×

Symplectic cohomologies and deformations. (English) Zbl 1418.32016

Summary: In this note we study the behavior of symplectic cohomology groups under symplectic deformations. Moreover, we show that for compact almost-Kähler manifolds \((X,J,g,\omega )\) with \(J\) \({\mathcal {C}}^\infty \)-pure and full the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern harmonic forms. Furthermore, we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott-Chern harmonic forms.

MSC:

32Q60 Almost complex manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58A12 de Rham theory in global analysis
53D05 Symplectic manifolds (general theory)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aeppli, A.: On the Cohomology structure of stein manifolds. In: Aeppli, A., Calabi, E., Röhrl, H. (eds.) Proceedings of the Conference on Complex Analysis. Springer, Berlin, Heidelberg (1965) · Zbl 0166.33902
[2] Angella, D., Kasuya, H.: Symplectic Bott-Chern cohomology of solvmanifolds, to appear in J. Symplectic Geom., arXiv:1308.4258 [math.SG] · Zbl 1384.22004
[3] Angella, D., Otiman, A., Tardini, N.: Cohomologies of locally conformally symplectic manifolds and solvmanifolds. Ann. Glob. Anal. Geom. 53(1), 67-96 (2018) · Zbl 1394.32022 · doi:10.1007/s10455-017-9568-y
[4] Angella, D., Tomassini, A.: On the \[\partial{\overline{\partial }} \]∂∂¯-lemma and Bott-Chern cohomology. Invent. Math. 192(1), 71-81 (2013) · Zbl 1271.32011 · doi:10.1007/s00222-012-0406-3
[5] Angella, D., Tomassini, A.: Symplectic manifolds and cohomological decomposition. J. Symplectic Geom. 12(2), 215-236 (2014) · Zbl 1305.53082 · doi:10.4310/JSG.2014.v12.n2.a1
[6] Angella, D., Tomassini, A.: On the cohomology of almost-complex manifolds. Int. J. Math. 23(2), 25 (2012) · Zbl 1238.53053 · doi:10.1142/S0129167X11007604
[7] Angella, D., Tomassini, A.: Inequalities à la Fröhlicher and cohomological decompositions. J. Noncommut. Geom. 9(2), 505-542 (2015) · Zbl 1325.32018 · doi:10.4171/JNCG/199
[8] Angella, D., Tomassini, A., Verbitsky, M.: On non-Kähler degrees of complex manifolds. Adv. in Geom. https://doi.org/10.1515/advgeom-2018-0026 · Zbl 1416.32011
[9] Angella, D., Tomassini, A., Zhang, W.: On decomposablity of almost-Kähler structures. Proc. Amer. Math. Soc. 142, 3615-3630 (2014) · Zbl 1298.53073 · doi:10.1090/S0002-9939-2014-12049-1
[10] Belluscio, G., Tomassini, A.: On the cohomology of almost Kähler manifolds. J. Geom. and Phys. 116, 146-151 (2017) · Zbl 1371.53067 · doi:10.1016/j.geomphys.2017.01.028
[11] Bott, R., Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114(1), 71-112 (1965) · Zbl 0148.31906 · doi:10.1007/BF02391818
[12] Brylinski, J.L.: A differential complex for Poisson manifolds. J. Differential Geom. 28, 93-114 (1988) · Zbl 0634.58029 · doi:10.4310/jdg/1214442161
[13] Cavalcanti, G.R.: New aspects of the \[dd^c\] ddc-lemma, Oxford University D. Phil thesis, arXiv:math/0501406v1 [math.DG]
[14] Chan, K., Suen, Y.-H.: A Frölicher-type inequality for generalized complex manifolds. Ann. Global Anal. Geom. 47(2), 135-145 (2015) · Zbl 1309.32007 · doi:10.1007/s10455-014-9439-8
[15] de Bartolomeis, \[P.: {\mathbb{Z}}_2\] Z2 and \[{\mathbb{Z}}\] Z-deformation theory for holomorphic and symplectic, complex, contact and symmetric manifolds. Prog. Math. 234, 75-104 (2005)
[16] Deligne, P., Griffiths, PhA, Morgan, J., Sullivan, D.P.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245-274 (1975) · Zbl 0312.55011 · doi:10.1007/BF01389853
[17] Drǎghici, T., Li, T.-J., Zhang, W.: Symplectic form and cohomology decomposition of almost complex four-manifolds. Int. Math. Res. Not. 2010(1), 1-17 (2010) · Zbl 1190.32021
[18] Fino, A., Tomassini, A.: On some cohomolgical properties of almost complex manifolds. J. Geom. Anal. 20, 107-131 (2010) · Zbl 1186.53085 · doi:10.1007/s12220-009-9098-3
[19] Hind, R., Medori, C., Tomassini, A.: On taming and compatible symplectic forms. J. Geom. Anal. 25, 2360-2374 (2015) · Zbl 1337.32039 · doi:10.1007/s12220-014-9516-z
[20] Macrí, M.: Cohomological properties of unimodular six dimensional solvable Lie algebras. (English summary). Differ. Geom. Appl. 31(1), 112-129 (2013) · Zbl 1263.53045 · doi:10.1016/j.difgeo.2012.10.002
[21] Mathieu, O.: Harmonic cohomology classes of symplectic manifolds. Comment. Math. Helv. 70(1), 1-9 (1995) · Zbl 0831.58004 · doi:10.1007/BF02565997
[22] Merkulov, S.A.: Formality of canonical symplectic complexes and Frobenius manifolds. Int. Math. Res. Not. 1998(14), 727-733 (1998) · Zbl 0931.58002 · doi:10.1155/S1073792898000439
[23] Li, T.-J., Zhang, W.: Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds. Comm. Anal. and Geom. 17, 651-684 (2009) · Zbl 1225.53066 · doi:10.4310/CAG.2009.v17.n4.a4
[24] Morozov, V.V.: Classification of nilpotent Lie algebras of sixth order. Izv. Vyss. Ucebn. Zaved. Matematika 4(5), 161-171 (1958) · Zbl 0198.05501
[25] Rinaldi, M.: Proprietà Coomologiche di Varietà Simplettiche, Master thesis, advisor Prof. A. Tomassini, Università di Parma (2014), http://www.bibliomath.unipr.it/TesiDipNuovo.html. Accessed 28 Mar 2014
[26] Tardini, N.: Cohomological aspects on complex and symplectic manifolds, complex and symplectic geometry. Springer INdAM Ser. 21, 31-247 (2017). (Springer) · Zbl 1402.53061
[27] Tardini, N., Tomassini, A.: On the cohomology of almost-complex and symplectic manifolds and proper surjective maps. Internat. J. Math. 27(12), 1650103 (2016). (20 pages) · Zbl 1358.53038 · doi:10.1142/S0129167X16501032
[28] Teleman, A.: The pseudo-effective cone of a non-Kählerian surface and applications. Math. Ann. 335(4), 965-989 (2006) · Zbl 1096.32011 · doi:10.1007/s00208-006-0782-3
[29] Tseng, L.S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: I. J. Differ. Geom. 91, 383-416 (2012) · Zbl 1275.53079 · doi:10.4310/jdg/1349292670
[30] Yan, D.: Hodge structure on symplectic manifolds. Adv. in Math. 120, 143-154 (1996) · Zbl 0872.58002 · doi:10.1006/aima.1996.0034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.