Abstract
We study the hard Lefschetz property on compact symplectic solvmanifolds, i.e., compact quotients M = ΓG of a simply-connected solvable Lie group G by a lattice Γ, admitting a symplectic structure.
References
De Andrés, L. C., Fernández, M., De León, M., et al.: Some six-dimensional compact symplectic and complex solvmanifolds, Rend. Mat. Appl., 12, 59–67 (1992)
De Bartolomeis, P., Tomassini, A.: On solvable generalized Calabi–Yau manifolds, Ann. Inst. Fourier, 56, 1281–1296 (2006)
Benson, C., Gordon, C. S.: Kähler and symplectic structures on nilmanifolds, Topology, 27, 513–518 (1988)
Benson, C., Gordon, C. S.: Kähler structures on compact solvmanifolds, Proc. Am. Math. Soc., 108, 971–980 (1990)
Brylinski, J. L.: A differential complex for Poisson manifolds, J. Differential. Geom., 28, 93–114 (1988)
Cavalcanti, G. R.: New aspects of the ddc-Lemma, Oxford Thesis D. Phil. Thesis, arXiv:math/05001406v1 (2005)
Chu, B. Y.: Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197, 145–159 (1974)
Fernández, M., Muñoz, V., Santisteban, J. A.: Cohomologically Kähler manifolds with no Kähler metrics, Int. J. Math. Math. Sci., 52, 3315–3325 (2003)
Figueroa-O’Farrill, J. M., Köhl, C., Spence, B.: Supersymmetry and the cohomology of (hyper)Kähler manifolds, Nuclear Physics B, 503, 614–626 (1997)
Griffiths, P. A., Harris, J.: Principles of Algebraic Geometry, New York, Wiley, 1978.
Hasegawa, K.: A note on compact solvmanifolds with Kähler structures, Osaka J. Math., 43, 131–135 (2006)
Hattori, A.: Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I, 8, 289–331 (1960)
Lejmi, M.: Stability under deformations of extremal almost-Kähler metrics in dimension 4, Mat. Res. Lett., 17, 601–612 (2010)
Li, T.-J., Zhang, W.: Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom., 17, 651–683 (2009)
Mathieu, O.: Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv., 70, 1–9 (1995)
Nakamura, I.: Complex parallelisable manifolds and their small deformations, J. Diff. Geom., 10, 85–112 (1975)
Merkulov, S. A.: Formality of canonical symplectic complexes and Frobenius manifolds, Int. Math. Res. Not., 14, 727–733 (1998)
Ovando, G.: Four dimensional symplectic Lie algebras, Beiträge Algebra Geom., 47, 419–434 (2006)
Tan, Q., Wang, H. Y., Zhou, J. R.: Primitive cohomology of real degree two on compact symplectic manifold, Manuscripta Math., 148, 535–556 (2015)
Tomasiello, A.: Reformulating supersymmetry with a generalized Dolbeault operator, J. High Energy Phys., 2, 010 (2008)
Tseng, L. S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: I, J. Differ. Geom., 91, 383–416 (2012)
Witten, E.: Constraints on Supersimmetry Breaking*, Nuclear Physics B, 202, 253–316 (1982)
Yan, D.: Hodge structure on symplectic manifolds, Adv. in Math., 120, 143–154 (1996)
Zumino, B.: Supersymmetry and Kähler manifolds, Phys. Lett. B, 87, 203–206 (1979)
Acknowledgements The first author would like to thank professor Hongyu Wang for stimulating discussions. We thank the referees for their time and comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflict of interest.
Additional information
Supported by PRC grant NSFC (Grant No. 11701226) (Tan); Natural Science Foundation of Jiangsu Province (Grant No. BK20170519) (Tan); Project PRIN “Variet reali e complesse: geometria, topologia e analisi armonica” and by GNSAGA of INdAM (Tomassini)
Rights and permissions
About this article
Cite this article
Tan, Q., Tomassini, A. Remarks on Some Compact Symplectic Solvmanifolds. Acta. Math. Sin.-English Ser. 39, 1874–1886 (2023). https://doi.org/10.1007/s10114-023-0416-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-023-0416-7