Annali di Matematica Pura ed Applicata ( IF 1 ) Pub Date : 2020-06-06 , DOI: 10.1007/s10231-020-01004-2 Teng Huang , Qiang Tan
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 \((-1)^{n}\chi (M^{2n})\ge 0\).
中文翻译:
L 2-Hard Lefschetz完全辛流形
对于一个完整的辛流形\(M ^ {2N} \) ,我们定义了\(L ^ {2} \)上-hard莱夫谢茨属性\(M ^ {2N} \) 。我们还证明,当且仅当每类\(L ^ {2} \)-调和形式时,完整辛流形\(M ^ {2n} \)满足\(L ^ {2} \)-硬Lefschetz性质包含\(L ^ {2} \)辛调和形式。作为应用,如果\(M ^ {2n} \)是满足辛辛的Lefschetz性质的闭合辛抛物流形,则其Euler特性满足不等式\((-1)^ {n} \ chi(M ^ {2n})\ ge 0 \)。