Abstract
In this paper, we define the generalized Lejmi’s P J operator on a compact almost Kähler 2n-manifold. We get that J is C ∞-pure and full if dim ker P J = b 2−1. Additionally, we investigate the relationship between J -anti-invariant cohomology introduced by T.-J. Li and W. Zhang and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau on a closed symplectic 4-manifold.
Similar content being viewed by others
References
Angella D., Tomassini A.: Symplectic manifolds and cohomological decomposition. J. Symplectic Geom. 2, 215–236 (2014)
Angella D., Tomassini A.: On cohomological decomposition of almost-complex manifolds and deformations. J. Symplectic Geom. 3, 403–428 (2011)
Besse A.L.: Einstein Manifolds. Springer, Berlin (1987)
Brylinski J.L.: A differential complex for poisson manifolds. J. Differ. Geom. 28, 93–114 (1988)
Barth W., Hulek K., Peters C., Van de Ven A.: Compact Complex Surfaces. Springer, Berlin (2004)
Donaldson S.K.: Two forms on four manifolds and elliptic equations. Nankai Tracts Math. 11, 153–172 (2006)
Donaldson S.K., Kronheimer P.B.: The Geometry of Four-Manifolds. Oxford Mathematical Monographs. Oxford Science Publications, New York (1990)
Draghici, T.: Private communication, July (2013)
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)
Fernández M., Muñoz V., Santisteban J.A.: Cohomologically Kähler manifolds with no Kähler metrics. Int. J. Math. Math. Sci. 52, 3315–3325 (2003)
Fino A., Tomassini A.: On some cohomological properties of almost complex manifolds. J. Geom. Anal. 20, 107–131 (2010)
Griffiths P.A., Harris J.: Principle of Algebraic Geometry. Wiley, New York (1978)
Guillemin V.: Symplectic Hodge theory and the dδ-lemma. MIT, Cambridge (2001)
Kodaira K., Morrow J.: Complex Manifolds. Holt, Rinehart and Winston, New York (1971)
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. arXiv:1204.5438v1, [math.DG]
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)
Lin, Y.: Symplectic harmonic theory and the Federer-Fleming deformation theorem. arXiv:1112.2442v3, [math.SG]
Tan, Q., Wang, H.Y., Zhu, P.: On tamed almost complex four-manifolds. Work in progress.
Tan, Q., Wang, H.Y., Zhang, Y., Zhu, P.: On cohomology of almost complex 4-manifolds. J. Geom. Anal. (2014). doi:10.1007/s12220-014-9477-2
Tseng L.S., Yau S.-T.: Cohomology and Hodge theory on symplectic manifolds: I. J. Differ. Geom. 91, 383–416 (2012)
Weil A.: Introduction à l’Étude des Variété Kählériennes. Publications de l’Instiut de Mathématique de l’Université de Nancago VI, Hermann (1958)
Yan D.: Hodge structure on symplectic manifolds. Adv. Math. 120, 143–154 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (China) Grants 11471145, 11401514 (Tan), 11371309 (Wang), 11426195 (Zhou).
Rights and permissions
About this article
Cite this article
Tan, Q., Wang, H. & Zhou, J. Primitive cohomology of real degree two on compact symplectic manifolds. manuscripta math. 148, 535–556 (2015). https://doi.org/10.1007/s00229-015-0761-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-015-0761-7