Tischler, Closed 2-forms and an embedding theorem for symplectic manifolds, J. Diff. Geom. 12 (1977), 229–235. Zbl0386.58001
  92. [92] D. Yan, Hodge Structure on Symplectic Manifolds, Adv. Math. 120 (1996), no. 1, 143–154. [Crossref] Zbl0872.58002

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