Weakly Lefschetz symplectic manifolds

Abstract

Abstract. For a symplectic manifold, the harmonic cohomology of symplectic divisors (introduced by Donaldson, 1996) and of the more general symplectic zero loci (introduced by Auroux, 1997) are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the s-Lefschetz property. In particular, we consider the symplectic blow-ups CP m of the complex projective space CP m along weakly Lefschetz symplectic submanifolds M ⊂ CP m . As an application we construct, for each even integer s ≥ 2, compact symplectic manifolds which are s-Lefschetz but not (s + 1)-Lefschetz.

Lefschetz decomposition for de Rham cohomology on weakly Lefschetz symplectic manifolds

“…The similar result was proved by Fernández et al [5]. Inspired by Tseng and Yau's work, we generalize Fernández et al's result by using very different approach.…”

“…Fernández et al [4] introduced s-Lefschetz property and dd Λ -lemma up to degree s. Definition 1 [4] Let (M, ω) be a symplectic manifold of dimension 2n and let 0 s n − 1.…”

“…For example, compact simply connected symplectic manifolds of dimension 6 are 1-Lefschetz. Actually, on a compact simply connected symplectic manifold of dimension 6, every de Rham cohomology class of degree k = 4 admits a symplectically harmonic representative ( [5,Corollary]). So we have the following decomposition on compact simply connected symplectic manifolds of dimension 6.…”

“…Gompf [6] constructed many closed symplectic manifolds which do not satisfy the Lefschetz property. So Fernández et al [4] introduced s-Lefschetz property which is weaker than hard Lefschetz property. We would observe that every compact symplectic manifold (M, ω) is s-Lefschetz for some integer s 0.…”

L2-hard Lefschetz complete symplectic manifolds

“…Yan in [27] given a simpler, more direct, proof of this fact, it T. Huang: School of Mathematical Sciences, University of Science and Technology of China; CAS Key Laboratory of Wu Wen-Tsun Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, Peoples Republic of China; e-mail: htmath@ustc.edu.cn;htustc@gmail.com Q. Tan: Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, Peoples Republic of China; e-mail: tanqiang@ujs.edu.cn follows the idea of the standard proof of the Hard Lefschetz Theorem (see [14]). In [13], the authors dealt with the symplectic manifolds satisfying a weak property following [12], they said that (M, ω) is an s-Lefschetz symplectic manifold, 0 < s < n − 1, if (1.1) is an epimorphism for all k < s. In [25], they discussed the hard Lefschetz condition on various cohomology groups and verify them for the Nakamura manifold of completely solvable type and the Kodaira-Thurston manifold. In [17], the author studied the L 2 cohomology of complete almost Kähler manifold.…”