Abstract
For a complete symplectic manifold \(M^{2n}\), we define the \(L^{2}\)-hard Lefschetz property on \(M^{2n}\). We also prove that the complete symplectic manifold \(M^{2n}\) satisfies \(L^{2}\)-hard Lefschetz property if and only if every class of \(L^{2}\)-harmonic forms contains a \(L^{2}\) symplectic harmonic form. As an application, we get if \(M^{2n}\) is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler characteristic satisfies the inequality \((-1)^{n}\chi (M^{2n})\ge 0\).
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Acknowledgements
We would like to thank Professor H.Y. Wang for drawing our attention to the symplectic parabolic manifold and giving generously helpful suggestions about these. We would also like to thank the anonymous referee for careful reading of my manuscript and helpful comments. This work is supported by Natural Science Foundation of China No. 11801539 (Huang), No. 11701226 (Tan) and Natural Science Foundation of Jiangsu Province BK20170519 (Tan).
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Huang, T., Tan, Q. L2-hard Lefschetz complete symplectic manifolds. Annali di Matematica 200, 505–520 (2021). https://doi.org/10.1007/s10231-020-01004-2
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DOI: https://doi.org/10.1007/s10231-020-01004-2