Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników

Czasopismo

2015 | 2 | 1 |

Tytuł artykułu

Formality and the Lefschetz property in symplectic and cosymplectic geometry

Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
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).

Słowa kluczowe

Twórcy

  • Fakultät für Mathematik, Universität Bielefeld, Postfach 100301, D-33501 Bielefeld
  • Universidad del País Vasco, Facultad de Ciencia y Tecnología, Departamento de Matemáticas, Apartado
    644, 48080 Bilbao, Spain
  • 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

Bibliografia

  • [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.
  • [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.
  • [3] D. Angella, Cohomological Aspects in Complex Non-Kähler Geometry, Lecture Notes inMathematics, 2095, Springer-Verlag, Berlin, 2014.
  • [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.
  • [5] V. I. Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Graduate Texts in Mathematics 60, Springer, 1997.
  • [6] M. Audin, Exemples de variétés presque complexes, Einseign. Math. (2) 37 (1991), no. 1–2, 175–190.
  • [7] M. Audin, Torus Actions on Symplectic Manifolds (Second revised edition) Progress in Mathematics 93, Birkhäuser, 2004.
  • [8] L. Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227–261. [Crossref]
  • [9] D. Auroux, Asymptotically holomorphic families of symplectic submanifolds, Geom. Funct. Anal. 7 (1997), no. 6, 971–995. [Crossref]
  • [10] I. K. Babenko and I. A. Taˇimanov, On nonformal simply-connected symplecticmanifolds, SiberianMath. Journal 41 (2) (2000), 204–217. [Crossref]
  • [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] O. Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology 43 (2004), 903–924. [Crossref]
  • [13] G. Bazzoni, M. Fernández and V. Muñoz, Non-formal co-symplectic manifolds, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4459–4481.
  • [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] G. Bazzoni and O. Goertsches, K-cosymplectic manifolds, Ann. Global Anal. Geom. 47 (2015), no. 3, 239–270. [Crossref]
  • [16] G. Bazzoni, G. Lupton and J. Oprea, Hereditary properties of co-Kähler manifolds, preprint http://arxiv.org/abs/1311.5675.
  • [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.
  • [18] G. Bazzoni and J. Oprea, On the structure of co-Kähler manifolds, Geom. Dedicata 170 (1) (2014), 71–85.
  • [19] C. Benson and C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518. [Crossref]
  • [20] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Mathematical Notes (2nd edition), London, 1981.
  • [21] D. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203, Birkhäuser, 2002.
  • [22] J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114.
  • [23] C. Bock, On low-dimensional solvmanifolds, preprint, http://arxiv.org/abs/0903.2926.
  • [24] A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2008.
  • [25] B. Cappelletti-Montano, A. de Nicola and I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys. 25 (10), 1343002 (2013). [Crossref]
  • [26] G. R. Cavalcanti, The Lefschetz property, formality and blowing up in symplectic geometry, Trans. Amer. Math. Soc. 359 (2007), no. 1, 333–348.
  • [27] G. R. Cavalcanti, M. Fernández and V. Muñoz, Symplectic resolutions, Lefschetz property and formality, Adv. Math. 218 (2008), no. 2, 576–599. [Crossref]
  • [28] D. Chinea, M. de León and J. C. Marrero, Topology of cosymplectic manifolds, J. Math. Pures Appl. 72 (1993), no. 6, 567–591.
  • [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.
  • [30] S. Console and M.Macrì, Lattices, cohomology and models of six-dimensional almost abelian solvmanifolds, preprint, http: //arxiv.org/abs/1206.5977.
  • [31] L. A. Cordero, M. Fernández and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), no. 3, 375–380. [Crossref]
  • [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]
  • [33] S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996), no. 4, 666–705.
  • [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]
  • [35] Y. Félix, S. Halperin and J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer, 2001.
  • [36] Y. Félix, J. Oprea and D. Tanré, Algebraic Models in Geometry, Oxford Graduate Texts in Mathematics, 17. Oxford University Press, 2008.
  • [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.
  • [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]
  • [39] M. Fernández and A. Gray, The Iwasawa manifold, Differential geometry, Peñíscola 1985, 157–159, Lecture Notes in Math., 1209, Springer, Berlin, 1986.
  • [40] M. Fernández and A. Gray, Compact symplectic four dimensional solvmanifolds not admitting complex structures, Geom. Dedicata 34 (1990), no. 4, 295–299.
  • [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.
  • [42] M. Fernández and V. Muñoz, Formality of Donaldson submanifolds, Math. Zeit. 250 (2005), no. 1, 149–175. [Crossref]
  • [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.
  • [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.
  • [45] M. Fernández, V. Muñoz and J. Santisteban, Cohomologically Kähler manifolds with no Kähler metrics, IJMMS. 52 (2003), 3315–3325.
  • [46] R. E. Gompf, A new construction fo symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595.
  • [47] P. Griflths and J. Morgan, Rational Homotopy Theory and Differential Forms (Second edition) Progress in Mathematics 16, Birkhäuser, 2013.
  • [48] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. [Crossref]
  • [49] M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9. Springer-Verlag, Berlin, 1986.
  • [50] Z.-D. Guan, Modification and the cohomology groups of compact solvmanifolds, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 74–81. [Crossref]
  • [51] Z.-D. Guan, Toward a Classification of Compact Nilmanifolds with Symplectic Structures, Int. Math. Res. Not. IMRN (2010), no. 22, 4377–4384.
  • [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]
  • [53] V. Guillemin, E. Miranda and A. R. Pires, Symplectic and Poisson geometry of b-manifolds, Adv.Math. 264 (2014), 864–896.
  • [54] K. Hasegawa, Minimal Models of Nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71. [Crossref]
  • [55] K. Hasegawa, A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), no. 1, 131–135.
  • [56] A. Hattori, Spectral sequences in the de Rhamcohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331.
  • [57] D. Huybrechts, Complex geometry. An introduction, Universitext. Springer-Verlag, Berlin, 2005.
  • [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]
  • [59] H. Kasuya, Cohomologically symplectic solvmanifolds are symplectic, J. Symplectic Geom. 9 (2011), no. 4, 429–434.
  • [60] H. Kasuya, Formality and hard Lefschetz property of aspherical manifolds, Osaka J. Math. 50 (2013), no. 2, 439–455.
  • [61] J. Kędra, Y. Rudyak and A. Tralle, Symplectically aspherical manifolds, J. Fixed Point Theory Appl. 3 (2008), no. 1, 1–21.
  • [62] K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751–798. [Crossref]
  • [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] D. Kotschick, On products of harmonic forms, Duke Math. J. 107 (2001), no. 3, 521–531.
  • [65] D. Kotschick and S. Terzić, On formality of generalized symmetric spaces, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 3, 491–505.
  • [66] H. Li, Topology of co-symplectic/co-Kähler manifolds, Asian J. Math., 12 (2008), no. 4, 527–543.
  • [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]
  • [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] P. Libermann and C. Marle, Symplectic Geometry and Analytical Mechanics, Kluwer, Dordrecht, 1987.
  • [70] G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), no. 1-3, 193–207. [Crossref]
  • [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.
  • [72] M.Macrì, Cohomological properties of unimodular six dimensional solvable Lie algebras, Differential Geom. Appl. 31 (2013), no. 1, 112–129. [Crossref]
  • [73] A. Mal’čev, On a class of homogeneous spaces, Izv. Akad. Nauk. Armyan. SSSR Ser. Mat. 13 (1949), 201–212.
  • [74] J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130.
  • [75] D.Martínez Torres, Codimension-one foliations calibrated by nondegenerate closed 2-forms, Pacific J.Math. 261 (2013), no. 1, 165–217.
  • [76] O. Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Hel. 70 (1995), no. 1, 1–9. [Crossref]
  • [77] D. McDuff, Examples of symplectic simply connected manifolds with no Kähler structure, J. Diff. Geom. 20 (1984), no. 1, 267–277.
  • [78] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Second edition. Oxford Mathematical Monographs, 1998.
  • [79] D. McDuff and D. Salamon, J-holomophic curves and symplectic topology, Colloquium Publications Volume 52, American Mathematical Society, 2004.
  • [80] S. A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices (1998), no. 14, 727–733. [Crossref]
  • [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.
  • [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.
  • [83] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), no. 2, 531–538. [Crossref]
  • [84] J. Oprea and A. Tralle, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, 1661, Springer-Verlag, Berlin, 1997.
  • [85] L. Ornea and M. Pilca, Remarks on the product of harmonic forms, Pacific J. Math. 250 (2011), no. 2, 353–363.
  • [86] S. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2–3, 311–333.
  • [87] H. Sawai and T. Yamada, Lattices on Benson-Gordon type solvable Lie groups, Topology Appl. 149 (2005), no. 1–3, 85–95.
  • [88] P. Seidel, Fukaya categories and Picard-Lefschetz theory, European Mathematical Society, Zürich, 2008.
  • [89] D. Sullivan, Infinitesimal Computations in Topology, Publications Mathématiques de l’I. H. É. S. 47 (1977), 269–331.
  • [90] W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468.
  • [91] D. Tischler, Closed 2-forms and an embedding theorem for symplectic manifolds, J. Diff. Geom. 12 (1977), 229–235.
  • [92] D. Yan, Hodge Structure on Symplectic Manifolds, Adv. Math. 120 (1996), no. 1, 143–154. [Crossref]

Typ dokumentu

Bibliografia

Identyfikatory

Identyfikator YADDA

bwmeta1.element.doi-10_1515_coma-2015-0006
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.