Motivic Hopf Algebras and Periods

Associated to Multiple Zeta Functions and Feynman Graphs

Lecture I: Work of Goncharov (+Deligne, Manin) on multiple zetas

Lecture II: Joint work with H. Esnault and D. Kreimer on Feynman graphs

1. Hopf algebra

\underrightarrow{lim} (ℚ[π_{1}(ℙ^{1} − \{0, 1, ∞\}, ∗)]/I^{N})^{∨}

in category of mixed Tate motives over ℤ

2. Renormalization Hopf algebra in the category of motives.

3. Periods

Comments

1. Compare and contrast the two Hopf algebras.

2. Problem: what is the physical meaning of co-representations of the renormalization Hopf algebra. Link with Dyson Schwinger equations.

3. Fantasy: The action of the renormalization group on the the renormalization hopf algebra in physics can in some way be related to the action of the Tannaka group of the category of motives.

Tannakian Categories

Tannakian category T abelian, ℚ-linear category with ⊗ (symmetric monoidal category). T rigid tensor category (Identity object 1 with X ⊗ 1 ≅ X and internal Hom \underline{Hom}(X, Y).) F : T → {f.d. ℚ-vector spaces} fibre functor (exact, faithful, compatible with ⊗) Example: G = Spec (A) affine algebraic group (A = commutative Hopf algebra). T = Rep_{k}(G) category of f.d. representations of G over a field k. (To fix ideas, k = ℚ) F (V) = V forgetful functor

Tannakian Categories Main Theorem

Main Theorem. Can recover G from (T , F). G = Aut^{⊗}F. Example: T = f.d. ℤ-graded vector spaces; F : T → f.d. vector spaces (forget grading.) G = 𝔾_{m}. g ∈ 𝔾_{m}(ℚ) = ℚ^{×},

ℚ = F (ℚ_{n}) ⟶^{g^{n}} F (ℚ_{n}) = ℚ.

Unipotent Algebraic Groups

Assume:

1. T Artinian, simple objects ℚ(n), n ∈ ℤ. Hom(ℚ(n), ℚ(m)) = ℚ · δ_{m,n}

2. Objects have weight filtrations W_{•}X with gr_{2n}^{W}X = ⊕Q(−n). Graded fibre functor

F_{gr} : T → f.d. graded vector sp. F_{gr}(M) := ⊕_{n}Hom_{T} (ℚ(n), gr_{−2n}^{W}M)

F := forget ∘ F_{gr}. In this case G = U ⋊ 𝔾_{m} with U unipotent. (U acts as identity on gr_{∗}^{W})

Motivic Hopf Algebra Assoc. to π_{1}

Voevodsky’s motivic category is a triangulated category, not a Tannakian category. In nice cases (number fields) we do know how to build Tannakian categories. Think of objects in our Tannakian category as being generalized cohomology groups. Ex: X variety, a ∈ X(k) point, n > 1, Y_{i} ⊂ X^{n}, 0 ≤ i ≤ n Y_{0} = {a} × X^{n−1}, Y_{1} = Δ_{X} × X^{n−2}, Y_{2} = X × Δ_{X} × X^{n−3}, …, Y_{n} = X^{n−1} × {a} Theorem(Beilinson). H^{n}(X^{n}, Y; ℚ) ≅ coker(ℚ → ℚ[π_{1}(X, a)]/I^{n})^{∨}. Here I ⊂ ℚ[π_{1}] is the augmentation ideal. This shows motivic nature of nilpotent quotients of π_{1}.

Motivic Hopf Algebra Assoc. to π_{1} (cont.)

n-step Nilpotent representations of π_{1} ↔ representations of the algebraic group Spec ((\underrightarrow{lim} ℚ[π_{1}]/I^{n})^{∨}) (\underrightarrow{lim} ℚ[π_{1}]/I^{n})^{∨} is a commutative Hopf algebra in the Tannakian category of motives. More generally, a, b ∈ X(k), can define the dual to the paths (\underrightarrow{lim} ℚ[_{a}P_{b}]/I^{n})^{∨} is a ring object in the Tannakian category, so can define the torsor of paths Spec ((\underrightarrow{lim} ℚ[_{a}P_{b}]/I^{n})^{∨})

Fibre Functors for Mixed Tate Motives

T Tannakian category of mixed Tate Motives. 3 fibre functors: F_{ω}(M) := ⊕_{n}Hom_{T} (ℚ(n), gr_{−2n}^{W}M) F_{Betti}(M) := H_{Betti}(M). (NB. in general, depends on embedding σ : k ↪ ℂ. We take k = ℚ) F_{DR}(M) = H_{DR}(M) de Rham cohomology. Comparison isomorphisms

comp_{Betti,ω} : M_{ω} ⊗ ℂ ⟶^{≅} M_{Betti} ⊗ ℂ.

Similarly comp_{DR,Betti}, etc. NB. Isom(F_{ω}, F_{Betti}) is a G = Gal(T )-torsor. G = U ⋊ 𝔾_{m} has no non-split torsors.

∴ ∃ a ∈ G(ℂ), M_{Betti} = comp_{Betti,ω}(a(M_{ω})) ⊂ M_{Betti} ⊗ ℂ

DR realization: cosimplicial scheme

∂_{i} : X^{n−1} → X^{n}

(x_{1}, …, x_{n−1}) ↦

\begin{cases}(a, x_{1}, …, x_{n−1}) & i = 0 \\ (x_{1}, …, x_{i}, x_{i}, x_{i+1}, …, x_{n−1}) & 1 ≤ i ≤ n − 1 \\ (x_{1}, …, x_{n−1}, a) & i = n \end{cases}

For X affine, resulting DR complex is quasi-isomorphic to Chen’s complex of iterated integrals. n ↦ ⊗_{n}Γ(X, Ω^{∗}); d⟨a⟩ = ⟨da⟩ + (−1)^{p}⟨∑(±)∂_{i}^{∗}a⟩ Ex: X = ℙ_{ℚ}^{1} − {0, 1, ∞}.

ℙ^{1} − {0, 1, ∞}; dch

Tangential basepoint at a puncture: Intuitively, a tangent vector at a puncture doesn’t determine a point of ℙ^{1} − {0, 1, ∞}, but it does determine the fibre of a covering upto canonical identification, i.e. it defines a functor from coverings to sets. Thus it can be used as a basepoint for π_{1}. dch ∈ (_{0→1}P_{1→0})_{Betti} (path from 0 to 1 with tangential basepoints)

Periods: Mixed Tate Hodge and de

Rham Structures

n ∈ ℤ, define 1 dim. ℚ-vect. spaces ℚ(n)_{Betti}, ℚ(n)_{DR} weight −2n, ℚ(n) = ℚ(1)^{⊗n}. ℚ(n)_{DR} ⊗ ℂ = ℚ(n)_{Betti} ⊗ ℂ = H^{−n,−n}(ℚ(n)) (Hodge filtration) General Mixed Tate Hodge structure H; successive extension of ℚ(n)’s. W_{∗}H increasing weight filtration; F^{∗}H_{ℂ} decreasing Hodge filtration H mixed Tate if gr^{W} H = ⊕_{i}ℚ(n_{i}). In this case H_{ℂ} = W_{2n}H_{ℂ} ⊕ F^{n+1}H_{ℂ}

Periods: Example

𝔾_{m}(ℂ) = ℂ^{×} = ℙ_{ℂ}^{1} − {0, ∞}

ℚ · S^{1} = H_{1}(𝔾_{m}, ℚ) = ℚ(1)_{Betti};

H^{1}(𝔾_{m}, ℚ) = ℚ(−1);

Q(−1)_{DR} = ℚ · \frac{dt}{t}; ℚ(−1)_{Betti} = ℚ · (S^{1})^{∨}.

Q(−1)_{DR} = 2πi · ℚ(−1)_{Betti} ⊂

ℚ(−1)_{Betti} ⊗ ℂ = ℚ(−1)_{DR} ⊗ ℂ

ℚ(−n)_{DR} = (2πi)^{n} · ℚ(−n)_{Betti}

Periods: Typical computation

ℙ^{n}, homogeneous coordinates A_{0}, …, A_{n} Δ : ∏ A_{i} = 0 coordinate simplex; X ⊂ ℙ^{n} hypersurface Assume vertices (0, …, 1, …, 0) ∉ X; define H := H^{n}(ℙ^{n} − X, Δ − X ∩ Δ) H is a Hodge structure (not necessarily mixed Tate). Lowest weight W_{0}H = ℚ(0) carried by the vertices. Homology: H^{∨} ↠ ℚ(0). Chose splitting ℚ ↪ H^{∨} Ex: σ = {(σ_{0}, …, σ_{n}) | σ_{i} ≥ 0} ⊂ ℙ^{n}(ℝ) If σ ∩ X = ∅, then [σ] ∈ H^{∨} = H_{n}(ℙ^{n} − X, Δ − X ∩ Δ) gives splitting.

Computation (cont.)

Relative DR-complex. V ↪ W closed. Ω_{W,V}^{∗} ⊂ Ω_{W}^{∗}. Ex: V ⊂ ℙ^{1} finite set of points. I ⊂ O_{ℙ^{1}} ideal Note n = dim W, Ω_{W,V}^{n} = Ω_{W}^{n}. For W affine, get Γ(W, Ω^{n}) → H_{DR}^{n}(W, V) W = ℙ^{n} − X, X : F = 0, deg F = d; Ω_{n} = ∑(±)A_{i}dA_{0} … dA_{i} … dA_{n}, ω = \frac{GΩ_{n}}{F^{r}} log divergent Feynman graph case; G graph, ℓ loops, n + 1 = 2ℓ edges F graph polynomial, ω_{G} = \frac{Ω_{n}}{F^{2}} Period = ∫_{σ} ω_{G}

Multiple Zeta Periods

σ = {(x_{1}, …, x_{n}) ∈ [0, 1]^{n} | 0 ≤ x_{1} ≤ … ≤ x_{n} ≤ 1}

⊂ (ℙ^{1})^{n}

pr_{i} : (ℙ^{1})^{n} → ℙ^{1}

ω := pr_{1}^{∗}(\frac{dt}{t − ε_{1}} ) ∧ … ∧ pr_{n}^{∗}(\frac{dt}{t − ε_{n}}); ε_{i} = 0, 1

Important theme: polar locus X = (ℙ^{1})^{n} − ∏_{i}(ℙ^{1} − {ε_{i}, ∞}) meets σ. Must blow up. Correct base is not (ℙ^{1})^{n} but rather \overline{M}_{0,n+3} → (ℙ^{1})^{n}. Moduli space of semi-stable genus 0 curves (Goncharov-Manin)

Discussion of \overline{M}_{0,n}

\overline{M}_{0,n} moduli space of semi-stable genus 0 curves. Tree of ℙ^{1}’s (connected, no loops) with only ordinary double points and with n smooth points marked. Group of automorphisms required to be finite (⇔ number of singular + marked points ≥ 3 on every irreducible component.)

Fine moduli space; C_{n} ⟶^{p_{n}} \overline{M}_{0,n} universal curve with n marked points. Remarkable fact:

C_{n}  ≅  \overline{M}_{0,n+1}
↘  ↙ forget n + 1-st pt.
\overline{M}_{0,n}

\overline{M}_{0,5}

Ex. \overline{M}_{0,4} ≅ ℙ^{1} (crossratio). \overline{M}_{0,5} = ℙ^{1} × ℙ^{1} with (0, 0), (1, 1), (∞, ∞) blown up. Note that chain σ on ℙ^{1} × ℙ^{1} meets polar locus for \frac{dt_{1}}{t_{1}−1} ∘ \frac{dt_{2}}{t_{2}} at points (0, 0), (1, 1). These points are blown up on \overline{M}_{0,5}.

Iterated Integrals

σ ↪ [0, 1]^{n} ↪ (ℙ^{1})^{n} ∫_{0}^{1} ω_{1} ∘ … ∘ ω_{n} := ∫_{σ} pr_{1}^{∗}ω_{1} ∧ … ∧ pr_{n}^{∗}ω_{n}

\underbrace{\frac{dt}{t − 1} ∘ \frac{dt}{t} ∘ … ∘ \frac{dt}{t}}_{n_{1}} ∘ … ∘ \underbrace{\frac{dt}{t − 1} ∘ \frac{dt}{t} ∘ … ∘ \frac{dt}{t}}_{n_{r}}

power series expansions for \frac{1}{t−1} yield multiple zeta value

Period = ∑_{a_{r}>...>a_{1}>0} \frac{1}{a_{1}^{n_{1}}a_{2}^{n_{2}} ⋯ a_{r}^{n_{r}}}

Deligne-Goncharov Construction

(\underrightarrow{lim} ℚ[π_{1}(ℙ^{1} − \{0, 1, ∞\}, tang. bpt.)]/I^{N})^{∨} ∈ Ob(T ) Action of G = Gal(T ) on this object yields ι : G_{ω} → H_{ω} ⊂ ℚ⟨⟨e_{0}, e_{1}⟩⟩ Measures linear algebra of G acting on π_{1}(ℙ^{1} − {0, 1, ∞}, tang. bpt.)_{mot}. Linear algebra identifies H_{ω} = V_{ω} ⋊ 𝔾_{m} and V_{ω}(ℂ) = ∏ := grouplike elts. in ℂ⟨⟨e_{0}, e_{1}⟩⟩ In particular, ⟨dch⟩ ∈ H_{ω}(ℂ) ℚ-form (_{1}P_{0})_{Betti} ⊂ (_{1}P_{0})_{ω} ⊗ ℂ equals image of ℚ-form (_{1}P_{0})_{ω} by ⟨dch⟩ · τ(2πi). Consequence: ⟨dch⟩ · τ(2πi) = ι(a) · v; v ∈ H_{ω}(ℚ) I.e.

τ(2πi)^{−1}⟨dch⟩ ∈ ι(U_{ω}) · v ⊂ V_{ω}

Deligne-Goncharov Construction (cont.)

Coefficient of e_{i_{1}} ⋯ e_{i_{n}} in ⟨dch⟩ is the corresponding iterated integral, e_{i_{j}} = 0, 1 This is the link between multiple zetas and the structure of G = Gal(T ) Theorem. Let D_{n} be the dim. of ℚ-vector space generated by coefficients of dch in degree n. Then generating series ∑ D_{n}t^{n} is term by term bounded by the coefficients of \frac{1}{1−t^{2}−t^{3}} . (Also proven by Terasoma).

Graph Hopf Algebra

Hopf algebra of graphs. (Kreimer, Connes-Kreimer) H = ⊕ℚ · Γ vector space on isomorphism classes of graphs Commutative multiplication: Γ · Γ′ := Γ ⨿ Γ′

Comultiplication H ⟶^{Δ} H ⊗ H

Δ(Γ) = ∑_{Γ′⊂Γ divergent subgraph} Γ′ ⊗ (Γ//Γ′)

(Γ//Γ′ means the quotient graph with components of Γ′ shrunk to points.) The precise meaning of “divergent subgraph” in the Kreimer picture is determined by the physical picture (the Lagrangian) One has to be careful that the co-multiplication be co-associative. Defn. Γ′ ⊂ Γ divergent, if 2h_{1}(Γ′) ≥ #(Edge(Γ′)

Motive Associated to a Graph

Construct M(Γ) so that Feynman amplitude for Γ is a period. Γ with n loops, N edges. Perturbative series leads to a certain integral of the form

F.A.(Γ) = ∫_{ℙ^{4n−1}(ℝ)} \frac{P Ω_{4n−1}}{Q_{1}Q_{2} ⋯ Q_{N}}

Here Q_{i} rank 4 quadrics (≥ 0), and P homogeneous of degree 2N − 4n. Ω = ∑_{i=1}^{4n} ±B_{i}dB_{1} ∧ … d̂B_{i} ∧ … ∧ dB_{4n}

Schwinger Picture

Q : ∑ A_{i}Q_{i} = 0 ↪ ℙ^{N−1} × ℙ^{4n−1} → ℙ^{4n−1}
↘ f ↓
ℙ^{N−1}

f family of projective quadrics of dim 4n − 2 Q_{i} ↔ M_{i}, 4n × 4n symmetric matrix M_{i} = M_{i} ⊗ I_{4}, M_{i} n × n symmetric Singular locus of f defined by Φ(A) = det(∑ A_{i}M_{i}) = 0 Φ(A) = Ψ(A)^{4}; Ψ(A) = det(∑ A_{i}M_{i})

Schwinger Picture (cont.)

Fibres of f are even dim. quadrics Q_{a} Middle dim. primitive cohomology H^{4n−2}(Q_{a}) ≅ ℚ rk 1 local system, monodromy {±1}, ramified over {Φ = 0}. Cohomology of resulting double cover of ℙ^{N−1}

\frac{GΩ_{N−1}}{\sqrt{Φ}} = \frac{GΩ_{N−1}}{Ψ^{2}}

Important case: N = 2n, G = 1.

F.A. = ∫_{ℙ^{4n−1}(ℝ)} \frac{Ω_{4n−1}}{Q_{1} ⋯ Q_{2n}} = \frac{1}{π^{2n}} ∫_{σ} \frac{Ω_{2n−1}}{Ψ^{2}}

σ = {(a_{1}, …, a_{2n}) ∈ ℙ^{2n−1}(ℝ) | a_{i} ≥ 0}

Graph Polynomial

ℙ^{N−1} ⊃ X_{Γ} : Ψ = Ψ_{Γ} = 0; Facts about Ψ_{Γ}(A):

1. Ψ = ∑\begin{matrix}T ⊂Γ \\ span. tree \end{matrix} ∏_{e∉T} A_{e}

2. Γ connected. View Γ as metrized graph with e having “length” A_{e} and vertices orthonormal. Then get relation with Laplacian of graph.

ℤ^{Edge(Γ)} ⟶^{d} ℤ^{V ert(Γ)}; Ψ = ∏ A_{e} · det(dd^{∗})

3. L : A_{e_{1}} = … = A_{e_{p}} = 0 coordinate linear space. Then

L ⊂ X_{Ψ} ⇔ h_{1}(e_{1} ∪ … ∪ e_{p}) > 0

M(Γ)

First try: M(Γ) = H^{N−1}(ℙ^{N−1} − X_{Γ})(N − 1) Criticism: ∫_{σ} \frac{Ω_{2n−1}}{Ψ^{2}} not defined because σ is a relative chain. try again:

M(Γ) =

H^{N−1}(ℙ^{N−1} − X_{Γ}, Δ − X_{Γ} ∩ Δ)(N − 1)

Δ : ∏ A_{e} = 0 coordinate simplex. Criticism: period still not defined.

σ ∩ X_{Γ} = ⋃_{L⊂X_{Γ}} L(ℝ)^{+}

Have to blow up to separate σ from X.

M(Γ) = H^{N−1}(P − Y, B − Y ∩ B)
ρ : P → ℙ^{N−1}, B = ρ^{−1}(Δ)

Example. Wheel with n-spokes W S_{n}. Hélène will talk about the structure of M(W S_{n}) = H^{2n−1}(ℙ^{2n−1} − X_{n})(2n − 1)

Algebra Structure

H_{mot} := ⊕_{Γ}M(Γ) (ind-motive) Product structure: A_{e} edge variables for Γ, B_{e} edge variables for Λ. Ψ_{Γ⨿Λ} = Ψ_{Γ}(A) · Ψ_{Λ}(B) #Edge(Γ) = N, #Edge(Λ) = S X_{Γ⨿Λ} ⊂ ℙ^{N+S−1}

ℙ^{N+S−1}−X_{Γ⨿Λ} ⟶^{h} (ℙ^{N−1}−X_{Γ})×(ℙ^{S−1}−X_{Λ})

Here h(A, B) = (A) × (B). h is a 𝔾_{m}-fibration, h(cA, c′B) = h(A, B).

∴ H^{N+S−1}(ℙ^{N+S−1} − X_{Γ⨿Λ}, …)(N + S − 1) ≅

H^{N−1}(ℙ^{N−1} − X_{Γ}, …)(N − 1)⊗ H^{S−1}(ℙ^{S−1} − X_{Λ}, …)(S − 1)

M(Γ) ⊗ M(Λ) ≅ M(Γ ⨿ Λ);
H_{mot} ⊗ H_{mot} → H_{mot}

Co-algebra Structure

Λ ⊂ Γ graphs. Λ = e_{1} ∪ … ∪ e_{p}; L : A_{1} = … = A_{p} = 0. Assume h_{1}(Λ) > 0 so L ⊂ X_{Γ}. Theorem. The normal cone of L ⊂ X_{Γ} is a union of the normal cones of X_{Λ} and X_{Γ//Λ}. I.e.

Ψ_{Γ} = Ψ_{Λ}(A_{1}, …, A_{p}) · Ψ_{Γ//Λ} + R where R has degree > deg Ψ_{Λ} in A_{1}, …, A_{p}. Reinterpretation. Consider the diagram

\begin{matrix}Y ∪ E & ⟶ & P \\ ↓ & & ↓^{π} \\ X & ⟶ & ℙ^{N−1} \end{matrix}  (1)

Here π is the blowup of L in ℙ^{N−1}, Y is the strict transform of X, and E ≅ L × ℙ^{p−1} is the exceptional divisor. Then

Y ∩ E = (L × X_{Λ}) ∪ (X_{Γ//Λ} × ℙ^{p−1}).

Co-algebra Structure (cont.)

Corollary. There is a well-defined residue map

M(Γ) ⟶^{Res} M(Λ) ⊗ M(Γ//Λ)

Proof. P := Blow(L ⊂ ℙ^{N−1}).

H^{N−1}(P −Y)(N −1) → H^{N−1}(P −Y −E)(N −1)

⟶^{Res} H^{N−2}(E − Y ∩ E)(N − 2)

But

M(Γ) ≅ H^{N−1}(P − Y − E)(N − 1)

and H^{N−2}(E − Y ∩ E)(N − 2) ≅ M(Λ) ⊗ M(Γ//Λ) QED.

Co-associativity

Must give a rule for which Λ ⊂ Γ will occur in the comultiplication. Always include Γ, ∅. Note M(∅) = ℚ(0) = 1. Co-associativity associated to renormalization: Λ ⊂ Γ such that 2h_{1}(Λ) ≥ #Edge(Λ). Minimal subgraphs: Λ ⊂ Γ such that h_{1}(Λ) > 0 and for any Ξ ⊊ Λ have h_{1}(Ξ) < h_{1}(Λ).

The Category of Graphs?

Objects graphs, morphisms Γ → Γ//Λ when h_{1}(Λ) = 0 (subforests). Link to H^{∗}(M_{g,n}) (Kontsevich) What about Γ → Λ × Γ//Λ when h_{1}(Λ) > 0? Link to H^{∗}(\overline{M}_{g,n})?? Remark. One can show (collaboration with Kreimer and K. Yeats) that if one works with oriented graphs, the differential Γ ↦ ∑_{e∈Γ} ±Γ/e in graph cohomology gives H the structure of differential Hopf algebra.

The Period

Renormalization theory (Feynman rules): homomorphism of algebras (not co-algebras)

ϕ : H → ℂ[log q]

On primitive graphs (N = 2n, no divergent subgraphs) this is our period

∫_{σ} \frac{Ω_{2n−1}}{Ψ_{Γ}^{2}} .

Broadhurst-Kreimer: W S_{n} ⇝ (∗)ζ(2n − 3), etc.

The Hodge Structure

N = 2n. Recall H_{Betti}^{N−1}(ℙ^{N−1} − X_{Γ}) has a Hodge filtration. Question: ω_{Γ} := \frac{Ω_{2n−1}}{Ψ_{Γ}^{2}} ∈ F^{?}H_{Betti}^{N−1}(ℙ^{N−1} − X_{Γ}).

For X smooth, would get ω_{Γ} ∈ F^{2n−2}. In W S_{n} case, by Broadhurst-Kreimer period ∈ ℚ^{×} · ζ(2n − 3). Expect an extension

0 → ℚ(0) → E → ℚ(3 − 2n) → 0

Recall F^{2n−3}ℂ(3 − 2n) = ℂ, so in this case expect ω_{Γ} ∈ F^{2n−3} Proposition. ω_{Γ} ∉ F^{2n−2} for any Γ. proof. X : det(T) = 0, T n × n symmetric. Z ⊂ ℙ^{2n−1} locus where rank(T) ≤ n − 2. Then codim. Z in ℙ^{2n−1} is ≤ 3. (Think of 2 × 2 case.

(\begin{matrix}a & b \\ b & c \end{matrix}) rk. 0 def. by a = b = c = 0. )

The Hodge Structure (cont.)

Coordinates z_{1}, z_{2}, z_{3} such that ψ_{Γ} ∈ (z_{1}, z_{2}, z_{3})^{2}. Locally, if we blow up Z,

ω_{Γ} = \frac{z_{1}^{2}dz_{1} ∧ d(z_{2}/z_{1}) ∧ d(z_{3}/z_{1}) …}{z_{1}^{4}ϕ(z_{1}, z_{2}/z_{1}, z_{3}/z_{1}, …)^{2}}

Pole of order 2 along exceptional divisor. Using Deligne’s pole order description of the Hodge filtration, we can conclude ω_{Γ} ∉ F^{2n−2}. Remark. Results of Broadhurst-Kreimer suggest that sometimes ω_{Γ} ∉ F^{2n−3}. Ex. Γ bipartite (3, 4) graph (all lines joining a point {a_{1}, a_{2}, a_{3}} to a point {b_{1}, b_{2}, b_{3}, b_{4}}.) Γ has 12 edges and 6 loops. Period = (∗)ζ(3, 5). Thus expect ω_{Γ} ∈ F^{8}H^{11}(ℙ^{11} − X_{Γ}). In this case, n = 6, 8 = 2n − 4.