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On the Symplectic Structures in Frame Bundles and the Finite Dimension of Basic Symplectic Cohomologies
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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We present a construction (and classification) of certain invariant 2-forms on the real symplectic group. They are used to define a symplectic form on the quotient by a maximal torus and to “lift” a symplectic structure from a symplectic manifold to the bundle of frames. This is a by-product of a failed attempt to prove certain finiteness theorems for basic symplectic cohomologies. In the last part of the paper we include a valid proof.
Keywords: symplectic cohomology; basic cohomology. 
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     author = {Andrzej Czarnecki},
     title = {On the {Symplectic} {Structures} in {Frame} {Bundles} and the {Finite} {Dimension} of {Basic} {Symplectic} {Cohomologies}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a28/}
}
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Andrzej Czarnecki. On the Symplectic Structures in Frame Bundles and the Finite Dimension of Basic Symplectic Cohomologies. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a28/

[1] Angella D., Cohomological aspects in complex non-Kähler geometry, Lecture Notes in Math., 2095, Springer, Cham, 2014 | DOI | MR

[2] Bak L., Czarnecki A., “A remark on the Brylinski conjecture for orbifolds”, J. Aust. Math. Soc., 91 (2011), 1–12, arXiv: 1001.2435 | DOI | MR

[3] Brylinski J.-L., “A differential complex for Poisson manifolds”, J. Differential Geom., 28 (1988), 93–114 | DOI | MR

[4] Cavalcanti G., New aspects of the ${\rm dd}^c$-lemma, arXiv: math.DG/0501406

[5] Courter R. C., “The dimension of maximal commutative subalgebras of $K_{n}$”, Duke Math. J., 32 (1965), 225–232 | DOI | MR

[6] Czarnecki A., Raźny P., “Examples of foliations with infinite dimensional special cohomology”, Ann. Mat. Pura Appl., 197 (2018), 399–409, arXiv: 1705.02216 | DOI | MR

[7] del Olmo M. A., Rodríguez M. A., Winternitz P., Zassenhaus H., “Maximal abelian subalgebras of ${\rm su}(p,q)$”, XVIIth International Colloquium on Group Theoretical Methods in Physics (Sainte-Adèle, PQ, 1988), World Sci. Publ., Teaneck, NJ, 1989, 401–404 | MR

[8] El Kacimi-Alaoui A., “Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications”, Compositio Math., 73 (1990), 57–106 | MR

[9] El Kacimi-Alaoui A., Sergiescu V., Hector G., “La cohomologie basique d'un feuilletage riemannien est de dimension finie”, Math. Z., 188 (1985), 593–599 | DOI | MR

[10] Fernández M., Ibáñez R., de León M., “On a Brylinski conjecture for compact symplectic manifolds”, Quaternionic structures in mathematics and physics (Trieste, 1994), Int. Sch. Adv. Stud. (SISSA), Trieste, 1998, 119–126 | MR

[11] He Z., Odd dimenisonal symplectic manifolds, Ph.D. Thesis, Massachusetts Institute of Technology, 2010 http://hdl.handle.net/1721.1/60189

[12] Hussin V., Winternitz P., Zassenhaus H., “Maximal abelian subalgebras of complex orthogonal Lie algebras”, Linear Algebra Appl., 141 (1990), 183–220 | DOI | MR

[13] Laffey T. J., “The minimal dimension of maximal commutative subalgebras of full matrix algebras”, Linear Algebra Appl., 71 (1985), 199–212 | DOI | MR

[14] Lalonde F., McDuff D., “Symplectic structures on fiber bundles”, Topology, 42 (2003), 309–347, arXiv: math.SG/0010275 | DOI | MR

[15] Molino P., Riemannian foliations, Progress in Mathematics, 73, Birkhäuser Boston, Inc., Boston, MA, 1988 | DOI | MR

[16] Pak H. K., “Transversal harmonic theory for transversally symplectic flows”, J. Aust. Math. Soc., 84 (2008), 233–245 | DOI | MR

[17] Patera J., Winternitz P., Zassenhaus H., “Maximal abelian subalgebras of real and complex symplectic Lie algebras”, J. Math. Phys., 24 (1983), 1973–1985 | DOI | MR

[18] Raźny P., “The Frölicher-type inequalities of foliations”, J. Geom. Phys., 114 (2017), 593–606, arXiv: 1605.03858 | DOI | MR

[19] Sugiura M., “Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras”, J. Math. Soc. Japan, 11 (1959), 374–434 ; Erratum, J. Math. Soc. Japan, 23 (1971), 379–383 | DOI | MR | DOI | MR

[20] Thurston W. P., “Some simple examples of symplectic manifolds”, Proc. Amer. Math. Soc., 55 (1976), 467–468 | DOI | MR

[21] Tseng L.-S., Yau S.-T., “Cohomology and {H}odge theory on symplectic manifolds: I”, J. Differential Geom., 91 (2012), 383–416, arXiv: 0909.5418 | DOI | MR

[22] Tseng L.-S., Yau S.-T., “Cohomology and Hodge theory on symplectic manifolds: II”, J. Differential Geom., 91 (2012), 417–443, arXiv: 1011.1250 | DOI | MR

[23] Weinstein A., “Fat bundles and symplectic manifolds”, Adv. Math., 37 (1980), 239–250 | DOI | MR

[24] Yan D., “Hodge structure on symplectic manifolds”, Adv. Math., 120 (1996), 143–154 | DOI | MR