Ring structure of the symplectic four-manifold from fibered three-manifold

. By the same token, the fourth and fifth terms are

m₃(A_i, A_j×A_k, A_l) =∗r

A_i∧L^−(p+1)(A_j∧A_k∧A_l)

−L^−(p+1)(A_i∧A_j∧A_k)∧A_l

−Ai∧L^−(p+1)

ω^p+1L^−(p+1)(Aj ∧A_k)∧A_l (5.44)

+L^−(p+1)

A_i∧ω^p+1L^−(p+1)(A_j ∧A_k)

∧A_l

,

−m₃(A_i, A_j, A_k×A_l) =∗r

−A_i∧L^−(p+1)(A_j∧A_k∧A_l) +L^−(p+1)(A_i∧A_j)∧A_k∧A_l +A_i∧L^−(p+1)

A_j ∧ω^p+1L^−(p+1)(A_k∧A_l) (5.45)

−L^−(p+1)(A_i∧A_j)∧ω^p+1L^−(p+1)(A_k∧A_l)

. Summing all the terms of (5.41)–(5.45) then gives zero. This shows that m₄ can be chosen to be zero. It is also straightforward to see that the higher maps mk fork >4 can also be chosen to be zero. Thus, we have shown that there is anA_∞-algebra for thep-filtered forms.

6. Ring structure of the symplectic four-manifold from fibered three-manifold

The purpose of this section is to present a pair of symplectic four-manifolds with the following properties:

• their de Rham cohomology rings are isomorphic and their primi-tive cohomologies have the same dimensions;

• their primitive cohomology has different ring structure; in partic-ular, the products in the componentP H_dd² _Λ⊗P H_dd² _Λ →P H_∂¹

− are different.

Both four-manifolds are topologically S¹×Y_τ where Y_τ = Σ×τ S¹ is a mapping torus of a closed surface Σ identified with a monodromy τ. The construction of such symplectic four-manifolds were described in

Section 4.4.2 and here we shall use the same notations and conventions as there.

According to Proposition 4.14, the dimensions of the de Rham co-homologies as well as the primitive coco-homologies of such symplectic four-manifolds are determined by two natural numbers. Consider the action of τ^∗ on H_d¹(Σ):

1) the first number is the dimension of the τ^∗-invariant subspace, which we denote by q+p;

2) the intersection pairing is not always non-degenerate on the τ^∗ -invariant subspace; the second number is the dimension of this kernel, which is q−p.

But in order to calculate the product structure of the primitive coho-mologies of the four-manifolds, we need to understand well their basis elements. So to begin, we shall first give a systematic construction of the basis elements of the de Rham cohomology of the three-manifold, Y_τ, which then will lead us to the basis elements of the primitive coho-mologies of X =S¹×Y_τ and their products.

6.1. Representatives of de Rham cohomologies of the fibered three-manifold.By symplectic linear algebra, we may choose a basis for theτ^∗-invariant subspace ofH_d¹(Σ):

[α₁],[α₂], . . . ,[α_p],[α_p+1], . . . ,[α_q],[β₁],[β₂], . . . ,[β_p]!

where p ≤ q and such that [α_j]·[β_j] = 1 = −[β_j]·[α_j] for 1 ≤ j ≤ p and other pairings vanish. Here, the pairing [γ]·[γ] is defined to be

Σ(γ∧γ)/

Σω_Σ for any [γ],[γ]∈H_d¹(Σ). Again, by symplectic linear algebra, we can extend this basis to a symplectic basis for H_d¹(Σ):

[α₁], . . . ,[αq],[β₁], . . . ,[βp]!

∪ [αq+1], . . . ,[αg], [β_p+1], . . . ,[β_q],[β_q+1], . . . ,[β_g]!

whereg is the genus of the surface Σ. A linear algebra argument shows that the image of τ^∗ −1 is always perpendicular to {[α_k]}^p_k=1 and {[β_k]}^q_k=1 with respect to the above symplectic pairing. It follows that the image ofτ^∗−1 is spanned by

[α_p+1], . . . ,[α_q],[α_q+1], . . . ,[α_g],[β_q+1], . . . ,[β_g]! . Thus, for example, there exists a [ζ_k] such that

τ^∗[ζ_k]−[ζ_k] = [α_k] for anyp < k≤g.

Now the de Rham cohomology ofY_τ is determined by the Wang exact sequence in (4.14) which involves the map τ^∗−1 on H_d¹(Σ). Hence, we can explicitly construct the basis elements of H_d¹(Y_τ) and H_d²(Y_τ) in terms of ker(τ^∗−1) and coker(τ^∗−1) as follows.

To start, consider the Jordan form of τ^∗. When 1 ≤ k ≤ q, the element [α_k] must be the upper-left-most element of some Jordan block

of eigenvalue 1. The size of the Jordan block is 1 if and only if 1 ≤ k ≤ p. That is to say, the interesting case is when k ∈ {p+ 1, . . . , q}. Let [γ_k,0],[γ_k,1], . . . ,[γ_k,k] be the canonical basis for the block, where γ_k,0 = α_k, γ_k,1 = ζ_k and with _k + 1 being the size of the Jordan block. The discussion in what immediately follows will be within a single Jordan block, and so for notational simplicity, we shall suppress the first subscript of γ and write γ_j for γ_k,j. With this understood, there exist functions {g_j}_j=0 on Σ such that

τ^∗(γ₀) =γ₀+dg₀, τ^∗(γ₁) =γ₁+γ₀+dg₁,

...

τ^∗(γ) =γ+γ−1+dg.

We remark that the terminal element [γ] does not belong to the image of τ^∗ −1. From the canonical basis, we can construct the following globally-defined differential forms on Yτ:

γ₀;γ˜₀ =γ₀+d(χ g₀),

γ₁;γ˜₁ =γ₁+φ γ₀+d(χ g₁) + (φ−1)d(χ g₀), γ₂;γ˜₂ =γ₂+φ γ₁+φ²−φ

2 γ₀+d(χ g₂) + (φ−1)d(χ g₁) +φ²−3φ+ 2

2 d(χ g₀), ...

where χ(φ) is an interpolating function defined on the interval φ ∈ (−0.1,1.1) and is equal to 0 on (−0.1,0.1) and 1 on (0.9,1.1). Here, we are covering the interval [0,1] using three charts: (−0.1,0.1), (0,1), and (0.9,1.1). The above ˜γ’s are invariant under the identification (x, φ)→ (τ(x), φ−1), and thus are well-defined one-forms onY_τ. In general, for j∈ {0, . . . , } where+ 1 is the size of the Jordan block, we have

γ_j ;γ˜_j =

j

i=0

f_i(φ)γ_j−i+f_i(φ−1)d(χ(φ)g_j−i) , where

f_i(φ) = 1 i!

i−1

#

m=0

(φ−m) = 1

i!φ(φ−1). . .(φ−i+ 1) and f₀(φ)≡1. The functions fi(φ) have the following properties:

• f_i(φ) is a polynomial in φof degree i;

• f_i(0) = 0 for any i >0; namely,f_i(φ) does not have any constant terms;

• fi(φ)−fi(φ−1) =fi−1(φ−1);

• f_i(φ) =_i

m=1(−1)^m+1f_i−m(φ)/m.

Taking the exterior derivative of ˜γ_j, we find d˜γ_j =dφ∧ ^j

i=1

(−1)ⁱ⁺¹ i γ˜_j−i (6.1)

for any j∈ {1, . . . , }. Notice that the sum above starts fromi= 1 and hence there are no terms of the form dφ∧γ˜ on the right-hand side of (6.1). Therefore, each α_k for k ∈ {1, . . . , q} leads to an element in H_d²(Yτ) defined by

dφ∧α_k whereα_k = ˜γk,_k,

and_k+ 1 is the size of the associated Jordan block.

Notice that for 1 ≤k≤p,k= 0, and therefore, α_k = ˜γk,0 = ˜αk. With Proposition 4.13 and the Wang exact sequence of (4.14), this construc-tion gives the following basis for the de Rham cohomologies of Y_τ from the canonical basis of τ^∗:

H_d¹(Yτ) =R^q+p+1 = span{dφ,α˜₁, . . . ,α˜q,β˜₁, . . . ,β˜p},

H_d²(Yτ) =R^q+p+1 = span{ω_Σ, dφ∧α˜₁, . . . , dφ∧α˜p, dφ∧β˜₁, . . . , dφ∧β˜p, dφ∧α_p+1, . . . , dφ∧α_q}.

6.2. Representatives of the primitive cohomologies of the sym-plectic four-manifold.We now consider the four-manifold X=S¹× Y_τ with the symplectic formω =dt∧dφ+ω_Σ. Its de Rham cohomologies are

H_d¹(X) =R^q+p+2 = span{dt, H_d¹(Yτ)},

H_d²(X) =R^2q+2p+2= span{dt∧H_d¹(Y_τ), H_d²(Y_τ)}, H_d³(X) =R^q+p+2 = span{dφ∧ω_Σ, dt∧H_d²(Y_τ)}.

Clearly, the Euler characteristic of X is zero. It is also straightforward to check that the signature of X is also zero.

We will use the exact sequence of Theorem 4.2 to construct a basis for the primitive cohomologiesP H_dd² _Λ(X) andP H_d²

+d^Λ(X). As noted in Proposition 4.14, they are both isomorphic to R^3q+p+1.

By Corollary 4.5, P H_dd² _Λ(X) has a component isomorphic to the coker(L : H_d⁰(X) → H_d²(X)), and P H_d²_+dΛ(X) has a component iso-morphic to the ker(L : H_d²(X) → H_d⁴(X)). It is not hard to see that both these components are isomorphic to the following basis elements

inH_d²(X):

dt∧(dφ−ω_Σ), dt∧α˜₁, . . . , dt∧α˜q, dt∧β˜₁, . . . , dt∧β˜p, dφ∧α˜₁, . . . , dφ∧α˜_p, dφ∧β˜₁, . . . , dφ∧β˜_p,

dφ∧α_p+1, . . . , dφ∧α_q.

The elements corresponding to dt ∧H_d¹(Y_Σ) may not be primitive in general and so do need to be modified. For instance, suppose thatτ^∗α= α+dg. Then,dt∧(α+d(χ g)) =dt∧(α+χdg+χg dφ) is not necessarily primitive. Note thatg is unique up to a constant. Thus, we may assume

that

Σ

g ω_Σ= 0, and thus g ω_Σ=dμ, (6.2)

for some 1-form μ on Σ. The standard computation on Lefschetz de-composition then shows that

dt∧(α+d(χ g))−d(χμ) =dt∧(α+χdg+χgdφ)−χgω_Σ−χdφ∧μ is a primitive 2-form, and is cohomologous to dt∧(α+d(χ g)).

Now the other component in Corollary 4.5 for P H_dd² _Λ(X) is ker(L : H_d¹(X) →H_d³(X)). This kernel is spanned by {α˜_p+1, . . . ,α˜_q}. The cor-responding elements of P H_dd² _Λ are those that when acted upon by ∂₋ give an element in the kernel. To explicitly construct them, we note that for any k∈ {p+ 1, . . . , q},

ω∧α˜_k =dt∧dφ∧γ˜_k,0+χg_k,0dφ∧ω_Σ

=−d

dt∧γ˜_k,1+χdφ∧μ_k,0 (6.3)

=−d

dt∧γ˜k,1+χdφ∧μk,0−d(χμk,1+χ(φ−1)μk,0) whereμ_k,0 andμ_k,1 are defined by (6.2). The expression inside the exte-rior derivative in line 2 above does not generally represent a primitive 2-form. Therefore, we added the exterior derivative ofχμ_k,1+χ(φ−1)μ_k,0 in line 3 to ensure primitivity. We thus obtain the following elements in P H_dd² _Λ fork∈ {p+ 1, . . . , q}:

dt∧γ˜_k,1+χdφ∧μ_k,0−d(χμ_k,1+χ(φ−1)μ_k,0).

The computation (6.3) shows that the∂₋action on the above expression is equal to ˜γ_k,0= ˜α_k.

For P H_d²

+d^Λ(X), the other component in Corollary 4.5 is coker(L : H_d¹(X)→H_d³(X)). This is spanned by{(dt∧dφ+ω_Σ)∧α_k}^q_k=p+1. These basis elements are cohomologous todt∧dφ∧α_k provided that allgi has zero integration againstω_Σ. The corresponding elements inP H_d²

+d^Λ(X)

are ∂₊α_k which by (6.1) are explicitly given by dφ∧ ^k

j=1

(−1)^j+1

j ˜γ_k,k−j , fork∈ {p+ 1, . . . , q}.

6.3. Two examples and their product structures.We shall present below two explicit constructions. The two four-manifolds will have the same de Rham cohomology ring, as well as identical primitive coho-mologies. However, their primitive product structures will be shown to be different. In particular, the image of the following pairing has differ-ent dimensions in these two constructions:

P H_dd² _Λ(X)⊗P H_dd² _Λ(X) → P H_∂¹

−(X) (B₂, B₂) → ∗^r

−d L⁻¹(B₂∧B₂) +∂₋B₂∧B₂+B₂∧∂₋B₂ . (6.4)

This is a symmetric bilinear operator. Note that the first term

−∗rd L⁻¹(B₂∧B₂) =−d L⁻²(B₂∧B₂) is not only∂₋-closed but also∂₊ -exact. We remark that on a compact, symplectic four-manifold, ∂₊B₀ is always ∂₋-exact for any function B₀. (See Proposition 3.16 of [19]).

Hence, the first term on the right-hand side of (6.4) does not have any contribution here.

6.3.1. Kodaira–Thurston nilmanifold.The first example is the Kodaira–Thurston nilmanifold, which we denote by X₁. The primitive cohomologies are computed in [18]. It is constructed from a torus with a Dehn twist. To be more precise, letT² =R²/Z², and let (a, b) be the co-ordinate forR². The monodromy mapτ is induced by (a, b) →(a, a+b).

It follows that H_d¹(T²) is spanned by da and db, andτ^∗(db) =da+db, τ^∗(da) =da. We take the area form on T² to be da∧db, and take the symplectic form on X=S¹×Y_τ to beω=dt∧dφ+da∧db.

The basis for the primitive cohomologies can be constructed from the recipe explained in the previous subsection. Note that ˜γ₀=da and

˜

γ₁=db+φ daare well-defined on Yτ. With this understood, we have P H_dd² _Λ(X₁) ={dt∧dφ−da∧db, dt∧γ˜₀, dφ∧˜γ₁, dt∧γ˜₁},

P H_∂¹₋(X₁) ={dt, dφ,γ˜₁}.

The only generator ofP H_dd² _Λ(X₁) which is not∂₋-closed is dt∧γ˜₁, and

∂₋(dt∧˜γ₁) =−γ˜₀. It follows that

(dφ∧γ˜₁)×(dt∧˜γ₁) =dφ and (dt∧˜γ₁)×(dt∧˜γ₁) = 2dt (6.5)

are the only non-trivial pairings between the generators ofP H_dd² _Λ(X₁).

Remark 6.1. This is the correspondence between the convention here and that of [18]:

e₁=dt, e₂ =dφ, e₃ = ˜γ₀, e₄ = ˜γ₁.

6.3.2. An example involving a genus two surface.Let Σ be a genus two surface. Fix a symplectic basis for itsH_d¹(Σ):{α₁, α₂, β₁, β₂}. Moreover, we may assume that the basis is an integral basis for the singular cohomology of Σ. Letω_Σ be an area form of Σ. Normalize it by

[ω_Σ] = [α₁∧β₁] = [α₂∧β₂].

A direct computation shows that τ^∗ preserves the intersection pairing.

It then follows from the theory of mapping class groups (see for example [7]) that τ^∗ does arise from a monodromy. Note that

S⁻¹τ^∗S=J,

Hence, the basis for the Jordan form is as follows:

γ₀=α₁, γ₁ =β₂, They give the following well-defined differential forms on Y_τ:

˜

We shall assume that g_iω_Σ =dμ_i for i∈ {0,1,2,3}. According to the discussion in the previous subsection,

ω=dt∧dφ+ω_Σ,

P H_dd² _Λ(X₂) ={dt∧dφ−ω_Σ, dt∧γ˜₀, dφ∧γ˜₃,

dt∧γ˜₁+χdφ∧μ₀−d(χμ₁+χ(φ−1)μ₀)}, P H_∂¹₋(X₂) ={dt, dφ,γ˜₃}.

Since P H_∂¹

−(X₂)∼=H_d³(X₂), its element can be captured by integrating againstH_d¹(X₂) ={dφ, dt,γ˜₀}. There is only one generator ofP H_dd² _Λ(X₂) that is not d-closed:

∂₋

dt∧˜γ₁+χdφ∧μ₀−d(χμ₁+χ(φ−1)μ₀)

=−˜γ₀.

We now consider the pairings between the generators ofP H_dd² _Λ(X₂).

If one of them is d-closed, the only non-trivial pairing is (dφ∧γ˜₃)×

dt∧γ˜₁+χdφ∧μ₀−d(χμ₁+χ(φ−1)μ₀)∼=dφ∧˜γ₀∧˜γ₃. The latter expression has non-zero integration againstdt. That is to say, the element in P H_∂¹

−(X₂) is proportional to dφ.

It remains to examine the square of dt∧γ˜₁+χdφ∧μ₀−d(χμ₁+ χ(φ−1)μ₀). As an element in H_d³(X₂), it is

−2

dt∧˜γ₁+χdφ∧μ₀−d(χμ₁+χ(φ−1)μ₀)

∧γ˜₀.

The expression has zero integration against ˜γ₀. We compute its integra-tion against dφ:

2

X₂

dφ∧dt∧˜γ₀∧γ˜₁, and it is not hard to see that

Σφγ˜₀ ∧γ˜₁ = 0 on each fiber Σφ of the fibration Y_τ → S¹. This implies the square of dt ∧γ˜₁ +χdφ ∧ μ₀ −d(χμ₁ +χ(φ−1)μ₀) can not be proportional to dt, though it can be proportional to dφ. We therefore can conclude that the image of P H_dd² _Λ(X₂)⊗P H_dd² _Λ(X₂) → P H_∂¹₋(X₂) is spanned only by dφ. We have thus shown that the images of P H_dd² _Λ ⊗P H_dd² _Λ → P H_∂¹₋ of the above two examples are of different dimensions.

Appendix A. Compatibility of filtered product with topological products

We here provide the proof of Theorem 5.2 demonstrating the com-patibility of the filtered product×as defined in Definition 5.1 with the wedge and Massey product. We begin first with a lemma which will be useful in the proof.

Lemma A.1. For any A_k∈Ω^k,

∗rdL^−(p+1)Ak− ∗rL^−(p+1)dAk= Π^p∗rdL^−(p+1)Ak− ∗rL^−(p+1)d(Π^pAk).

(A.1)

Note that the second term of the right-hand side vanishes whenk > n+p, and the first term vanishes when k≤n+p.

Proof. Case (i): whenk≤n+p. We invoke (2.11) to write A_k as Ak= Π^pAk+ω^p+1∧(L^−(p+1)Ak).

After takingL^−(p+1)◦d, it becomes

L^−(p+1)dA_k=L^−(p+1)d(Π^pA_k) +L^−(p+1))

ω^p+1∧d(L^−(p+1)A_k)* . Since k ≤ n+p, the last term is equal to d(L^−(p+1)A_k). This finishes the proof for this case.

Case (ii): when k > n+p. We write A_k as

Ak=ω^k−n∧B₂n−k+ω^k−n+1∧A_2n−k−2

where B_2n−k ∈ P^n−k and A_2n−k−2 ∈Ω^2n−k−2. A straightforward com-putation shows that

dL^−(p+1)A_k =ω^{k−n−p−1}∧(∂₊B_2n−k) +ω^k−n−p∧(∂₋B_2n−k +dA_2n−k−2) in Ω^k−2p−1,

⇒ ∗rdL^−(p+1)A_k =ω^p∧(∂₊B_2n−k) +ω^p+1∧(∂₋B_2n−k+dA_2n−k−2) in Ω^2n−k+2p+1.

Hence,ω^p∧(∂₊B_2n−k) is Π^p∗rdL^−(p+1)A_k. Meanwhile, it is not hard to see that ω^k−n−p∧(∂₋B_2n−k+dA_2n−k−2) is L^−(p+1)dA_k. This finishes

the proof of the lemma. q.e.d.

We now give the proof of Theorem 5.2. We shall consider the four different cases separately.

(1) F^pH₊^j ×F^pH₊^k →F^pH₊^j+k, j+k≤n+p

Lemma A.2. For j ≤ n+p, k ≤ n+p and j +k ≤ n+p, the product F^pH₊^j ×F^pH₊^k →F^pH₊^j+k induced by (5.11) is compatible with the topological products.

Proof. (Wedge product) Given [ξ_j]∈H_d^j and [ξ_k]∈H_d^k, it is not hard to see that

Π^p(ξ_j∧ξ_k) = Π^p

(Π^pξ_j)∧(Π^pξ_k)

= (Π^pξ_j)×(Π^pξ_k).

(Massey product) Consider two elements [A_j] ∈ F^pH₊^j and [A_k] ∈ F^pH₊^k. SinceA_jandA_kared₊-closed,g(A_j) =L^−(p+1)dA_jandg(A_k) = L^−(p+1)dA_k. Moreover,

dAj =ω^p+1∧g(Aj) and dAk=ω^p+1∧g(Ak).

(A.2)

With (A.2), the Massey product (5.17) is

g(A_j), g(A_k)p = (L^−(p+1)dA_j)∧A_k+ (−1)^jA_j∧(L^−(p+1)dA_k).

(A.3)

We now calculate g(Aj×A_k). According to Theorem 5.3,Aj×A_k is d₊-closed. It follows thatg(A_j×A_k) =L^−(p+1)d(A_j×A_k). With (A.2),

d(A_j×A_k) =d(A_j ∧A_k−ω^p+1∧L^−(p+1)(A_j ×A_k))

=ω^p+1∧

(L^−(p+1)dA_j)∧A_k+ (−1)^jA_j∧(L^−(p+1)dA_k) +dL^−(p+1)(A_j∧A_k)

.

Since j+k≤n+p,L^p+1 is injective on Ω^j+k−2p−1. Thus, g(Aj×A_k) =g(Aj), g(A_k)p+d(L^−(p+1)(Aj∧A_k)).

This completes the proof of the lemma. q.e.d.

(2) F^pH₊^j ×F^pH₊^k →F^pH₋^{2n+2p+1−j−k},j+k > n+p

Lemma A.3. Forj ≤n+p,k≤n+pandj+k > n+p, the product F^pH₊^j ×F^pH₊^k →F^pH₋^{2n+2p+1−j−k}induced by (5.12) is compatible with the topological products.

Proof. (Wedge product) Given [ξ_j] ∈ H_d^j and [ξ_k] ∈ H_d^k, let A_j = Π^pξj, ηj−2p−2 = L^−(p+1)ξj, A_k = Π^pξ_k and η_k−2p−2 = L^−(p+1)ξ_k. Namely,

ξ_j =A_j+ω^p+1∧η_j−2p−2 and ξ_k=A_k+ω^p+1∧η_k−2p−2. It follows fromdξj = 0 and dξk = 0 that

dA_j =−ω^p+1∧dη_j−2p−2 and dA_k =−ω^p+1∧dη_k−2p−2. Sincej≤n+p,L^−(p+1)dAj =−dηj−2p−2andL^−(p+1)dA_k=−dη_k−2p−2. According to (5.12), f(ξ_j)×f(ξ_k) is equal to

Aj×A_k

= Π^p∗r

)−dL^−(p+1)(Aj∧A_k)−(dηj−2p−2)∧A_k−(−1)^jAj∧(dη_k−2p−2)* . (A.4)

The next task is to compute f(ξ_j∧ξ_k) =−∗rdL^−(p+1)(ξ_j∧ξ_k). Since j+k > n+p, it is also equal to −Π^p∗rdL^−(p+1)(ξ_j∧ξ_k). We write

ξ_j∧ξ_k =A_j∧A_k+ω^p+1∧ζ_j+k−2p−2

whereζ_j+k−2p−2 =A_j∧η_k−2p−2+η_j−2p−2∧A_k+ω^p∧η_j−2p−2∧η_k−2p−2. Note that

dζ_j+k−2p−2= (dη_j−2p−2)∧A_k+ (−1)^jA_j∧(dη_k−2p−2).

By applying Π^p on (A.1) forξj∧ξk, we find that f(ξ_j∧ξ_k) = Π^p∗r

)L^−(p+1)d(ξ_j∧ξ_k)−dL^−(p+1)(A_j ∧A_k)

−d

L^−(p+1)(ω^p+1∧ζ_j+k−2p−2)* .

The first term on the right-hand side is zero. The third term can be calculated with the help of (2.23):

−Π^p∗r

)d

L^−(p+1)(ω^p+1∧ζj+k−2p−2)*

=−Π^p∗r(dζj+k−2p−2) +d₋(Π^p∗rζj+k−2p−2)

=−Π^p∗r

)(dηj−2p−2)∧Ak+ (−1)^jAj∧(dηk−2p−2)*

+d₋(Π^p∗rζj+k−2p−2).

To sum up,f(ξ_j∧ξ_k) is equal to

−Π^p∗r

)(dη_j−2p−2)∧A_k+ (−1)^jA_j ∧(dη_k−2p−2)−dL^−(p+1)(A_j ∧A_k)* +d₋(Π^p∗rζj+k−2p−2)

(A.5)

It follows from (A.5) and (A.4) that (5.12) is compatible with the wedge product.

(Massey product) Given [A_j]∈F^pH₊^j and [A_k]∈F^pH₊^k,g(A_j), g(A_k)p

is completely the same as that in the proof of Lemma A.2, and the Massey product is given by (A.3).

We now calculate g(Aj ×A_k). According to (5.12) and (5.14), g(A_j×A_k) =∗rΠ^p∗r

−d L^−(p+1)(A_j∧A_k)

+ (L^−(p+1)dAj)∧A_k+ (−1)^jAj∧(L^−(p+1)dA_k)

. (A.6)

The first term of the right-hand side can be computed with the help of (A.1) forAj∧Ak:

− ∗rΠ^p∗rd L^−(p+1)(Aj ∧Ak)

=L^−(p+1)d(A_j∧A_k)−dL^−(p+1)(A_k∧A_k)

=L^−(p+1)L^p+1

(L^−(p+1)dA_j)∧A_k+ (−1)^kA_j∧(L^−(p+1)dA_k)

−dL^−(p+1)(Ak∧Ak)

where the last inequality uses (A.2). By plugging it into (A.6) and ap-plying (2.12), we have

g(A_j ×A_k) = (L^−(p+1)dA_j)∧A_k+ (−1)^jA_j∧(L^−(p+1)dA_k)

−dL^−(p+1)(A_k∧A_k).

This together with (A.3) shows that (5.12) is compatible with the Massey

product. q.e.d.

(3) F^pH₊^j ×F^pH₋^k →F^pH₋^k−j, j≤k

Lemma A.4. For j ≤ k ≤ n+p, the product F^pH₊^j ×F^pH₋^k → F^pH₋^k−j induced by (5.8) is compatible with the topological products.

Proof. (Wedge product) Suppose that [ξ_j]∈H_d^j and [ξ]∈H_d where = 2n+ 2p+ 1−k. Since k ≤n+p, > n+p, ξ = ω^p+1∧η_2n−k−1 for some η_2n−k−1 ∈Ω^2n−k−1. Let Aj = Π^pξj and ηj−2p−2 =L^−(p+1)ξj. Namely,

ξ_j =A_j+ω^p+1∧η_j−2p−2 and ξ =ω^p+1∧η_2n−k−1. Since dξ_k= 0 and dξ_j = 0,

dAj =−ω^p+1∧dηj−2p−2 and ω^p+1∧dη_2n−k−1 = 0.

According to (5.14), f(ξj) =Aj and f(ξ) =−∗rdη₂n−k−1. By (5.8), f(ξ_j)×f(ξ) =−(−1)^j∗r(A_j∧dη_2n−k−1).

(A.7)

We now compute f(ξ_j∧ξ) =−∗rdL^−(p+1)(ξ_j∧ξ). Due to (A.1),

−∗rd L^−(p+1)(ξ_j ∧ξ) =−Π^p∗rd L^−(p+1)(ξ_j∧ξ)− ∗rL^−(p+1)d(ξ_j∧ξ) where the second term vanishes. Then writeξ_j∧ξasω^p+1∧ξ_j∧η_2n−k−1 and apply (2.23) forζ =ξ_j∧η_2n−k−1:

f(ξ_j∧ξ) =−Π^p∗r

d(ξ_j∧η_2n−k−1)

+d₋(Π^p∗rζ)

=−Π^p∗r

)(−1)^j(A_j +ω^p+1∧η_j−2p−2)∧dη_2n−k−1*

+d₋(Π^p∗rζ)

=−(−1)^jΠ^p∗r(Aj∧dη_2n−k−1) +d₋(Π^p∗rζ).

(A.8)

The last equality uses the fact that ω^p+1 ∧dη_2n−k−1. Due to (2.22),

∗r(A_j ∧dη_2n−k−1) ∈ F^pΩ^k−j. That is to say, the first Π^p-operator in (A.8) acts as the identity map.

Whenk−j=n+p, Π^p∗rζ belongs to Π^pΩ^n+p+1 ={0}. By (A.7) and (A.8), we conclude that (5.8) is compatible with the wedge product.

(Massey product) Given [A_j] ∈ F^pH₊^j and [ ¯A_k] ∈ F^pH₋^k, let B_j−2p = L^−pA_j. Since d₊A_j = 0, dA_j =L^p+1∧(L^−(p+1)dA_j). By (5.14),

g(A_j) =L^−(p+1)dA_j and g( ¯A_k) =∗rA¯_k. Due to (2.22),L^p+1(g( ¯A_k)) = 0. And the Massey product (5.17) is

g(A_j), g( ¯A_k)p =∂₋B_j−2p,∗rA¯_kp= (−1)^jA_j∧(∗rA¯_k).

According to (5.8) and (5.14),

g(A_j ×A¯_k) =∗r(A_j ×A¯_k) = (−1)^jA_j∧(∗rA¯_k).

This completes the proof of the lemma. q.e.d.

Since all the products are graded anti-commutative, the product F^pH₋^j ×F^pH₊^k → F^pH₋^j−k induced by (5.9) is also compatible with the topological products.

(4) F^pH₋^j ×F^pH₋^k →0

Lemma A.5. For j ≤ n+p and k ≤ n+p, the product F^pH₋^j × F^pH₋^k →0defined by (5.10) is compatible with the topological products.

Proof. By simple degree counting, it is compatible with the wedge product. It follows from (2.22) that L^p+1g =−L^p+1∗r = 0. Hence, the Massey productg(·), g(·)p = 0 on F^pH₋^∗. q.e.d.

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Department of Mathematics National Taiwan University Taipei 10617, Taiwan E-mail address: cjtsai@ntu.edu.tw

Department of Mathematics University of California Irvine, CA 92697 USA E-mail address: lstseng@math.uci.edu

Department of Mathematics Harvard University Cambridge, MA 02138 USA E-mail address: yau@math.harvard.edu

Dans le document COHOMOLOGY AND HODGE THEORY ON SYMPLECTIC MANIFOLDS: III Chung-Jun Tsai, Li-Sheng Tseng & Shing-Tung Yau Abstract (Page 48-61)