The Diffiety Institute Preprint Series

Preprint DIPS–6/2006 October 31, 2006

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

A Spectral Sequence Associated with a Symplectic Manifold

by

A. M. Vinogradov and C. Di Pietro

DIPS-6/2006

                                                 arXiv:math.SG/0611138

A Spectral Sequence Associated with a Symplectic Manifold

A. M. VINOGRADOV AND C. DI PIETRO

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 on M. Consider the ideal Λ_{L}(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 {E_{r}^{p,q}, d_{r}^{p,q}} 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^{1}(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^{k}(M) for the de Rham cohomology of M and D = ∑_{k} D_{k} for the algebra of multivectors on M. The isomorphism Γ_{1} : V ∈ D_{1} ↦ V ⌟ Ω ∈ Λ^{1} 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^{k} → Λ^{k+2} , ⊤ω = ω ∧ Ω,
⊥: Λ^{k} → Λ^{k−2} , ⊥ω = P ⌟ ω,

acting on Λ are basic for our purposes. Put ⊤Λ = im ⊤ and Λ_{ϵ} = ker ⊥. Elements of Λ_{ϵ} are called effective forms. Another very useful fact is the Hodge–Lepage expansion (see, for instance, [2]):

Proposition 1. Any ω ∈ Λ^{k}, k ≥ 0, admits a unique expansion of the form

ω = ω_{0} + ⊤ω_{1} + ⊤^{2}ω_{2} ⋯ + ⊤^{n}ω_{n}

with ω_{i} ∈ Λ_{ϵ}^{k−2i}. In particular, Λ splits into the direct sum Λ_{ϵ} ⊕ ⊤Λ. There exists a symplectic analogue of the Hodge star–operator

∗: Λ^{k} → Λ^{2n−k}, k ≥ 0,

uniquely characterized by the following property:

η ∧ ∗ ω = (⊥^{k}(η ∧ ω))Ω^{n}, η, ω ∈ Λ^{k}.

The map δ = {δ_{k}}_{k}, δ_{k} = (−1)^{k+1} ∗ d∗: Λ^{k} → Λ^{k−1}, is a (−1)–degree differential in Λ and [⊥, d] = δ (see [1]). The introduced operators are subject to the following graded commutation relations:

[d, δ] = 0 ; [⊤, d] = 0 ; [⊤, δ] = d ; [⊥, δ] = 0,  (1)

the first of which shows that (Λ, d, δ) is a bicomplex (see [1]). Proposition 2. The Hodge–Lepage expansion of dω, ω ∈ Λ_{ϵ}, is of the form

dω = (dω)_{0} + ⊤(dω)_{1}, (dω)_{0}, (dω)_{1} ∈ Λ_{ϵ}.

Corollary 1. Λ_{ϵ} is δ–closed.

2. The term E_{0}

An explicit description of the symplectic filtration is based on the following fact of linear algebra. Proposition 3. Λ_{L} = ⊤Λ. It shows that the operator ⊤^{k} respects the symplectic filtration by shifting it by 2k. Moreover, it commute with d and, so, induces an automorphism τ = {τ_{r}^{k}}, τ_{r}^{k} colonE_{r}^{p,q} → E_{r}^{p+k,q+k}, of the symplectic spectral sequence. Proposition 4.

(1) The term E_{0}^{p,q} is trivial if (p, q) ∈ ℤ^{2} lies outside the triangle with vertexes at (0, 0), (0, n) and (n, n).

(2) The term E_{0}^{0,q} is naturally isomorphic to Λ_{ϵ}^{q}.

(3) τ_{0}^{p}: E_{0}^{0,q−p} → E_{0}^{p,q} is an isomorphism, if 0 ≤ p ≤ q ≤ n. The last assertion of Propostion 4 is inductively generalized to terms E_{r}, r > 0. Proposition 5. Let r > 0. Then τ_{r}^{k}: E_{r}^{p,q} → E_{r}^{p+k,q+k} is an isomorphism, if ∑_{i=1}^{r}(i − 1) ≤ p ≤ q ≤ q + k ≤ n − (2 + ∑_{i=1}^{r}(i − 2)). In particular, τ_{1}^{p}: E_{1}^{0,q−p} → E_{1}^{p,q} is an isomorphism, if 0 ≤ p ≤ q < n.

3. The term E_{1}

The exact sequence constructed in this section gives a useful description of the first term of the symplectic spectral sequence. It is composed of the following two families of ℝ–homomorphisms;

ϕ_{p,q}: H^{q−p}(M) → E_{1}^{p,q},  0 ≤ p ≤ q ≤ n
[ω]_{im d} ↦ [⊤^{p}ω]_{im d_{0}}
ψ_{p,q}: E_{1}^{p,q} → H^{q−p−1}(M),  0 ≤ p ≤ q < n
[⊤^{p}ρ]_{im d_{0}} ↦ [η]_{im d}

with ⊤η = dρ. Put A^{k} = {ω | ω ∈ Λ^{k}; dω ∈ Λ_{ϵ}^{k+1}}, C^{k} = A^{k}/im d ∩ Λ^{k} and consider ℝ– homomorphisms

ϕ_{p,n}: C^{n−p} → E_{1}^{p,n}, [ω]_{im d} ↦ [⊤^{p}ω]_{im d_{0}}
ψ_{p,n}: E_{1}^{p,n} → C^{n−(p+1)}, [⊤^{p}ρ]_{im d_{0}} ↦ [η]_{im d}.

The above defined homomorphisms together with the multiplication by the cohomology class of Ω homomorphism τ: H^{∗}(M) → H^{∗+2}(M) form the following sequences

\begin{aligned}⋯ → H^{q−p}(M) ⟶^{ϕ_{p,q}} E_{1}^{p,q} & ⟶^{ψ_{p,q}} H^{q−(p+1)}(M) \\ & ⟶^{τ} H^{q+1−p}(M) ⟶^{ϕ_{p,q+1}} E_{1}^{p,q+1} → … \end{aligned}  (2)

whose left and right ends are

0 → H^{0}(M) ⟶^{ϕ_{p,p}} E_{1}^{p,p} ⟶^{ψ_{p,p}} 0 ⟶^{τ} H^{1}(M) ⟶^{ϕ_{p,p+1}} E_{1}^{p,p+1} → …

and

\begin{aligned}⋯ → H^{n−(p+2)}(M) ⟶^{τ} C^{n−p} & ⟶^{ϕ_{p,n}} E_{1}^{p,n} \\ & ⟶^{ψ_{p,n}} C^{n−(p+1)} ⟶^{τ^{p+1}} H^{n+p+1}(M) → 0 , \end{aligned}

respectively. Notice that sequence (2) involves the terms of the p–th column of E_{1}.

Theorem 1. The sequence (2) is exact.

4. Stabilization theorems

Theorem 1 is key in studying stability of the symplectic spectral sequence. For instance, if Ω is exact, then the homomorphism τ: H^{k}(M) → H^{k+2}(M) in (2) is trivial. This fact and Theorem 1 lead to the following result.

Theorem 2. If Ω is exact, then the symplectic spectral sequence stabilizes at the term E_{2}. Moreover, if a term E_{2}^{p,q} is different from zero, then either p = 0, or q = n and E_{2}^{0,q} ≅ H^{q}(M), if q ≤ n, and E_{2}^{p,n} ≅ H^{n+p}(M), if p ≥ 0.

Corollary 2. If Ω is the standard symplectic form on M = T^{∗}N, dim N = n, then the corresponding symplectic spectral sequence stabilizes at the second term and E_{2}^{0,q} ≅ H^{q}(N), E_{2}^{p,q} = 0, for p > 0.

Theorem 2 is generalized as follows.

Theorem 3. Let t > 1 be the minimal integer such that Ω^{t} = dρ. Then the symplectic spectral sequence stabilizes at the term E_{t−r+1} where r is an integer such that 0 ≤ r ≤ max{0, 2t − (n + 1)}.

The estimate in the previous theorem can not be, generally, improved as the following example shows .

Example 1. The symplectic spectral sequence associated with (C, Ω) × (ℝ^{2h}, Ω_{ℝ^{2h}}), where (C, Ω) is a closed symplectic 2m–fold and (ℝ^{2h}, Ω_{ℝ^{2h}}) is the standard symplectic manifold, is not stable in terms E_{h}, if h ≤ m + 1 = t, and becomes stable by starting from the term E_{m+2}.

If cohomology classes [Ω^{t}], t ≤ n, are all nontrivial, then M is closed. Closed symplectic manifolds are characterized by the fact that E_{∞}^{p,p} = E_{2}^{p,p} ≅ ℝ, if 0 ≤ p ≤ n. Moreover, it holds

Theorem 4. The symplectic spectral sequence for a closed symplectic manifold stabilizes at the term E_{2}.

5. The Brylinski conjecture

Definition 1. A form η is called symplectically harmonic iff η ∈ ker d ∩ ker δ.

In [1] Brylinski conjectured that each cohomology class of a closed symplectic manifold contains at least one symplectically harmonic form. It was, however, disproved by counterexamples found by Mathieu (see [3]). So, the problem of characterization of simplectic manifolds for which the Brylinski conjecture holds arises. Call such symplectic manifolds harmonic. In view of (1) and Proposition 2 we have the following useful technical characterization of harmonic symplectic manifolds.

Proposition 6. A symplectic manifolds is harmonic iff any cohomology class is represented by a form η whose Hodge–Lepage expansion terms are all closed.

This Proposition helps to prove

Theorem 5. A closed symplectic manifold is harmonic iff τ_{2}^{n−q}: E_{2}^{0,q} → E_{2}^{n−q,n} is an isomorphism for all 0 ≤ q ≤ n.

Corollary 3. For a closed harmonic manifold mappings τ_{2}^{n−p−q}: E_{2}^{p,q} → E_{2}^{n−q,n−p} are isomorphisms for p + q ≤ n and 0 ≤ p ≤ q. In particular, E_{2} is symmetric with respect to the line p + q = n.

6. Associated diffiety

Let J^{k}(M, n) be the manifold of k–jets of n–dimensional submanifolds of a manifold M. The k–th prolongation M_{(k)} of a symplectic manifold (M, Ω) is a submanifold in J^{k}(M, n) composed of k–th jets of Lagrangian submanifolds of M. By restricting the Cartan distribution on J^{∞}(M, n) to M_{(∞)} we obtain a diffiety which locally coincides with the infinite prolongation of the equation dρ = 0, ρ ∈ Λ^{1}(N), dim N = n. A natural projection M_{(∞)} → M induces a morphism of the symplectic spectral sequence of M to the C–spectral sequence of the diffiety M_{(∞)}. This allows to find out some useful interpretations for various terms and differentials of the symplectic spectral sequence. For instance, consider the following functional defined on compact Lagrangian submanifolds associated with an element θ ∈ E_{1}^{0,n}:

L ↦ ∫_{L} ω|_{L},  θ = [ω],  ω ∈ Λ^{n} mod ⊤Λ.

Then d_{1}^{0,n}(θ) = 0 is the Euler-Lagrange equation for extremals of this functional.

7. Generalisations

There are numerous analogues of the symplectic spectral sequence due to the fact that the underling construction is of a rather general nature. Below we list some of its ”neighbors”.

• Let N be a submanifold of a symplectic manifold M. By restricting the symplectic filtration in Λ(M) to Λ(N) we get a spectral sequence converging to the de Rham cohomology of N. In particular, this way a spectral sequence is associated with a Hamilton–Jacobi equation E. This spectral sequence is the ”classical part” of the C–spectral sequence associated with E.

• Let M be a contact manifold and I ⊂ Λ(M) the ideal composed of differential forms that vanish on all Legendre submanifolds of M. Powers of I form the contact filtration in Λ(M). This way one gets the contact spectral sequence associated with a contact manifold.

• By restricting the contact filtration to a submanifold N of a contact manifold M one gets a spectral sequence which is the classical part of the C–spectral sequence associated with N interpreted as an (overdetermined) system of first order scalar differential equations (see [2, 5]).

• Let (M, P) be a Poisson manifold, P being the Poisson bivector. The Poisson filtration in the algebra D(M) of multivectors fields on M is that formed by powers of the principal ideal generated by P. The Poisson differential d_{P} , d_{P} (Q) = [[P, Q]], Q ∈ D(M), respects this filtration and the corresponding to it spectral sequence is called Poisson. If P is nondegerate the Poisson spectral sequence is, in a sense, dual to the corresponding symplectic one.

Constructions and results of this note naturally generalize to all these spectral sequences. Details will be given in a separate publication.

References

[1] J.L. Brylinski A differential complex for Poisson manifolds J. of Diff. Geo. 28 (1988), 93–114 [2] V.V. Lychagin Contact geometry and non linear second–order differential equations, Russian

Math. Surveys 34:1 (1979), 149–180

[3] O. Mathieu Harmonic Cohomology classes of symplectic manifolds, Comment. Math. Helvetici 70 (1995), 1–9

[4] J. McCleary A User’s Guide to Spectral Sequences, Cambridge studies in advanced mathematics vol. 58 in Mathematics Springer vol. 220

[5] A.M. Vinogradov, J.S. Krasil’shchik Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Translations of Mathematical Monographs, vol. 182, AMS, 1999

[6] A. Weinstein Lectures on Symplecitc Manifolds, Conference Board of the Mathematics Sciences by the American Mathematical Society

A. M. Vinogradov, Dipartimento di Matematica e Informatica, Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy; INFN, Gruppo Collegato di Salerno, Italy

 E-mail address: vinograd@unisa.it

C. Di Pietro, Dipartimento di Matematica e Informatica, Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy;

 E-mail address: cdipietr@unisa.it