A spectral sequence associated with a symplectic manifold

Abstract

With a symplectic manifold a spectral sequence converging to its de Rham cohomology is associated. A method of computation of its terms is presented together with some stabilization results. As an application a characterization of symplectic harmonic manifolds is given and a relationship with the C–spectral sequence is indicated. Let (M,Ω) be a 2n–dimensional symplectic manifold and Λ(M) be the algebra of differential forms onM . Consider the ideal ΛL(M) of Λ(M), composed of all differential forms that vanish when restricted to any Lagrangian submanifold of M . This ideal is differentially closed and its powers constitute the symplectic filtration in the de Rham complex of M. The corresponding spectral sequence {Ep,q r , d r } is called the symplectic spectral sequence associated with (M,Ω). A motivation for this construction comes from the theory of C–spectral sequences (see [5]). Moreover, if M = T ∗N , then the symplectic spectral sequence is nothing but the “classical part” of the C–spectral sequence associated with the differential equation dρ = 0, ρ ∈ Λ(N). 1. Notations and preliminaries In this section the notation is fixed and all necessary facts concerning symplectic manifolds (see [1, 2, 6] for further details) are collected. Throughout the paper (M,Ω) stands for a 2n–dimensional symplectic manifold , Λ = ∑ k Λ k for the algebra of differential forms on M , H(M) = ∑ k H (M) for the de Rham cohomology of M and D = ∑ k Dk for the algebra of multivectors on M . The isomorphism Γ1 : V ∈ D1 → V Ω ∈ Λ of C∞(M)–modules extends uniquely to a C∞(M)–algebra isomorphism Γ: D → Λ. P = Γ−1(Ω) is called the corresponding to Ω Poisson bivector. C∞(M)–linear operators : Λ → Λ , ω = ω ∧ Ω, ⊥ : Λ → Λk−2 , ⊥ω = P ω, acting on Λ are basic for our purposes. Put Λ = im and Λ = ker⊥. Elements of Λ are called effective forms. Another very useful fact is the Hodge–Lepage expansion (see, for instance, [2]):