A genus two construction

Consider a genus two surface Σ₂. Fix a symplectic base for its H_d¹(Σ): {a₁, b₁, a₂, b₂}. Moreover, we may assume that the basis actually form an integral basis for the singular cohomology of Σ.

Let ω_Σ be an area form of Σ. Normalize it by

[ω_Σ] = [a₁∧ b₁] = [a₂∧ b₂] .

Let τ be a monodromy of Σ whose action on H_d¹(Σ) are shows that τ^∗ preserves the intersection pairing. It follows from the theory of mapping class group (see for example [8]) that τ^∗ does arise from a monodromy, but is not unique. Note that

S⁻¹τ^∗S = J ,

The basis for its Jordan form is

γ0= a1 , γ1 = b2 ,

They associate the following differential forms on Yτ:

˜

γ0 = a1+ χdg0+ χ⁰g0dφ ,

˜

γ1 = b2+ φ a1+ χ(dg1+ (φ − 1)dg0) + χ⁰(g1+ (φ − 1)g0)dφ ,

˜

γ₂ = a₂+ φ b₂+φ²− φ

2 a₁+ χ(dg₂+ (φ − 1)dg₁+φ²− 3φ + 2 2 dg₀) + χ⁰(g₂+ (φ − 1)g₁+φ²− 3φ + 2

2 g₀)dφ ,

˜

γ₃ = (b₁− a₂) + φ a₂+φ²− φ

2 b₂+φ³− 3φ²+ 2φ

6 a₁

+ χ(dg₃+ (φ − 1)dg₂+φ²− 3φ + 2

2 dg₁+ φ³− 6φ²+ 11φ − 6

6 dg₀)

+ χ⁰(g3+ (φ − 1)g2+φ²− 3φ + 2

2 g1+ φ³− 6φ²+ 11φ − 6

6 g0)dφ

We 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 ∧ ˜γ₀, dφ ∧ ˜γ₃,

dt ∧ ˜γ₁+ χ⁰dφ ∧ µ₀− d(χ⁰µ₁+ χ⁰(φ − 1)µ₀)} , P H_∂¹₋(X₂) = {dt, dφ, ˜γ₃} .

Since P H_∂¹₋(X₂) ∼= H_d³(X₂), its element can captured by integrating against H_d¹(X₂) = {dφ, dt, ˜γ₀}.

There is only one generator of P H_dd² Λ(X2) that is not d-closed:

∂− dt ∧ ˜γ1+ χ⁰dφ ∧ µ0− d(χ⁰µ1+ χ⁰(φ − 1)µ0)
= −˜γ0 .

We now consider the pairings between the generators of P H_dd² Λ(X₂). If one of them is d-closed, the only non-trivial pairing is

(dφ ∧ ˜γ3) × dt ∧ ˜γ1+ χ⁰dφ ∧ µ0− d(χ⁰µ1+ χ⁰(φ − 1)µ0)
∼= dφ ∧ ˜γ0∧ ˜γ3 .

The latter expression has nonzero integration against dt. That is to say, the element in P H_∂¹₋(X₂) is proportional to dφ.

It remains to examine the square of dt ∧ ˜γ1+ χ⁰dφ ∧ µ0 − d(χ⁰µ1 + χ⁰(φ − 1)µ0). As an element in H_d³(X₂), it is

−2 dt ∧ ˜γ₁+ χ⁰dφ ∧ µ₀− d(χ⁰µ₁+ χ⁰(φ − 1)µ₀)
∧ ˜γ₀ .

The expression has zero integration against ˜γ0. We compute its integration against dφ.

2 Z

X2

dφ ∧ dt ∧ ˜γ0∧ ˜γ1 , and it is not hard to see thatR

Σφγ˜0∧ ˜γ1= 0 on each fiber Σφof the fibration Yτ → S¹. Hence, the square of dt ∧ ˜γ₁ + χ⁰dφ ∧ µ₀ − d(χ⁰µ₁+ χ⁰(φ − 1)µ₀) can only be proportional to dφ. It follows that the image of P H_dd² Λ(X2) ⊗ P H_dd² Λ(X2) → P H_∂¹₋(X2) is spanned by dφ. And we have 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.

A Compatibility of filtered product with topological products

We here provide the proof of Theorem 5.2 demonstrating the compatibility 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)A_k− ∗_rL^−(p+1)dA_k = Π^p∗_rdL^−(p+1)A_k− ∗_rL^−(p+1)d(Π^pA_k) . (A.1) Note that the second term of the right hand side vanishes when k > n + p, and the first term vanishes when k ≤ n + p.

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

A_k= ω^k−n∧ B_2n−k+ ω^k−n+1∧ A⁰_2n−k−2

where B_2n−k∈ P^n−k and A⁰_2n−k−2 ∈ Ω^2n−k−2. A straightforward computation shows that dL^−(p+1)A_k= ω^{k−n−p−1}∧ (∂₊B_2n−k) + ω^k−n−p∧ (∂₋B_2n−k+ dA⁰_2n−k−2) in Ω^k−2p−1 ,

⇒ ∗_rdL^−(p+1)A_k= ω^p∧ (∂₊B_2n−k) + ω^p+1∧ (∂₋B_2n−k+ dA⁰_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 ∧ (∂−B2n−k+ dA⁰_2n−k−2) is L^−(p+1)dAk. This finishes the proof of the lemma.

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 [ξj] ∈ H_d^j and [ξk] ∈ H_d^k, it is not hard to see that Π^p(ξj∧ ξ_k) = Π^p (Π^pξj) ∧ (Π^pξ_k)
= (Π^pξj) × (Π^pξ_k) .

(Massey product) Consider two elements [Aj] ∈ F^pH₊^j and [Ak] ∈ F^pH₊^k. Since Aj and Ak

are d₊-closed, g(A_j) = L^−(p+1)dA_j and g(A_k) = L^−(p+1)dA_k. Moreover,

dA_j = ω^p+1∧ g(A_j) and dA_k= ω^p+1∧ g(A_k) . (A.2) With (A.2), the Massey product (5.17) is

hg(A_j), g(A_k)i_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 that g(A_j× A_k) = L^−(p+1)d(A_j× A_k). With (A.2),

d(Aj× A_k) = d(Aj∧ A_k− ω^p+1∧ L^−(p+1)(Aj× 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(A_j × A_k) = hg(A_j), g(A_k)i_p+ d(L^−(p+1)(A_j∧ A_k)) . This completes the proof of the lemma.

(2) F^pH₊^j × F^pH₊^k → F^pH2n+2p+1−j−k

− , j + k > n + p

Lemma A.3. For j ≤ n + p, k ≤ n + p and j + k > n + p, the product F^pH₊^j × F^pH₊^k → F^pH2n+2p+1−j−k

− induced by (5.12) is compatible with the topological products.

Proof. (Wedge product) Given [ξj] ∈ H_d^j and [ξk] ∈ H_d^k, let Aj = 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 from dξj = 0 and dξk= 0 that

dAj = −ω^p+1∧ dη_j−2p−2 and dA_k= −ω^p+1∧ dη_k−2p−2 .

Since j ≤ n + p, L^−(p+1)dAj = −dηj−2p−2 and L^−(p+1)dA_k= −dη_k−2p−2. According to (5.12), f (ξ_j) × f (ξ_k) is equal to

A_j× A_k

= Π^p∗_r − dL^−(p+1)(A_j∧ A_k) − (dη_j−2p−2) ∧ A_k− (−1)^jA_j∧ (dη_k−2p−2) . (A.4) The next task is to compute f (ξj ∧ ξ_k) = −∗rdL^−(p+1)(ξj∧ ξ_k). Since j + k > n + p, it is also equal to −Π^p∗_rdL^−(p+1)(ξ_j∧ ξ_k). We write

ξj∧ ξ_k = Aj∧ A_k+ ω^p+1∧ ζ_j+k−2p−2

where ζj+k−2p−2 = Aj ∧ η_k−2p−2+ ηj−2p−2∧ A_k+ ω^p∧ η_j−2p−2∧ η_k−2p−2. Note that dζj+k−2p−2 = (dηj−2p−2) ∧ Ak+ (−1)^jAj∧ (dη_k−2p−2) .

By applying Π^p on (A.1) for ξj∧ ξ_k, we find that

f (ξj∧ ξ_k) = Π^p∗_rL^−(p+1)d(ξj∧ ξ_k) − dL^−(p+1)(Aj∧ 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∗_rd L^−(p+1)(ω^p+1∧ ζ_j+k−2p−2)


= − Π^p∗_r(dζj+k−2p−2) + d−(Π^p∗_rζj+k−2p−2)

= − Π^p∗_r(dη_j−2p−2) ∧ A_k+ (−1)^jA_j∧ (dη_k−2p−2) + d−(Π^p∗_rζ_j+k−2p−2) .

To sum up, f (ξj∧ ξ_k) is equal to

− Π^p∗_r(dη_j−2p−2) ∧ A_k+ (−1)^jAj ∧ (dη_k−2p−2) − dL^−(p+1)(Aj ∧ A_k) + d−(Π^p∗_rζ_j+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 [Aj] ∈ F^pH₊^j and [A_k] ∈ F^pH₊^k, hg(Aj), g(A_k)ip 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(A_j × A_k). According to (5.12) and (5.14), g(Aj× A_k) = ∗rΠ^p∗_rh

− d L^−(p+1)(Aj∧ A_k)

+ (L^−(p+1)dA_j) ∧ A_k+ (−1)^jA_j∧ (L^−(p+1)dA_k)i .

(A.6)

The first term of the right hand side can be computed with the help of (A.1) for A_j∧ A_k:

− ∗_rΠ^p∗_rd L^−(p+1)(Aj∧ A_k)

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

= L^−(p+1)L^p+1 h

(L^−(p+1)dAj) ∧ Ak+ (−1)^kAj∧ (L^−(p+1)dAk) i

− dL^−(p+1)(Ak∧ A_k) where the last inequality uses (A.2). By plugging it into (A.6) and applying (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 togehter with (A.3) shows that (5.12) is compatible with the Massey product.

(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<sub>`</sub>] ∈ H<sub>d</sub><sup>`</sup>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 A_j = Π^pξ_j and ηj−2p−2= L^−(p+1)ξj. Namely,

ξj = Aj+ ω^p+1∧ η_j−2p−2 and ξ`= ω^p+1∧ η_2n−k−1

Since dξ_k= 0 and dξj = 0,

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

According to (5.14), f (ξj) = Aj and f (ξ_`) = −∗rdη_2n−k−1. By (5.8),

f (ξj) × f (ξ_{`</sub>) = −(−1)<sup>j</sup>∗<sub>r</sub>(Aj ∧ dη<sub>2n−k−1</sub>) . (A.7) We now compute f (ξj ∧ ξ<sub>`}) = −∗rdL^−(p+1)(ξj∧ ξ_`). Due to (A.1),

−∗_rd L^−(p+1)(ξj ∧ ξ_{`</sub>) = −Π<sup>p</sup>∗<sub>r</sub>d L<sup>−(p+1)</sup>(ξj∧ ξ<sub>`}) − ∗rL^−(p+1)d(ξj∧ ξ_`)

where the second term vanishes. Then write ξ_j∧ ξ_` as ω^p+1∧ ξ_j∧ η_2n−k−1 and apply (2.23) for ζ = ξj∧ η_2n−k−1:

f (ξj∧ ξ_`) = −Π^p∗_r d(ξj ∧ η_2n−k−1)
+ d₋(Π^p∗_rζ)

= −Π^p∗_r(−1)^j(Aj+ ω^p+1∧ η_j−2p−2) ∧ dη2n−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.

When k − 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

hg(A_j), g( ¯A_k)i_p = h∂−B_j−2p, ∗_rA¯_ki_p = (−1)^jA_j ∧ (∗_rA¯_k) . According to (5.8) and (5.14),

g(Aj× ¯Ak) = ∗r(Aj× ¯Ak) = (−1)^jAj∧ (∗_rA¯k) . This completes the proof of the lemma.

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 → 0 defined by (5.10) is compatible with the topological products.

Proof. By the 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 product hg( · ), g( · )i_p = 0 on F^pH₋^∗.

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Department of Mathematics, National Taiwan University Taipei 10617, Taiwan

Email address: [email protected]

Department of Mathematics, University of California, Irvine Irvine, CA 92697, USA

Email address: [email protected]

Department of Mathematics, Harvard University Cambridge, MA 02138, USA

Email address: [email protected]

在文檔中 Cohomology and Hodge theory on symplectic manifolds: III (頁 58-68)