Abstract
In this paper, we calculate the dimension of the J-anti-invariant cohomology subgroup \(H_J^-\) on \({\mathbb {T}}^4\). Inspired by the concrete example, \({\mathbb {T}}^4\), we get that: on a closed symplectic 4-dimensional manifold \((M, \omega )\), \(h^-_J=0\) for generic \(\omega \)-compatible almost complex structures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Audin, M.: Symplectic and almost complex manifolds, with an appendix. In: Gauduchon, P. (ed.) Holomorphic Curves in Symplectic Geometry. Progress in Mathmatics, pp. 41–74. Birkhäuser, Basel (1994)
Bär, C.: On nodal sets for Dirac and Laplace operators. Commun. Math. Phys. 188, 709–721 (1997)
Barth, W., Hulek, K., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Springer, Berlin (2004)
Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds, Oxford Mathematical Monographs. Oxford Science Publications, New York (1990)
Draghici, T., Li, T.-J., Zhang, W.: Symplectic forms and cohomology decomposition of almost complex four-manifolds. Int. Math. Res. Not. 1, 1–17 (2010)
Draghici, T., Li, T.-J., Zhang, W.: On the \(J\)-anti-invariant cohomology of almost complex \(4\)-manifolds. Q. J. Math. 64, 83–111 (2013)
Gauduchon, P.: Le théorème de l’excetricité nulle. C. R. Acad. Sc. Paris. Série A 285, 387–390 (1977)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)
Hirsch, M.W.: Differential Topology, Graduate Texts in Mathematics, vol. 33. Springer, New York (1976)
Kim, J.: A closed symplectic four-manifold has almost Kähler metrics of negative scalar curvature. Ann. Glob. Anal. Geom. 33, 125–136 (2008)
Kodaira, K., Morrow, J.: Complex Manifolds. Holt, Rinehart and Winston, New York (1971)
Lejmi, M.: Strictly nearly Kähler 6-manifolds are not compatible with symplectic forms. C. R. Math. Acad. Sci. Paris 34, 759–762 (2006)
Lejmi, M.: Stability under deformations of extremal almost-Kähler metrics in dimension \(4\). Math. Res. Lett. 17, 601–612 (2010)
Lejmi, M.: Stability under deformations of Hermitian-Einstein almost-Kähler metrics. Ann. Inst. Fourier 64, 2251–2263 (2014)
Li, T.-J., Zhang, W.: Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds. Commun. Anal. Geom. 17, 651–684 (2009)
McDuff, D., Salamon, D.: Introduction to symplectic topology, Oxford Mathematical Monographs, 2nd edn. Oxford University Press, Oxford (1998)
Salamon, S.: Special structures on four-manifolds. Riv. Mat. Univ. Parma 4(17*), 109–123 (1993)
Tan, Q., Wang, H.Y., Zhang, Y., Zhu, P.: On cohomology of almost complex \(4\)-manifolds. J. Geom. Anal. 25, 1431–1443 (2015)
Tan, Q., Wang, H.Y., Zhou, J.R.: Primitive cohomology of real degree two on compact symplectic manifolds. Manuscr. Math. 148, 535–556 (2015)
Taubes, C.H.: Self-dual connections on \(4\)-manifolds with indefinite intersection matrix. J. Differ. Geom. 19, 517–560 (1984)
Wang, H.Y.: The existence of nonminimal solutions to the Yang-Mills equation with group \(SU(2)\) on \(S^2\times S^2\) and \(S^1\times S^3\). J. Differ. Geom. 34, 701–767 (1991)
Acknowledgements
The first author would like to thank Professor Xiaojun Huang for his support. The first author also would like to thank Professors Tedi Draghici and Hongyu Wang for their patient guidance. The second author would like to thank East China Normal University and Professor Qing Zhou for hosting his visit in the spring semester in 2014. The authors dedicate this paper to the memory of Professor Ding Weiyue in deep appreciation for his long-term support of their work. The authors are grateful to the referees for their valuable comments and suggestions. Especially, the referees suggest a more direct argument about the openness part and point out the mistake in the proof of Lemma 3.3. Supported by NSFC (China) Grants 11471145, 11401514 (Tan), 11371309 (Wang), 11426195 (Zhou).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tan, Q., Wang, H. & Zhou, J. A Note on the Deformations of Almost Complex Structures on Closed Four-Manifolds. J Geom Anal 27, 2700–2724 (2017). https://doi.org/10.1007/s12220-017-9779-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-017-9779-2
Keywords
- Almost Kähler four-manifold
- Deformations of almost complex structures
- Dimension of J-anti-invariant cohomology