Symplectic Parabolicity andL2 Symplectic Harmonic Forms*

Abstract

In this paper, we study the symplectic cohomologies and symplectic harmonic forms which introduced by Tseng and Yau. Based on this, we get if (M 2n , ω) is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler number satisfies the inequality (−1) n χ(M 2n ) ≥ 0. (2000): 53D05. AMS Classification

A note on Euler number of locally conformally Kähler manifolds

“…Inspired by Kähler geometry, Tan-Wang-Zhou gave the definition of symplectic parabolic manifold [15]. By a well known result that a closed symplectic manifold satisfies the hard Lefschetz property if only if de Rham cohomology consists with the new symplectic cohomology, they proved that if (M, ω) is a 2n-dimensional closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then the Euler number satisfies (−1) n χ(M 2n ) ≥ 0.…”

L2-hard Lefschetz complete symplectic manifolds

“…In [24], they defined the L 2 -dd Λ Lemma on a complete non-compact almost Kähler manifold. We say that the L 2 -dd Λ Lemma holds if the following properties are equivalent:…”

“…In [24], they prove that the L 2 -dd Λ lemma holds on universal covering space of a closed symplectic manifold which satisfies the hard Lefschetz property. Using the useful Proposition 3.9, we could prove that the any class of L 2 -harmonic forms on the universal covering space (M ,g,J,ω) contains a L 2 symplectic harmonic form, see Proposition 3.11.…”

“…([24, Definition 3.3]) Let (M, g, J, ω) be a complete non-compact almost Kähler manifold with bounded geometry. Let α ∈ Ω k (2) be a dand d Λ -closed differential form.…”

“…([24, Proposition 3.4]) Let (M, g, J, ω) be a 2n-dimensional closed almost Kähler manifold. We denote by π : (M,g,J,ω) → (M, g, J, ω) the universal covering map.…”