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Complex Manifolds

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Formality and the Lefschetz property in symplectic and cosymplectic geometry

Giovanni Bazzoni
  • Corresponding author
  • Fakultät für Mathematik, Universität Bielefeld, Postfach 100301, D-33501 Bielefeld
  • Other articles by this author:
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/ Marisa Fernández
  • Corresponding author
  • Universidad del País Vasco, Facultad de Ciencia y Tecnología, Departamento de Matemáticas, Apartado 644, 48080 Bilbao, Spain
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Vicente Muñoz
  • Corresponding author
  • Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
  • Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera 15, 28049 Madrid, Spain
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Published Online: 2015-07-29 | DOI: https://doi.org/10.1515/coma-2015-0006

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).

MSC: 53C15; 55S30; 53D35; 55P62; 57R17

References

  • [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. Google Scholar

  • [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. Google Scholar

  • [3] D. Angella, Cohomological Aspects in Complex Non-Kähler Geometry, Lecture Notes inMathematics, 2095, Springer-Verlag, Berlin, 2014. Google Scholar

  • [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. Google Scholar

  • [5] V. I. Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Graduate Texts in Mathematics 60, Springer, 1997. Google Scholar

  • [6] M. Audin, Exemples de variétés presque complexes, Einseign. Math. (2) 37 (1991), no. 1–2, 175–190. Google Scholar

  • [7] M. Audin, Torus Actions on Symplectic Manifolds (Second revised edition) Progress in Mathematics 93, Birkhäuser, 2004. Google Scholar

  • [8] L. Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227–261. CrossrefGoogle Scholar

  • [9] D. Auroux, Asymptotically holomorphic families of symplectic submanifolds, Geom. Funct. Anal. 7 (1997), no. 6, 971–995. CrossrefGoogle Scholar

  • [10] I. K. Babenko and I. A. Taˇimanov, On nonformal simply-connected symplecticmanifolds, SiberianMath. Journal 41 (2) (2000), 204–217. CrossrefGoogle Scholar

  • [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. Google Scholar

  • [12] O. Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology 43 (2004), 903–924. CrossrefGoogle Scholar

  • [13] G. Bazzoni, M. Fernández and V. Muñoz, Non-formal co-symplectic manifolds, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4459–4481. Google Scholar

  • [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. Google Scholar

  • [15] G. Bazzoni and O. Goertsches, K-cosymplectic manifolds, Ann. Global Anal. Geom. 47 (2015), no. 3, 239–270. CrossrefGoogle Scholar

  • [16] G. Bazzoni, G. Lupton and J. Oprea, Hereditary properties of co-Kähler manifolds, preprint http://arxiv.org/abs/1311.5675. Google Scholar

  • [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. Google Scholar

  • [18] G. Bazzoni and J. Oprea, On the structure of co-Kähler manifolds, Geom. Dedicata 170 (1) (2014), 71–85. Google Scholar

  • [19] C. Benson and C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518. CrossrefGoogle Scholar

  • [20] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Mathematical Notes (2nd edition), London, 1981. Google Scholar

  • [21] D. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203, Birkhäuser, 2002. Google Scholar

  • [22] J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114. Google Scholar

  • [23] C. Bock, On low-dimensional solvmanifolds, preprint, http://arxiv.org/abs/0903.2926. Google Scholar

  • [24] A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2008. Google Scholar

  • [25] B. Cappelletti-Montano, A. de Nicola and I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys. 25 (10), 1343002 (2013). CrossrefGoogle Scholar

  • [26] G. R. Cavalcanti, The Lefschetz property, formality and blowing up in symplectic geometry, Trans. Amer. Math. Soc. 359 (2007), no. 1, 333–348. Google Scholar

  • [27] G. R. Cavalcanti, M. Fernández and V. Muñoz, Symplectic resolutions, Lefschetz property and formality, Adv. Math. 218 (2008), no. 2, 576–599. CrossrefGoogle Scholar

  • [28] D. Chinea, M. de León and J. C. Marrero, Topology of cosymplectic manifolds, J. Math. Pures Appl. 72 (1993), no. 6, 567–591. Google Scholar

  • [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. Google Scholar

  • [30] S. Console and M.Macrì, Lattices, cohomology and models of six-dimensional almost abelian solvmanifolds, preprint, http: //arxiv.org/abs/1206.5977. Google Scholar

  • [31] L. A. Cordero, M. Fernández and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), no. 3, 375–380. CrossrefGoogle Scholar

  • [32] P. Deligne, P. Griflths, J. Morgan and D. Sullivan, Real Homotopy Theory of Kähler Manifolds, Invent. Math. 29 (1975), no. 3, 245–274. CrossrefGoogle Scholar

  • [33] S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996), no. 4, 666–705. Google Scholar

  • [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. CrossrefGoogle Scholar

  • [35] Y. Félix, S. Halperin and J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer, 2001. Google Scholar

  • [36] Y. Félix, J. Oprea and D. Tanré, Algebraic Models in Geometry, Oxford Graduate Texts in Mathematics, 17. Oxford University Press, 2008. Google Scholar

  • [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. Google Scholar

  • [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. CrossrefGoogle Scholar

  • [39] M. Fernández and A. Gray, The Iwasawa manifold, Differential geometry, Peñíscola 1985, 157–159, Lecture Notes in Math., 1209, Springer, Berlin, 1986. Google Scholar

  • [40] M. Fernández and A. Gray, Compact symplectic four dimensional solvmanifolds not admitting complex structures, Geom. Dedicata 34 (1990), no. 4, 295–299. Google Scholar

  • [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. Google Scholar

  • [42] M. Fernández and V. Muñoz, Formality of Donaldson submanifolds, Math. Zeit. 250 (2005), no. 1, 149–175. CrossrefGoogle Scholar

  • [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. Google Scholar

  • [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. Google Scholar

  • [45] M. Fernández, V. Muñoz and J. Santisteban, Cohomologically Kähler manifolds with no Kähler metrics, IJMMS. 52 (2003), 3315–3325. Google Scholar

  • [46] R. E. Gompf, A new construction fo symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. Google Scholar

  • [47] P. Griflths and J. Morgan, Rational Homotopy Theory and Differential Forms (Second edition) Progress in Mathematics 16, Birkhäuser, 2013. Google Scholar

  • [48] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. CrossrefGoogle Scholar

  • [49] M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9. Springer-Verlag, Berlin, 1986. Google Scholar

  • [50] Z.-D. Guan, Modification and the cohomology groups of compact solvmanifolds, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 74–81. CrossrefGoogle Scholar

  • [51] Z.-D. Guan, Toward a Classification of Compact Nilmanifolds with Symplectic Structures, Int. Math. Res. Not. IMRN (2010), no. 22, 4377–4384. Google Scholar

  • [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. CrossrefGoogle Scholar

  • [53] V. Guillemin, E. Miranda and A. R. Pires, Symplectic and Poisson geometry of b-manifolds, Adv.Math. 264 (2014), 864–896. Google Scholar

  • [54] K. Hasegawa, Minimal Models of Nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71. CrossrefGoogle Scholar

  • [55] K. Hasegawa, A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), no. 1, 131–135. Google Scholar

  • [56] A. Hattori, Spectral sequences in the de Rhamcohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331. Google Scholar

  • [57] D. Huybrechts, Complex geometry. An introduction, Universitext. Springer-Verlag, Berlin, 2005. Google Scholar

  • [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. CrossrefGoogle Scholar

  • [59] H. Kasuya, Cohomologically symplectic solvmanifolds are symplectic, J. Symplectic Geom. 9 (2011), no. 4, 429–434. Google Scholar

  • [60] H. Kasuya, Formality and hard Lefschetz property of aspherical manifolds, Osaka J. Math. 50 (2013), no. 2, 439–455. Google Scholar

  • [61] J. Kędra, Y. Rudyak and A. Tralle, Symplectically aspherical manifolds, J. Fixed Point Theory Appl. 3 (2008), no. 1, 1–21. Google Scholar

  • [62] K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751–798. CrossrefGoogle Scholar

  • [63] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in “Élie Cartan et les Math. d’Aujourd’Hui”, Astérisque horssérie, 1985, 251–271. Google Scholar

  • [64] D. Kotschick, On products of harmonic forms, Duke Math. J. 107 (2001), no. 3, 521–531. Google Scholar

  • [65] D. Kotschick and S. Terzić, On formality of generalized symmetric spaces, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 3, 491–505. Google Scholar

  • [66] H. Li, Topology of co-symplectic/co-Kähler manifolds, Asian J. Math., 12 (2008), no. 4, 527–543. Google Scholar

  • [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. CrossrefGoogle Scholar

  • [68] P. Libermann, Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact, Colloque Géom. Diff. Globale (Bruxelles 1958), 37–59, 1959. Google Scholar

  • [69] P. Libermann and C. Marle, Symplectic Geometry and Analytical Mechanics, Kluwer, Dordrecht, 1987. Google Scholar

  • [70] G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), no. 1-3, 193–207. CrossrefGoogle Scholar

  • [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. Google Scholar

  • [72] M.Macrì, Cohomological properties of unimodular six dimensional solvable Lie algebras, Differential Geom. Appl. 31 (2013), no. 1, 112–129. CrossrefGoogle Scholar

  • [73] A. Mal’čev, On a class of homogeneous spaces, Izv. Akad. Nauk. Armyan. SSSR Ser. Mat. 13 (1949), 201–212. Google Scholar

  • [74] J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. Google Scholar

  • [75] D.Martínez Torres, Codimension-one foliations calibrated by nondegenerate closed 2-forms, Pacific J.Math. 261 (2013), no. 1, 165–217. Google Scholar

  • [76] O. Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Hel. 70 (1995), no. 1, 1–9. CrossrefGoogle Scholar

  • [77] D. McDuff, Examples of symplectic simply connected manifolds with no Kähler structure, J. Diff. Geom. 20 (1984), no. 1, 267–277. Google Scholar

  • [78] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Second edition. Oxford Mathematical Monographs, 1998. Google Scholar

  • [79] D. McDuff and D. Salamon, J-holomophic curves and symplectic topology, Colloquium Publications Volume 52, American Mathematical Society, 2004. Google Scholar

  • [80] S. A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices (1998), no. 14, 727–733. CrossrefGoogle Scholar

  • [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. Google Scholar

  • [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. Google Scholar

  • [83] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), no. 2, 531–538. CrossrefGoogle Scholar

  • [84] J. Oprea and A. Tralle, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, 1661, Springer-Verlag, Berlin, 1997. Google Scholar

  • [85] L. Ornea and M. Pilca, Remarks on the product of harmonic forms, Pacific J. Math. 250 (2011), no. 2, 353–363. Google Scholar

  • [86] S. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2–3, 311–333. Google Scholar

  • [87] H. Sawai and T. Yamada, Lattices on Benson-Gordon type solvable Lie groups, Topology Appl. 149 (2005), no. 1–3, 85–95. Google Scholar

  • [88] P. Seidel, Fukaya categories and Picard-Lefschetz theory, European Mathematical Society, Zürich, 2008. Google Scholar

  • [89] D. Sullivan, Infinitesimal Computations in Topology, Publications Mathématiques de l’I. H. É. S. 47 (1977), 269–331. Google Scholar

  • [90] W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468. Google Scholar

  • [91] D. Tischler, Closed 2-forms and an embedding theorem for symplectic manifolds, J. Diff. Geom. 12 (1977), 229–235. Google Scholar

  • [92] D. Yan, Hodge Structure on Symplectic Manifolds, Adv. Math. 120 (1996), no. 1, 143–154. CrossrefGoogle Scholar

About the article

Received: 2015-04-14

Accepted: 2015-07-11

Published Online: 2015-07-29


Citation Information: Complex Manifolds, Volume 2, Issue 1, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2015-0006.

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