GAUSS-MANIN CONNECTIONS FOR ARRANGEMENTS, IV NONRESONANT EIGENVALUES

DANIEL C. COHEN^{†} AND PETER ORLIK^{‡}

Abstract. An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the Gauss-Manin connection for the moduli space of arrangements of fixed combinatorial type in the cohomology of the complement with coefficients in the local system determined by the weights. For nonresonant weights, we solve the eigenvalue problem for the endomorphisms arising in the 1-form associated to the Gauss-Manin connection.

1. Introduction

Let A = {H_{1}, …, H_{n}} be an arrangement of n ordered hyperplanes in ℂ^{ℓ}, with complement M = M(A) = ℂ^{ℓ} \ ⋃_{j=1}^{n} H_{j}. Assume that A contains ℓ linearly independent hyperplanes. A complex rank one local system on M is determined by a collection of weights λ = (λ_{1}, …, λ_{n}) ∈ ℂ^{n}. Associated to λ, we have a representation ρ : π_{1}(M) → ℂ^{∗}, given by γ_{j} ↦ exp(−2π i λ_{j}) for any meridian loop γ_{j} about the hyperplane H_{j} of A, and an associated local system L on M. For weights which are nonresonant in the sense of Schechtman, Terao, and Varchenko [13], the local system cohomology vanishes in all but one dimension, H^{q}(M; L) = 0 for q ≠ ℓ. Parallel translation of fibers over curves in the moduli space of all arrangements combinatorially equivalent to A gives rise to a Gauss-Manin connection on the vector bundle over this moduli space with fiber H^{ℓ}(M; L). This connection arises in a variety of applications, including the Aomoto-Gelfand theory of hypergeometric integrals [2, 8, 12], and the representation theory of Lie algebras and quantum groups [14, 16]. As such, it has been studied by a number of authors, including Aomoto [1], Schechtman and Varchenko [14, 16], Kaneko [10], and Kanarek [9].

Denote the combinatorial type of A by T . The moduli space of all arrangements of type T is determined by the set of dependent collections of subsets of hyperplanes in the projective closure of A in ℂℙ^{ℓ}, see [15]. Let B(T ) be a smooth, connected component of this moduli space. There is a fiber bundle p : M(T ) → B(T ) whose fibers, p^{−1}(b) = M_{b}, are complements of arrangements A_{b} of type T . Since B(T ) is connected, M_{b} is diffeomorphic to M. The fiber bundle p : M(T ) → B(T ) is locally trivial. Consequently, given a local system on the fiber, there is an associated flat vector bundle H → B(T ), with fiber H^{ℓ}(M_{b}; L_{b}) at b ∈ B(T ). For nonresonant weights, Terao [15] showed that the Gauss-Manin connection on this vector bundle has connection 1-form

2000 Mathematics Subject Classification. 32S22, 14D05, 52C35, 55N25.

Key words and phrases. hyperplane arrangement, local system, Gauss-Manin connection.

†Partially supported by National Security Agency grant MDA904-00-1-0038.

‡Partially supported by National Security Agency grant MDA904-02-1-0019.

(1.1)  ∇ = ∑ Θ_{T′} ⊗ Ω_{λ}(T′, T ),

where Θ_{T′} is a logarithmic 1-form on the closure of B(T ) with a simple pole along the divisor corresponding to the codimension one degeneration T′ of T , and Ω_{λ}(T′, T ) is an endomorphism of H^{ℓ}(M; L). For general position arrangements, this Gauss-Manin connection was found by Aomoto and Kita [2]. Terao [15] computed this connection for a larger class of arrangements. In [4], we determined the “Gauss-Manin endomorphisms” Ω_{λ}(T′, T ) for all arrangements. The aim of this paper is to solve the eigenvalue problem for these endomorphisms.

Identify the hyperplanes of A with their indices. An edge of A is a nonempty intersection of hyperplanes in A. An edge is dense if the subarrangement of hyperplanes containing it is irreducible: the hyperplanes cannot be partitioned into nonempty sets so that, after a change of coordinates, hyperplanes in different sets are in different coordinates, see [13]. For an edge X, define λ_{X} = ∑_{X⊆H_{j}} λ_{j}. Let A_{∞} = A ∪ H_{n+1} be the projective closure of A, the union of A and the hyperplane at infinity in ℂℙ^{ℓ}, see [12]. Set λ_{n+1} = − ∑_{j=1}^{n} λ_{j}. Schechtman, Terao, and Varchenko [13], refining work of Esnault, Schechtman, and Viehweg [6], found conditions on the weights which insure that the local system cohomology groups vanish except in the top dimension. They proved that if M is the complement of an arrangement A in ℂ^{ℓ} of combinatorial type T with ℓ linearly independent hyperplanes and L is a rank one local system on M whose weights λ satisfy the condition

λ_{X} ∉ ℤ_{≥0} for every dense edge X of A_{∞},

then H^{q}(M; L) = 0 for q ≠ ℓ and dim H^{ℓ}(M; L) = |χ(M)|, where χ(M) is the Euler characteristic of M. These conditions depend only on the type T , so we call weights satisfying them T -nonresonant.

Throughout this paper, we assume that A contains ℓ linearly independent hyperplanes, hence n ≥ ℓ, and that λ is T -nonresonant. We consider only codimension one degenerations of combinatorial types and refer to these as simply degenerations.

Theorem. Let T′ be a degeneration of T , and let λ be a collection of generic T - nonresonant weights for the rank one local system L. Then the Gauss-Manin endomorphism Ω_{λ}(T′, T ) is diagonalizable. The spectrum of Ω_{λ}(T′, T ) is contained in the set {0, λ_{S}}, where λ_{S} = ∑_{j∈S} λ_{j} for some S ⊂ {1, …, n + 1}.

The set S is part of a pair (S, r), called the principal dependence of the degeneration T′ of T , see Theorem 3.2. It follows from our results in Sections 2, 4, and 5 that weights λ which satisfy λ_{S} ≠ 0 are sufficiently generic. Our results also yield an algorithm for determining the multiplicities of the eigenvalues, see Remark 5.3.

Let G denote the combinatorial type of a general position arrangement of n hyperplanes in ℂ^{ℓ}. The cohomology of the complement of an arrangement of type G is the rank ℓ truncation, A^{•}(G), of the exterior algebra on n generators e_{j}, j ∈ [n], where [n] = {1, …, n}, corresponding to the hyperplanes. The Orlik-Solomon algebra A^{•}(A) ≃ H^{•}(M(A); ℂ) is generated by one dimensional classes a_{j}, j ∈ [n]. It is the quotient of A^{•}(G) by a homogeneous ideal, I^{•}(A), hence it is a finite dimensional graded ℂ-algebra [11]. It is known that A^{•}(A) depends only on the combinatorial type T of A so we may write A^{•}(T ).

Weights λ yield an element a_{λ} = ∑_{j=1}^{n} λ_{j}a_{j} in A^{1}(T ), and multiplication by a_{λ} gives A^{•}(T ) the structure of a cochain complex. The resulting cohomology H^{•}(T ) = H^{•}(A^{•}(T ), a_{λ}) is a combinatorial analog of H^{•}(M(A); L). If the weights are T - nonresonant, then H^{•}(M(A); L) ≃ H^{•}(A^{•}(T ), a_{λ}) and the only (possibly) nonzero group H^{ℓ}(T ) has the βnbc basis of Falk and Terao [7]. This basis provides an explicit surjection τ : H^{ℓ}(G) → H^{ℓ}(T ). Our results in [4] yield a commutative diagram of endomorphisms for each degeneration T′ of T :

(1.2)  \begin{matrix}H^{ℓ}(G) & ⟶^{τ} & H^{ℓ}(T ) \\ ↓^{Ω̃_{λ}(T′,T )} & ↓^{Ω_{λ}(T′,T )} \\ H^{ℓ}(G) & ⟶^{τ} & H^{ℓ}(T ) \end{matrix}

The endomorphism Ω̃_{λ}(T′, T ) of H^{ℓ}(G) is induced by an endomorphism ω_{λ}^{•}(T′, T ) of A^{ℓ}(G), see [5], (3.3), and Theorem 3.1.

Here is a brief outline of the paper. In Section 2, we recall the Aomoto complex and the “formal Gauss-Manin connection matrices” of [5] which are essential in our arguments. We recall the moduli space of combinatorially equivalent arrangements in Section 3 and identify the principal dependence (S, r) of the degeneration T′ of T . Using the principal dependence, we construct a realizable type T (S, r) and an endomorphism Ω_{λ}(S, r) of H^{ℓ}(G). In Section 4, we determine the eigenstructure of the endomorphism Ω_{λ}(S, r). In Section 5, we show that Ω̃_{λ}(T′, T ) may be replaced by Ω_{λ}(S, r) in (1.2) and thereby determine the eigenstructure of Ω_{λ}(T′, T ). We conclude with several examples to illustrate the main result.

2. General position

In this section, we record a number of constructions in the Orlik-Solomon complex of a general position arrangement which will be used subsequently.

Let G = G_{n}^{ℓ} be the combinatorial type of a general position arrangement of n hyperplanes in ℂ^{ℓ}, where n ≥ ℓ. The Orlik-Solomon algebra A^{•}(G) is the rank ℓ truncation of an exterior algebra on n generators. Let T = {i_{1}, …, i_{q}} ⊂ [n]. If order matters, we call T a q-tuple and write T = (i_{1}, …, i_{q}) and e_{T} = e_{i_{1}} ⋯ e_{i_{q}}. The algebra A^{•}(G) is generated (as an algebra) by {e_{j} | 1 ≤ j ≤ n}, and has (additive) basis {e_{T} }, where e_{T} = 1 if T = ∅, and T ≠ ∅ is an increasingly ordered tuple of cardinality at most ℓ.

Define a map ∂ : A^{q}(G) → A^{q−1}(G) by ∂(e_{T}) = ∑_{k=1}^{q}(−1)^{k−1}e_{T_{k}}, where T_{k} = (i_{1}, …, î_{k}, …, i_{q}) if T = (i_{1}, …, i_{q}). Then ∂ ∘ ∂ = 0, providing A^{•}(G) with the structure of a chain complex

(2.1)  (A^{•}(G), ∂) : A^{0}(G) ⟵^{∂} A^{1}(G) ⟵ ⋯ ⟵ A^{ℓ−1}(G) ⟵^{∂} A^{ℓ}(G)

It is well known that the homology of this complex is concentrated in the top dimension, H_{q}(A(G), ∂) = 0 for q ≠ ℓ. The dimension of the unique nontrivial homology group is β(n, ℓ) = dim H_{ℓ}(A^{•}(G), ∂) = ∑_{k=0}^{ℓ}(−1)^{k}\binom{n}{k} = \binom{n−1}{ℓ}.

Weights λ = (λ_{1}, …, λ_{n}) ∈ ℂ^{n} determine an element e_{λ} = ∑_{j=1}^{n} λ_{j}e_{j} in A^{1}(G). Since A^{•}(G) is a quotient of an exterior algebra, we have e_{λ}e_{λ} = 0. Consequently, multiplication by e_{λ} defines a cochain complex

(2.2)  (A^{•}(G), e_{λ}) : A^{0}(G) ⟶^{e_{λ}} A^{1}(G) ⟶ ⋯ ⟶ A^{ℓ−1}(G) ⟶^{e_{λ}} A^{ℓ}(G)

If λ ≠ 0, it is well known that the cohomology of this complex is concentrated in the top dimension, H^{q}(A^{•}(G), e_{λ}) = 0 for q ≠ ℓ, and that dim H^{ℓ}(A^{•}(G), e_{λ}) = β(n, ℓ).

The endomorphism ω_{λ}(T′, T ) of A^{ℓ}(G) which induces the map Ω̃_{λ}(T′, T ): H^{ℓ}(G) → H^{ℓ}(G) of (1.2) is the specialization at λ of a “formal Gauss-Manin connection endomorphism” given in [5] and in (3.3). The latter is a linear combination of endomorphisms ω_{S}^{•} of the Aomoto complex (A_{R}^{•}(G), e_{y}) of G, a universal complex for the cohomology H^{•}(A^{•}(G), e_{λ}). The Aomoto complex has terms A_{R}^{q}(G) = A^{q}(G) ⊗ R, where R = ℂ[y_{1}, …, y_{n}] is the polynomial ring, and the boundary map is given by multiplication by e_{y} = ∑_{j=1}^{n} y_{j}e_{j}.

The endomorphisms ω_{S}^{•} correspond to subsets S of [n + 1], the index set of the projective closure of the general position arrangement in ℂℙ^{ℓ}, G_{∞}. The symmetric group Σ_{n+1} on n + 1 letters acts on A^{•}(G) by permuting the hyperplanes of G_{∞}, and on R by permuting the variables y_{j}, where y_{n+1} = − ∑_{j=1}^{n} y_{j}. In the basis {e_{j} | 1 ≤ j ≤ n} for the Orlik-Solomon algebra, the action of σ ∈ Σ_{n+1} is given by σ(e_{i}) = e_{σ(i)} if σ(n + 1) = n + 1, and by

σ(e_{i}) = \begin{cases}−e_{σ(n+1)} & if σ(i) = n + 1, \\ e_{σ(i)} − e_{σ(n+1)} & if σ(i) ≠ n + 1, \end{cases}

if σ(n + 1) ≠ n + 1. Denote the induced action on the Aomoto complex by ϕ_{σ} : A_{R}^{•}(G) → A_{R}^{•}(G),

ϕ_{σ}(e_{i_{1}} ⋯ e_{i_{p}} ⊗ f(y_{1}, …, y_{n})) = σ(e_{i_{1}}) ⋯ σ(e_{i_{p}}) ⊗ f(y_{σ(1)}, …, y_{σ(n)}).

Lemma 2.1. For each σ ∈ Σ_{n+1}, the map ϕ_{σ} is a cochain automorphism of the Aomoto complex (A_{R}^{•}(G), e_{y}).

If T = (i_{1}, …, i_{p}) ⊂ [n] is a p-tuple, then (j, T) = (j, i_{1}, …, i_{p}) is the (p + 1)-tuple which adds j with 1 ≤ j ≤ n to T as its first entry. For S = {s_{1}, …, s_{k}} ⊂ [n + 1], let σ_{S} denote the permutation ( \begin{matrix}1 & 2 & ··· & k \\ s_{1} & s_{2} & ··· & s_{k} \end{matrix} ). Write S ≡ T if S and T are equal sets.

Definition 2.2. Let T be a p-tuple, S a q + 1 element subset of [n + 1], and j ∈ [n]. If S = S_{0} = [q + 1], define the endomorphism ω_{S_{0}}^{•} : (A_{R}^{•}(G), e_{y}) → (A_{R}^{•}(G), e_{y}) by

ω_{S_{0}}^{p}(e_{T}) = \begin{cases}y_{j}∂e_{(j,T)} & if p = q and S_{0} ≡ (j, T), \\ e_{y}∂e_{T} & if p = q + 1 and S_{0} ≡ T, \\ 0 & otherwise. \end{cases}

If S ≠ S_{0}, define ω_{S}^{•} = ϕ_{σ_{S}} ∘ ω_{S_{0}}^{•} ∘ ϕ_{σ_{S}}^{−1}.

One can check that this agrees with the case by case definition in [5, Def. 4.1].

Proposition 2.3 ([5, Prop. 4.2]). For every subset S of [n + 1], the map ω_{S}^{•} is a cochain homomorphism of the Aomoto complex (A_{R}^{•}(G), e_{y})).

For S_{0} = [s], 1 ≤ q ≤ ℓ, and 1 ≤ r ≤ min(q, s − 1), consider the sets V_{S_{0}}^{q,r} and W_{S_{0}}^{q,r} of elements in A_{R}^{q}(G) given by

\begin{aligned}V_{S_{0}}^{q,r} & = \{e_{J}e_{K} | |J| ≤ r − 1\} ⋃\{η_{S_{0}}e_{J}e_{K} | |J| = r − 1\} and \\ W_{S_{0}}^{q,r} & = \{e_{S_{0}}e_{K} (if q ≥ s)\} ⋃\{(∂e_{J})e_{K} | |J| ≥ r + 1\} ⋃\{η_{S_{0}}e_{J}e_{K} | |J| ≥ r\}, \end{aligned}

where J ⊂ S_{0}, K ⊂ [n] \ S_{0}, and η_{S} = ∑_{i∈S} y_{i}e_{i}. Let B_{S_{0}}^{q,r} = V_{S_{0}}^{q,r} ⋃ W_{S_{0}}^{q,r}. If S ⊂ [n + 1] and S ≠ S_{0}, define B_{S}^{q,r} = {ϕ_{σ_{S}}(v) | v ∈ B_{S_{0}}^{q,r}}. Define V_{S}^{q,r} and W_{S}^{q,r} analogously. Given weights λ = (λ_{1}, …, λ_{n}), let B_{S}^{q,r}(λ) = {v|_{y_{i}↦λ_{i}} | v ∈ B_{S}^{q,r}} denote the specialization of B_{S}^{q,r} at λ, a sets of vectors in A^{q}(G). Define V_{S}^{q,r}(λ) and W_{S}^{q,r}(λ) analogously. We will abuse notation and write η_{S} = ∑_{i∈S} λ_{i}e_{i} when working in the Orlik-Solomon algebra. Note that ∂η_{S} = λ_{S} = ∑_{i∈S} λ_{i}.

Lemma 2.4. If λ_{S} ≠ 0, the set of vectors B_{S}^{q,r}(λ) spans the vector space A^{q}(G).

Proof. It suffices to consider the case S = S_{0}.

First, we show that the set {∂e_{J} | |J| = r + 1} ⋃{η_{S}e_{J} | |J| = r − 1} spans A^{r}(G_{s}^{s}), where G_{s}^{s} is a general position arrangement of s hyperplanes (indexed by S) in ℂ^{s}. For this arrangement, both the chain complex (A^{•}(G_{s}^{s}), ∂) of (2.1) and the cochain complex (A^{•}(G_{s}^{s}), e_{λ}) = (A^{•}(G_{s}^{s}), η_{S}) of (2.2) are acyclic, and

dim im[∂ : A^{r+1}(G_{s}^{s}) → A^{r}(G_{s}^{s})] = β(s, r) = \binom{s − 1}{r},
dim im[η_{S} : A^{r−1}(G_{s}^{s}) → A^{r}(G_{s}^{s})] = \binom{s}{r} − β(s, r) = \binom{s − 1}{r − 1}.

Note that dim A^{r}(G_{s}^{s}) = \binom{s}{r} = \binom{s−1}{r} + \binom{s−1}{r−1}.

Suppose x ∈ span{∂e_{J} | |J| = r + 1} ∩ span{η_{S}e_{J} | |J| = r − 1}. Then ∂x = 0, and x = η_{S}y for some y ∈ A^{r−1}(G_{s}^{s}). So ∂x = ∂(η_{S}y) = λ_{S}y − η_{S}∂y = 0. Since λ_{S} ≠ 0, we can write y = cη_{S}∂y, where c = 1/λ_{S}. But this implies that x = η_{S}y = 0. Consequently, {∂e_{J} | |J| = r + 1} ⋃{η_{S}e_{J} | |J| = r − 1} spans A^{r}(G_{s}^{s}).

Using this, a straightforward exercise shows that the set of vectors B_{S}^{q,r}(λ) spans the vector space A^{q}(G) = A^{q}(G_{n}^{ℓ}). □

3. Principal dependence

Let T be the combinatorial type of the arrangement A of n hyperplanes in ℂ^{ℓ} with n ≥ ℓ ≥ 1. We consider the family of all arrangements of type T . Recall that A is ordered by the subscripts of its hyperplanes and we assume that A, and hence every arrangement of type T , contains ℓ linearly independent hyperplanes.

Choose coordinates u = (u_{1}, …, u_{ℓ}) on ℂ^{ℓ}. The hyperplanes of an arrangement of type T are defined by linear polynomials α_{i} = b_{i,0} + ∑_{j=1}^{ℓ} b_{i,j}u_{j} (i = 1, …, n). We embed the arrangement in projective space and add the hyperplane at infinity as last in the ordering, H_{n+1}. The moduli space of all arrangements of type T may be viewed as the set of matrices

(3.1)  b = (\begin{matrix}b_{1,0} & b_{1,1} & ⋯ & b_{1,ℓ} \\ b_{2,0} & b_{2,1} & ⋯ & b_{2,ℓ} \\ ⋮ & ⋮ & ... & ⋮ \\ b_{n,0} & b_{n,1} & ⋯ & b_{n,ℓ} \\ 1 & 0 & ⋯ & 0 \end{matrix})

whose rows are elements of ℂℙ^{ℓ}, and whose (ℓ + 1) × (ℓ + 1) minors satisfy certain dependency conditions, see [12, Prop. 9.2.2].

Given S ⊂ [n+1], let N_{S}(T ) = N_{S}(b) denote the submatrix of (3.1) with rows specified by S. Let rank N_{S}(T ) be the size of the largest minor with nonzero determinant. Define the multiplicity of S in T by

(3.2)  m_{S}(T ) = |S| − rank N_{S}(T ).

Call S dependent (in type T ) if m_{S}(T ) > 0. For such S, the linear polynomials {α_{j} | j ∈ S} are dependent. For q ≤ n + 1, let Dep(T )_{q} denote the dependent sets of cardinality q, and let Dep(T ) = ⋃_{q} Dep(T )_{q}. If T′ is a combinatorial type for which Dep(T ) ⊂ Dep(T′), let Dep(T′, T ) = Dep(T′) \ Dep(T ). Terao [15] showed that the combinatorial type T is determined by Dep(T )_{ℓ+1}, but dependent sets of both smaller and larger cardinality arise in our considerations, see Example 3.4.

Let

Dep(T )_{q}^{∗} = {S ∈ Dep(T )_{q} | ⋂_{j∈S} H_{j} ≠ ∅}

and let Dep(T )^{∗} = ⋃_{q} Dep(T )_{q}^{∗}. If S ∈ Dep(T )^{∗}, then codim(⋂_{j∈S} H_{j}) < |S|. If T′ is a combinatorial type for which Dep(T )^{∗} ⊂ Dep(T′)^{∗}, let Dep(T′, T )^{∗} = Dep(T′)^{∗} \ Dep(T )^{∗}. If |S| ≥ ℓ + 2, then S ∈ Dep(T ) but S ∈ Dep(T )^{∗} if and only if every subset of S of cardinality ℓ + 1 is dependent. It is convenient to work with these smaller collections of dependent sets.

Define endomorphisms of A_{R}^{•}(G) by

(3.3)  ω^{•}(T ) = ∑_{S∈Dep(T )} m_{S}(T ) · ω_{S}^{•} and ω^{•}(T′, T ) = ∑_{S∈Dep(T′,T )} m_{S}(T′) · ω_{S}^{•}.

These are cochain homomorphisms of the Aomoto complex by Proposition 2.3. Since Dep(T )_{q}^{∗} = Dep(T )_{q} for q ≤ ℓ + 1, we have

(3.4) ω^{•}(T ) = ∑_{S∈Dep(T )^{∗}} m_{S}(T ) · ω_{S}^{•} and ω^{•}(T′, T ) = ∑_{S∈Dep(T′,T )^{∗}} m_{S}(T′) · ω_{S}^{•}.

Theorem 3.1 ([5]). The endomorphism Ω̃_{λ}(T′, T ) is induced by the specialization ω_{λ}(T′, T ) := ω^{ℓ}(T′, T )|_{y_{j}↦λ_{j}} of the endomorphism ω^{ℓ}(T′, T ).

Denote the cardinality of S by s = |S|. For 1 ≤ r ≤ min(ℓ, s − 1), consider the combinatorial type T (S, r) defined by

T ∈ Dep(T (S, r))^{∗} ⟺ |T ∩ S| ≥ r + 1.

This type is realized by a pencil of hyperplanes indexed by S with a common subspace of codimension r, together with n − s hyperplanes in general position. Note that for r = 1 the hyperplanes in S coincide, so T (S, r) is a multi-arrangement.

Theorem 3.2. Let T′ be a degeneration of a realizable combinatorial type T . For each set S_{i} ∈ Dep(T′, T )^{∗}, let r_{i} be minimal so that Dep(T (S_{i}, r_{i}))^{∗} ⊂ Dep(T′)^{∗}. Given the collection {(S_{i}, r_{i})}, there is a unique pair (S, r) with r = min{r_{i}}, S_{i} ⊂ S for every pair (S_{i}, r_{i}) where r_{i} = r, and Dep(T (S, r))^{∗} ⊂ Dep(T′)^{∗}.

Proof. Terao [15] classified the three codimension one degeneration types in the moduli space of an arrangement whose only dependent set is the minimally dependent set T of size q + 1.

I: |S ∩ T | ≤ q − 1 for all S ∈ Dep(T′, T )^{∗}; II: {(m, T_{k}) | m ∉ T } for each fixed k, 1 ≤ k ≤ |T |; III: {(m, T_{k}) | 1 ≤ k ≤ |T |} for each fixed m ∉ T.

If q = 1, then Type II does not appear. Recall that T_{k} = (i_{1}, …, î_{k}, …, i_{q+1}) if T = (i_{1}, …, i_{q+1}), and note that m ∈ [n + 1] in cases II and III above.

It follows from our analysis of the corresponding types in general [5] that if a Type II degeneration is present, then the value of r decreases and there is a unique set of maximal cardinality with minimal r. In the other types, r remains constant, but a unique dependent set of T increases in T′. □

Definition 3.3. Let T′ be a degeneration of T . We call the pair (S, r) which satisfies the conditions of Theorem 3.2 the principal dependence of the degeneration.

Example 3.4. Let T be the combinatorial type of the arrangement A of 4 lines in ℂ^{2} depicted in Figure 1. Here Dep(T )^{∗} = {123}.

Figure 1. A line arrangement and three degenerations

[figure: 3; 4; 1; 2; A; 3; 4; 1; 2; A_{1}; 3; 4; 1; 2; A_{3}; 3; 4; 12; A_{2}]

The combinatorial types T_{i} of the (multi)-arrangements A_{i} shown in Figure 1 are degenerations of T . For these degenerations, the collections {(S_{i}, r_{i})} and corresponding principal dependencies (S, r) are given in the table below.

{(S_{i}, r_{i})}  (S, r)
T_{1} (345, 2)  (345, 2)
T_{2} (12, 1), (124, 2), (125, 2)  (12, 1)
T_{3} (124, 2), (134, 2), (234, 2), (1234, 2) (1234, 2)

For the combinatorial type T (S, r), write ω^{•}(S, r) = ω^{•}(T (S, r)), see (3.4). In Theorem 5.1 below, we show that the Gauss-Manin endomorphism Ω_{λ}(T′, T ) of (1.1) is induced by the specialization of ω^{ℓ}(S, r) at T -nonresonant weights λ, ω_{λ}^{ℓ}(S, r). First, we solve the eigenvalue problem for the latter endomorphism.

4. Diagonalization

The purpose of this section is to solve the eigenvalue problem for ω_{λ}^{q}(S, r), the endomorphism of the Orlik-Solomon algebra obtained by specializing ω^{q}(S, r) at generic weights λ = (λ_{1}, …, λ_{n}). This allows calculation of the eigenstructure of the induced endomorphism in cohomology, Ω_{λ}(S, r), which is related to the Gauss-Manin endomorphism in Theorem 5.1. First, we establish several technical results concerning the endomorphism ω^{q}(S, r) of the Aomoto complex itself. Recall that these endomorphisms are given explicitly by

ω^{•}(S, r) = ∑_{K∈Dep(T (S,r))^{∗}} m_{K}(S, r) · ω_{K}^{•},

where m_{K}(S, r) is the multiplicity of K in type T (S, r), see (3.2), and ω_{K}^{•} is given in Definition 2.2. It follows from Proposition 2.3 that ω^{•}(S, r) is a chain map. Note that ω^{q}(S, r) = 0 for q < r.

Given (S, r), define

Ψ_{S,r}^{q} = ∑_{{T ⊂S \atop |T |=r+1}} ω^{q}(T, r).

Note that Ψ_{S,r}^{r} = ω^{r}(S, r). For q ≥ r, the endomorphisms ω^{q}(S, r) satisfy the following recursion.

Lemma 4.1. For q ≥ r, we have

Ψ_{S,r}^{q} = ∑_{k=0}^{s−r−1} \binom{r + k − 1}{k}ω^{q}(S, r + k).

Proof. If T ⊂ [n] satisfies |T | = r + 1, then Dep(T (T, r))^{∗} = {K | K ⊇ T }, and it is readily checked that m_{K}(T, r) = 1 for each such K. Hence, ω^{q}(T, r) = ∑_{K⊇T} ω_{K}^{q}, and we have

Ψ_{S,r}^{q} = ∑_{{T ⊂S \atop |T |=r+1}} ∑_{K⊇T} ω_{K}^{q} = ∑_{|K∩S|≥r+1} ω_{K}^{q}.

If |K ∩ S| = r + p, then ω_{K}^{q} occurs \binom{r+p}{r+1} times in this sum, so

Ψ_{S,r}^{q} = ∑_{|K∩S|≥r+1} ω_{K}^{q} = ∑_{p≥1} ∑_{|K∩S|=r+p} \binom{r + p}{r + 1}ω_{K}^{q}.

If K ∈ Dep(T (S, j))^{∗}, then |K ∩ S| ≥ j + 1, and m_{K}(S, j) = |K ∩ S| − j. It follows that ω^{q}(S, j) = ∑_{|K∩S|≥j+1}(|K ∩ S| − j)ω_{K}^{q}. Hence,

∑_{k=0}^{s−r−1} \binom{r + k − 1}{k}ω^{q}(S, r + k) = ∑_{k=0}^{s−r−1} ∑_{i≥k+1} ∑_{|K∩S|=r+i} \binom{r + k − 1}{k}(i − k)ω_{K}^{q}

Rewriting this last sum, we obtain

∑_{k=0}^{s−r−1} \binom{r + k − 1}{k}ω^{q}(S, r + k) = ∑_{p≥1} ∑_{|K∩S|=r+p} ∑_{j=0}^{p−1} \binom{r + j − 1}{j}(p − j)ω_{K}^{q}.

A straightforward inductive argument shows that ∑_{j=0}^{p−1} \binom{r+j−1}{j}(p−j) = \binom{r+p}{r+1}, which completes the proof. □

Given S, recall that η_{S} = ∑_{i∈S} y_{i}e_{i} and y_{S} = ∑_{i∈S} y_{i} = ∂η_{S}.

Lemma 4.2. Let J ⊂ S and L ⊂ [n] \ S. Then

Ψ_{S,r}^{q}(e_{J}e_{K}) = \begin{cases}0 & if |J| ≤ r − 1, \\ \binom{r+p}{r}y_{S}e_{J}e_{K} − \binom{r+p−1}{r−1}η_{S}(∂e_{J})e_{K} & if |J| = r + p, where p ≥ 0. \end{cases}

Proof. Given (J, L), it follows from Definition 2.2 that ω_{K}^{q}(e_{J}e_{L}) ≠ 0 only for the following K:

(4.1)  \begin{matrix}(J, L), & (J_{k}, L, n + 1), & (J, L_{k}, n + 1), \\ (i, J, L), & (J, L, n + 1), & (i, J_{k}, L, n + 1), & (i, J, L_{k}, n + 1), \end{matrix}

where i ∉ (J, L).

If |J| ≤ r − 1, then |K ∩ S| ≤ r for each of the above K, so T ⊄ K for all T ⊂ S with |T | = r + 1. It follows that ω^{q}(T, r)(e_{J}e_{L}) = 0 for each such T. Consequently, Ψ_{S,r}^{q}(e_{J}e_{L}) = 0.

Let T ⊂ S be a subset of cardinality r + 1, and note that Ψ_{T,r}^{q} = ∑_{K⊃T} ω_{K}^{q}, so Ψ_{S,r}^{q} = ∑_{T ⊂S} Ψ_{T,r}^{q}, where the sum is over all T ⊂ S with |T | = r + 1. Given such a T, if |T ∩ J| ≤ r − 1, then none of the sets K recorded in (4.1) contains T. It follows that Ψ_{T,r}^{q}(e_{J}e_{L}) = 0 if |T ∩ J| ≤ r − 1.

Suppose |J| = r. If |J ∩ T | = r, then T ≡ (i, J) for some i ∈ S \ J, and

Ψ_{T,r}^{q}(e_{J}e_{L}) = ω_{(i,J,L)}(e_{J}e_{L}) + ∑_{k=1}^{q−r} ω_{(i,J,L_{k},n+1)}(e_{J}e_{L})

= y_{i}∂(e_{i}e_{J}e_{L}) + ∑_{k=1}^{q−r}(−1)^{r+k}y_{i}e_{i}e_{J}e_{L_{k}}

= y_{i}e_{J}e_{L} − y_{i}e_{i}∂(e_{J}e_{L}) + (−1)^{r}y_{i}e_{i}e_{J}∂e_{L} = y_{i}e_{J}e_{L} − y_{i}e_{i}(∂e_{J})e_{L}.

Therefore, using the identity y_{J}e_{J} = η_{J}∂e_{J}, we have

\begin{aligned}Ψ_{S,r}^{q}(e_{J}e_{L}) & = ∑_{T ⊂S} Ψ_{T,r}^{q}(e_{J}e_{L}) = ∑_{i∈S\J}(y_{i}e_{J}e_{L} − y_{i}e_{i}(∂e_{J})e_{L}) \\ & = (y_{S} − y_{J})e_{J}e_{L} − (η_{S} − η_{J})(∂e_{J})e_{L} = y_{S}e_{J}e_{L} − η_{S}(∂e_{J})e_{L}. \end{aligned}

Now, assume that |J| = r + p for some p ≥ 1. As above, we have Ψ_{T,r}^{q}(e_{J}e_{L}) = 0 if |T ∩ J| ≠ r, r + 1. If |T ∩ J| = r + 1, then T ⊆ J and all of the sets K of (4.1) contain T. In this instance, Ψ_{T,r}^{q}(e_{J}e_{L}) = ψ(e_{J}e_{L}), where

ψ = ω_{(J,L)} + ω_{(J,L,n+1)} + ∑_{k=1}^{q}(ω_{((J,L)_{k},n+1)} + ∑_{i∉(J,L)} ω_{(i,(J,L)_{k},n+1)})

Writing J ≡ (T, J′), a calculation reveals that Ψ_{T,r}^{q}(e_{J}e_{L}) = ψ(e_{J}e_{L}) = y_{T} e_{J}e_{L}.

If |T ∩ J| = r, then T \ T ∩ J = {t} for some t ∈ S \ J. For such T, of the sets K from (4.1), only (t, J, L), (t, J_{k}, L, n + 1) for j_{k} ∉ T, and (t, J, L_{k}, n + 1) contain T. This observation, and a calculation, yields

Ψ_{T,r}^{q}(e_{J}e_{L}) = (ω_{(t,J,L)} + ∑_{j_{k}∉T} ω_{(t,J_{k},L,n+1)} + ∑_{k=1}^{q−r−p} ω_{(t,J,L_{k},n+1)})(e_{J}e_{L})

= (ω_{(t,J,L)} + ∑_{k=1}^{r+p} ω_{(t,(J,L)_{k},n+1)} − ∑_{j_{k}∈T} ω_{(t,J_{k},L,n+1)})(e_{j}e_{L})

= y_{t}e_{J}e_{L} − ∑_{j_{k}∈T}(−1)^{k−1}y_{t}e_{t}e_{J_{k}}e_{L}.

Summing over all T ⊂ S with |T ∩ J| = r, we obtain

∑_{|T ∩J|=r}Ψ_{T,r}^{q}(e_{J}e_{L}) = ∑_{t∈S\J} ∑_{{A⊂[r+p] \atop |A|=r}}(y_{t}e_{J}e_{L} − ∑_{i=1}^{r}(−1)^{a_{i}−1}y_{t}e_{t}e_{J_{a_{i}}}e_{L})

= \binom{r + p}{r}(y_{S} − y_{J})e_{J}e_{L} − ∑_{t∈S\J} ∑_{k=1}^{r+p}(−1)^{k−1}\binom{r + p − 1}{r − 1}y_{t}e_{t}e_{J_{k}}e_{L}

= \binom{r + p}{r}(y_{S} − y_{J})e_{J}e_{L} − ∑_{t∈S\J} \binom{r + p − 1}{r − 1}y_{t}e_{t}(∂e_{J})e_{L}

= \binom{r + p}{r}y_{S}e_{J}e_{L} − \binom{r + p − 1}{r}y_{J}e_{J}e_{L} − \binom{r + p − 1}{r − 1}η_{S}(∂e_{J})e_{L}.

Recall that Ψ_{T,r}^{q}(e_{J}e_{L}) = y_{T} e_{J}e_{L} for T ⊂ J. Summing over all T ⊂ J, we obtain ∑_{T ⊂J} Ψ_{T,r}^{q}(e_{J}e_{L}) = \binom{r+p−1}{r}y_{J}e_{J}e_{L}. Therefore,

\begin{aligned}Ψ_{S,r}^{q}(e_{J}e_{L}) & = (∑_{T ⊂S} Ψ_{T,r}^{q})(e_{J}e_{L}) = ( ∑_{|T |=r} Ψ_{T,r}^{q} + ∑_{|T |=r+1} Ψ_{T,r}^{q})(e_{J}e_{L}) \\ & = \binom{r + p}{r}y_{S}e_{J}e_{L} − \binom{r + p − 1}{r − 1}η_{S}(∂e_{J})e_{L} \end{aligned}

if |J| = r + p. □

Let λ = (λ_{1}, …, λ_{n}) be a collection of weights, and consider the endomorphism ω_{λ}^{q}(S, r) : A^{q}(G) → A^{q}(G) of the Orlik-Solomon algebra obtained by specializing ω^{q}(S, r) at λ. Given S, we abuse notation and write η_{S} = ∑_{i∈S} λ_{i}e_{i}. Recall the spanning set B_{S}^{q,r}(λ) = V_{S}^{q,r}(λ) ⋃ W_{S}^{q,r}(λ) of A^{q}(G) from Lemma 2.4.

Theorem 4.3. Let λ be a collection of weights satisfying λ_{S} ≠ 0. Then the specialization, ω_{λ}^{q}(S, r), of ω^{q}(S, r) at λ is diagonalizable, with eigenvalues 0 and λ_{S}.

1. The 0-eigenspace is spanned by the set of vectors V_{S}^{q,r}(λ) and has dimension

∑_{p=0}^{r} \binom{s}{p}\binom{n − s}{q − p} − \binom{s − 1}{r}\binom{n − s}{q − r}.

2. The λ_{S}-eigenspace is spanned by the set of vectors W_{S}^{q,r}(λ) and has dimension

∑_{p=r+1}^{min(q,s)} \binom{s}{p}\binom{n − s}{q − p} + \binom{s − 1}{r}\binom{n − s}{q − r}.

Proof. By Lemma 2.4, the set of vectors B_{S}^{q,r}(λ) = V_{S}^{q,r}(λ) ⋃ W_{S}^{q,r}(λ) spans the vector space A^{q}(G_{n}^{ℓ}). So to establish this result, it suffices to show that these vectors are eigenvectors of the endomorphism ω_{λ}^{q}(S, r), and that the dimensions of the eigenspaces are as asserted. We will prove this by induction on q − r.

For ease of notation, we will suppress dependence on λ in the proof, and, for instance, write simply ω^{q}(S, r) = ω_{λ}^{q}(S, r) and Ψ_{S,r}^{q} = Ψ_{S,r}^{q}|_{y_{j}↦λ_{j}}. Using Lemma 2.1, it suffices to consider the case S ⊂ [n]. Let J ⊂ S, K ⊂ [n] \ S, and recall that

\begin{aligned}V_{S}^{q,r} & = \{e_{J}e_{K} | |J| ≤ r − 1\} ⋃\{η_{S}e_{J}e_{K} | |J| = r − 1\} and \\ W_{S}^{q,r} & = \{e_{S}e_{K} (if q ≥ s)\} ⋃\{(∂e_{J})e_{K} | |J| ≥ r + 1\} ⋃\{η_{S}e_{J}e_{K} | |J| ≥ r\}. \end{aligned}

In the case q − r = 0, we have V_{S}^{r,r} = {e_{J}e_{K} | |J| ≤ r − 1} ⋃{η_{S}e_{J} | |J| = r − 1}, W_{S}^{r,r} = {∂e_{J} | |J| = r + 1}, and ω^{r}(S, r) = Ψ_{S,r}^{r}. By Lemma 4.2, if |J| ≤ r − 1, then Ψ_{S,r}^{r}(e_{J}e_{K}) = 0. If |J| = r − 1, then, using Lemma 4.2 again, we have

Ψ_{S,r}^{r}(η_{S}e_{J}) = ∑_{i∈S} λ_{i}Ψ_{S,r}^{r}(e_{i}e_{J}) = ∑_{i∈S} λ_{i}(λ_{S}e_{i}e_{J} − η_{S}∂(e_{i}e_{J}))

= λ_{S}η_{S}e_{J} − ∑_{i∈S} λ_{i}η_{S}(e_{J} − e_{i}∂e_{J})

= λ_{S}η_{S}e_{J} − λ_{S}η_{S}e_{J} + η_{S}η_{S}∂e_{J} = 0.

Thus, every element of E^{r}(0) = span V_{S}^{r,r} is a 0-eigenvector of ω^{r}(S, r). A straightforward exercise reveals that dim E^{r}(0) = ∑_{k=0}^{r} \binom{s}{k}\binom{n−s}{r−k}−\binom{s−1}{r}. If |J| = r +1, then, using Lemma 4.2 again,

\begin{aligned}Ψ_{S,r}^{r}(∂e_{J}) & = ∑_{k=1}^{r+1}(−1)^{k−1}Ψ_{S,r}^{r}(e_{J_{k}}) = ∑_{k=1}^{r+1}(−1)^{k−1}(λ_{S}e_{J_{k}} − η_{S}∂e_{J_{k}}) \\ & = λ_{S}∂e_{J} − η_{S}∂^{2}e_{J} = λ_{S}∂e_{J} \end{aligned}

Thus, every element of E^{r}(λ_{S}) = span W_{S}^{r,r} is a λ_{S}-eigenvector of ω^{r}(S, r). Note that dim E^{r}(λ_{S}) = \binom{s−1}{r}. Since dim E^{r}(0) + dim E^{r}(λ_{S}) = dim A^{r}(G_{n}^{ℓ}), the above calculations establish Theorem 4.3 in the case q − r = 0.

If q − r ≥ 1, then by induction, for each k ≥ 1, ω^{q}(S, r + k) is diagonalizable, with eigenvalues 0 and λ_{S}, and corresponding eigenspaces E^{r+k}(0) = span V_{S}^{r+k,r} and E^{r+k}(λ_{S}) = span W_{S}^{r+k,r}. In the determination of the eigenstructure of ω^{q}(S, r), we will use the recursion provided by Lemma 4.1 in the following form:

(4.2)  ω^{q}(S, r) = Ψ_{S,r}^{q} − ∑_{k=1}^{s−r−1} \binom{r + k − 1}{k}ω^{q}(S, r + k).

First, consider the 0-eigenspace of the endomorphism ω^{q}(S, r). If |J| ≤ r − 1, then by (4.2), Lemma 4.2, and induction, we have

ω^{q}(S, r)(e_{J}e_{K}) = Ψ_{S,r}^{q}(e_{J}e_{K}) − ∑_{k=1}^{s−r−1} \binom{r + k − 1}{k}ω^{q}(S, r + k)(e_{J}e_{K}) = 0.

If |J| = r − 1, then ω^{q}(S, r + k)(η_{S}e_{J}e_{K}) = 0 for k ≥ 1 by Lemma 4.2. Using (4.2) and Lemma 4.2, we have

ω^{q}(S, r)(η_{S}e_{J}e_{K}) = Ψ_{S,r}^{q}(η_{S}e_{J}e_{K}) − ∑_{k=1}^{s−r−1} \binom{r + k − 1}{k}ω^{q}(S, r + k)(η_{S}e_{J}e_{K})

= Ψ_{S,r}^{q}(η_{S}e_{J}e_{K}) = ∑_{i∈S} λ_{i}Ψ_{S,r}^{q}(e_{i}e_{J}e_{K})

= ∑_{i∈S}[λ_{i}λ_{S}e_{i}e_{J}e_{K} − λ_{i}η_{S}∂(e_{i}e_{J})e_{K}]

= λ_{S}η_{S}e_{J}e_{K} − ∑_{i∈S} λ_{i}η_{S}e_{J}e_{K} + ∑_{i∈S} λ_{i}η_{S}e_{i}(∂e_{J})e_{K}

= λ_{S}η_{S}e_{J}e_{K} − λ_{S}η_{S}e_{J}e_{K} + η_{S}η_{S}(∂e_{J})e_{K} = 0.

Next, consider the λ_{S}-eigenspace. If q ≥ s, we must show that e_{S}e_{K} is an eigenvector of ω^{q}(S, r) corresponding to the eigenvalue λ_{S} for each K ⊂ [n] \ S with |K| = q − s. By induction, we have ω^{q}(S, r + k)(e_{S}e_{K}) = λ_{S}e_{S}e_{K} for each k ≥ 1. By Lemma 4.2, we have Ψ_{S,r}^{q}(e_{S}e_{K}) = \binom{s}{r}λ_{S}e_{S}e_{K} − \binom{s−1}{r−1}η_{S}∂e_{S}e_{K}. Since η_{S}∂e_{S} = λ_{S}e_{S}, we have Ψ_{S,r}^{q}(e_{S}e_{K}) = \binom{s−1}{r}λ_{S}e_{S}e_{K}. Hence, by (4.2), we have

\begin{aligned}ω^{q}(S, r)(e_{S}e_{K}) & = Ψ_{S,r}^{q}(e_{S}e_{K}) − ∑_{k=1}^{s−r−1} \binom{r + k − 1}{k}ω^{q}(S, r + k)(e_{S}e_{K}) \\ & = \binom{s − 1}{r}λ_{S}e_{S}e_{K} − ∑_{k=1}^{s−r−1} \binom{r + k − 1}{k}λ_{S}e_{S}e_{K} = λ_{S}e_{S}e_{K}, \end{aligned}

using the binomial identities

(4.3)  ∑_{k=0}^{p} \binom{N + k}{k} = \binom{N + p + 1}{p} = \binom{N + p + 1}{N + 1},

with N = r − 1 and p = s − r − 1.

If |J| ≥ r + 1, we must show that ω^{q}(S, r)(∂e_{J}e_{K}) = λ_{S}∂e_{J}e_{K}. Suppose |J| = r + p + 1 for some p ≥ 0. Then, by Lemma 4.2, we have

Ψ_{S,r}^{q}(∂e_{J}e_{K}) = ∑_{i=1}^{r+p+1}(−1)^{i−1}Ψ_{S,r}^{q}(e_{J_{i}}e_{K})

= ∑_{i=1}^{r+p+1}(−1)^{i−1} [\binom{r + p}{r}λ_{S}e_{J_{i}}e_{K} − \binom{r + p − 1}{r − 1}η_{S}(∂e_{J_{i}})e_{K}]

= \binom{r + p}{r}λ_{S}(∂e_{J})e_{K} − \binom{r + p − 1}{r − 1}η_{S}(∂^{2}e_{J})e_{K}

= \binom{r + p}{r}λ_{S}(∂e_{J})e_{K}.

By induction, we have

ω^{q}(S, r + k)((∂e_{J})e_{K}) = \begin{cases}λ_{S}(∂e_{J})e_{K} & if 1 ≤ k ≤ p, \\ 0 & if p + 1 ≤ k ≤ s − r − 1. \end{cases}

So using the recursion (4.2) and the identities (4.3), we obtain

\begin{aligned}ω^{q}(S, r)((∂e_{J})e_{K}) & = \binom{r + p}{r}λ_{S}(∂e_{J})e_{K} − ∑_{k=1}^{p} \binom{r + k − 1}{k}λ_{S}(∂e_{J})e_{K} \\ & = λ_{S}(∂e_{J})e_{K}. \end{aligned}

If |J| ≥ r, we must show that ω^{q}(S, r)(η_{S}e_{J}e_{K}) = λ_{S}η_{S}e_{J}e_{K}. Suppose |J| = r + p for some p ≥ 0. Then, by Lemma 4.2, we have

Ψ_{S,r}^{q}(η_{S}e_{J}e_{K}) = ∑_{i∈S} λ_{i}Ψ_{S,r}^{q}(e_{i}e_{J}e_{K})

= ∑_{i∈S} y_{i} [\binom{r + p + 1}{r}λ_{S}e_{i}e_{J}e_{K} − \binom{r + p}{r − 1}η_{S}∂(e_{i}e_{J})e_{K}]

= \binom{r + p + 1}{r}λ_{S}η_{S}e_{J}e_{K} − \binom{r + p}{r − 1} ∑_{i∈S} λ_{i}η_{S}(e_{J} − e_{i}∂e_{J})e_{K}

= [\binom{r + p + 1}{r} − \binom{r + p}{r − 1}] λ_{S}η_{S}e_{J}e_{K} + \binom{r + p}{r − 1}η_{S}η_{S}(∂e_{J})e_{K}

= \binom{r + p}{r}λ_{S}η_{S}e_{J}e_{K}.

By induction, we have

ω^{q}(S, r + k)(η_{S}e_{J}e_{K}) = \begin{cases}λ_{S}η_{S}e_{J}e_{K} & if 1 ≤ k ≤ p, \\ 0 & if p + 1 ≤ k ≤ s − r − 1. \end{cases}

So using the recursion (4.2) and the identities (4.3), we obtain ω^{q}(S, r)(η_{S}e_{J}e_{K}) = λ_{S}η_{S}e_{J}e_{K} as above.

Thus the vectors in the sets V_{S}^{q,r}(λ) and W_{S}^{q,r}(λ) are eigenvectors of ω^{q}(S, r) corresponding to the eigenvalues 0 and λ_{S} as asserted. Since these vectors span A^{q}(G_{n}^{ℓ}) by Lemma 2.4, it remains to compute the dimensions of the eigenspaces E^{q}(0) = span V_{S}^{q,r} and E^{q}(λ_{S}) = span W_{S}^{q,r} corresponding to these eigenvalues.

If |J| = p and |J| + |K| = q, then span{e_{J}e_{K} | J ⊂ S, K ⊂ [n] \ S} has dimension

\binom{s}{p}\binom{n − s}{q − p}.

If |J| = p + 1 and |J| − 1 + |K| = q, then span{(∂e_{J})e_{K} | J ⊂ S, K ⊂ [n] \ S} has dimension

dim im[∂ : A^{p+1}(G_{s}^{s}) → A^{p}(G_{s}^{s})] · \binom{n − s}{q − p} = \binom{s − 1}{p}\binom{n − s}{q − p}.

If |J| = p − 1 and |J| + 1 + |K| = q, then span{η_{S}e_{J}e_{K} | J ⊂ S, K ⊂ [n] \ S} has dimension

dim ker[η_{S} : A^{p}(G_{s}^{s}) → A^{p+1}(G_{s}^{s})] · \binom{n − s}{q − p} = [\binom{s}{p} − \binom{s − 1}{p}] \binom{n − s}{q − p}.

Using these calculations, it is readily checked that

dim E^{q}(0) = ∑_{p=0}^{r} \binom{s}{p}\binom{n − s}{q − p} − \binom{s − 1}{r}\binom{n − s}{q − r}, and

dim E^{q}(λ_{S}) = ∑_{p=r+1}^{min(q,s)} \binom{s}{p}\binom{n − s}{q − p} + \binom{s − 1}{r}\binom{n − s}{q − r}.

The fact that dim E^{q}(0) + dim E^{q}(λ_{S}) = dim A^{q}(G_{n}^{ℓ}) = \binom{n}{q} may be checked using the binomial identities

∑_{p=0}^{k} \binom{m}{p}\binom{N}{k − p} = \binom{m + N}{k} and ∑_{p=0}^{m} \binom{m}{p}\binom{N}{k + p} = \binom{m + N}{m + r} = \binom{m + N}{N − k}

with m = s, N = n − s, and k = q in the case q < s, and m = s, N = n − s, and k = n − s − q in the case q ≥ s. □

If n = ℓ and λ ≠ 0, the complex (A^{•}(G), e_{λ}) is acyclic. So assume that n > ℓ. Then, for λ ≠ 0, the cohomology of this complex is concentrated in dimension ℓ, and dim H^{ℓ}(G) = \binom{n−1}{ℓ}. Let ρ = ρ_{G} : A^{ℓ}(G) → H^{ℓ}(G) denote the projection. Since ω_{λ}^{•}(S, r) is a chain map, the kernel of this projection, ker(ρ) ⊂ A^{ℓ}(G), is an invariant subspace for ω_{λ}^{ℓ}(S, r).

Lemma 4.4. Let T : V → V be an endomorphism of a finite dimensional (complex) vector space, and V′ an invariant subspace. If T is diagonalizable, then the induced endomorphism T′′ on the quotient V′′ = V/V′ is also diagonalizable, and the spectrum of T′′ is contained in the spectrum of T.

Proof. Let T′ denote the restriction of T to V′, and let π : V → V′′ be the projection. The vector space V admits a basis B = {v_{1}, …, v_{k}, v_{k+1}, …, v_{n}} for which B′ = {v_{1}, …, v_{k}} is a basis for the subspace V′ and B′′ = {π(v_{k+1}), …, π(v_{n})} is a basis for the quotient V′′. The matrix of T relative to the basis B is

A = (\begin{matrix}A′ & ∗ \\ 0 & A′′ \end{matrix}) ,

where A′ is the matrix of T′ relative to B′ and A′′ is the matrix of the induced endomorphism T′′ relative to B′′.

Let r_{1}, …, r_{m} be the distinct eigenvalues of T. Since T is diagonalizable, the minimal polynomial p of T factors as p(t) = (t − r_{1}) ⋯ (t − r_{m}). The polynomial p annihilates the matrix A of T, p(A) = 0. Using the block decomposition of A above, it follows that p also annihilates the matrix A′′ of T′′, p(A′′) = 0. Consequently, the minimal polynomial p′′ of T′′ divides p. Hence, p′′ is of the form (t − r_{i_{1}}) ⋯ (t − r_{i_{j}}), T′′ is diagonalizable, and the eigenvalues of T′′ are among the eigenvalues of T. □

For an arrangement A of arbitrary combinatorial type T , and T -nonresonant weights λ, we recall the βnbc basis of [7] for the single nonvanishing cohomology group H^{ℓ}(T ) = H^{ℓ}(M; L). Recall that the hyperplanes of A = {H_{j}}_{j=1}^{n} are ordered. A circuit is an inclusion-minimal dependent set of hyperplanes in A, and a broken circuit is a set T for which there exists H < min(T) so that T ∪ {H} is a circuit. A frame is a maximal independent set, and an nbc frame is a frame which contains no broken circuit. Since A contains ℓ linearly independent hyperplanes, every frame has cardinality ℓ. The set of nbc frames is a basis for A^{ℓ}(T ). An nbc frame B = (H_{j_{1}}, …, H_{j_{ℓ}}) is a βnbc frame provided that for each k, 1 ≤ k ≤ ℓ, there exists H ∈ A such that H < H_{j_{k}} and (B \ {H_{j_{k}}}) ∪ {H} is a frame. Note that these constructions depend only on the combinatorial type T of A, and let βnbc(T ) be the set of all βnbc frames of an arrangement of type T .

Definition 4.5. Given B = (H_{j_{1}}, …, H_{j_{ℓ}}) in βnbc(T ), define ξ(B) ∈ A^{ℓ}(T ) by ξ(B) = ∧_{p=1}^{ℓ}a_{λ}(X_{p}), where X_{p} = ⋂_{k=p}^{ℓ} H_{j_{k}} and a_{λ}(X) = ∑_{X⊆H_{i}} λ_{i}a_{i}. Denote the cohomology class of ξ(B) in H^{ℓ}(T ) = H^{ℓ}(A^{•}(T ), a_{λ}) by the same symbol. The set {ξ(B) | B ∈ βnbc(T )} is the βnbc basis for H^{ℓ}(T ).

Theorem 4.6. Let S ⊂ [n + 1] be a subset of cardinality s, and fix r, 1 ≤ r ≤ min(ℓ, s − 1). For G-nonresonant weights λ satisfying λ_{S} ≠ 0, the endomorphism Ω_{λ}(S, r) of H^{ℓ}(G) induced by ω_{λ}^{ℓ}(S, r) is diagonalizable, with eigenvalues 0 and λ_{S}. The dimension of the λ_{S}-eigenspace is

∑_{p=r+1}^{min(ℓ,s)} \binom{s}{p}\binom{n − s − 1}{ℓ − p} + \binom{s − 1}{r}\binom{n − s − 1}{ℓ − r},

and the dimension of the 0-eigenspace is

∑_{p=0}^{r} \binom{s}{p}\binom{n − s − 1}{ℓ − p} − \binom{s − 1}{r}\binom{n − s − 1}{ℓ − r}.

Proof. By Theorem 4.3 and Lemma 4.4, the endomorphism Ω_{λ}(S, r) is diagonalizable, with spectrum contained in {0, λ_{S}}.

Let I = {I = (i_{1}, …, i_{ℓ}) | 1 ≤ i_{1} < i_{2} ⋯ < i_{ℓ} ≤ n}. Then {e_{I} | I ∈ I} is the nbc basis of A^{ℓ}(G) and {ξ_{I} = λ_{i_{1}} ⋯ λ_{i_{ℓ}}e_{I} | I ∈ I, 1 ∉ I} is the βnbc basis of H^{ℓ}(G). The projection ρ : A^{ℓ}(G) → H^{ℓ}(G) is given by

ρ(e_{I}) = \begin{cases}(λ_{i_{1}} ⋯ λ_{i_{ℓ}})^{−1}ξ_{I} & if 1 ∉ I, \\ −(λ_{i_{1}} ⋯ λ_{i_{ℓ}})^{−1} ∑_{j∉I} ξ_{j}ξ_{I_{1}} & if 1 ∈ I. \end{cases}

Using Lemma 2.1, we can assume that S ⊂ [2, n]. Since ρ ∘ ω_{λ}^{ℓ}(S, r) = Ω_{λ}(S, r) ∘ ρ, if v is an eigenvector of ω_{λ}^{ℓ}(S, r) and ρ(v) ≠ 0, then ρ(v) is an eigenvector of Ω_{λ}(S, r). Let J ⊂ S and K ⊂ [2, n] \ S. Note that 1 ∉ K. Then one can check that the 0-eigenspace of Ω_{λ}(S, r) is spanned by

{ρ(e_{J}e_{K}) | |J| ≤ r − 1} ⋃{ρ(η_{S}e_{J}e_{K}) | |J| = r − 1},

that the λ_{S}-eigenspace of Ω_{λ}(S, r) is spanned by

{ρ(e_{S}e_{K}) | if ℓ ≥ s} ⋃{ρ((∂e_{J})e_{K}) | |J| ≥ r + 1} ⋃{ρ(η_{S}e_{J}e_{K}) | |J| ≥ r},

and that the dimensions of these eigenspaces are as asserted. □

Example 4.7. Let n = 5, ℓ = 2, S = {3, 4, 5}, and r = 1. By Theorem 4.6, for G-nonresonant weights satisfying λ_{S} ≠ 0, the endomorphism Ω_{λ}(S, r) of H^{2}(G) ≃ ℂ^{6} is diagonalizable, the λ_{S}-eigenspace is 5-dimensional, and the 0-eigenspace is 1dimensional (note that \binom{p}{q} = 0 if p < q). Calculating as in the proof of Theorem 4.6, we find that the λ_{S}-eigenspace has basis

ρ(λ_{2}λ_{3}λ_{5}(∂e_{3,5})e_{2}) = λ_{5}ξ_{2,3} − λ_{3}ξ_{2,5},  ρ(−λ_{3}η_{3,4,5}e_{3}) = ξ_{3,4} + ξ_{3,5},
ρ(λ_{2}λ_{4}λ_{5}(∂e_{4,5})e_{2}) = λ_{5}ξ_{2,4} − λ_{4}ξ_{2,5},  ρ(λ_{5}η_{3,4,5}e_{5}) = ξ_{3,5} + ξ_{4,5},
ρ(λ_{3}λ_{4}λ_{5}∂e_{3,4,5}) = λ_{5}ξ_{3,4} − λ_{4}ξ_{3,5} + λ_{3}ξ_{4,5},

and the 0-eigenspace has basis ρ(λ_{1}λ_{2}e_{1,2}) = ξ_{2,3} + ξ_{2,4} + ξ_{2,5}.

5. Nonresonant eigenvalues

In this section, we prove that the Gauss-Manin endomorphism Ω_{λ}(T′, T ) of (1.1) is diagonalizable and determine its eigenvalues. We accomplish this by showing that the endomorphism Ω̃_{λ}(T′, T ) in the commutative diagram (1.2) may be replaced by the endomorphism Ω_{λ}(S, r), whose eigenstructure was computed in Theorem 4.6.

For an arbitrary type T , let I^{•}(T ) be the corresponding Orlik-Solomon ideal, so that A^{•}(T ) ≃ A^{•}(G)/I^{•}(T ). The natural projection of A^{•}(G) onto A^{•}(T ) is a chain map π : (A^{•}(G), e_{λ}) → (A^{•}(T ), a_{λ}) which, for T -nonresonant weights λ, induces the projection τ : H^{ℓ}(G) → H^{ℓ}(T ) upon passage to cohomology. If ρ_{G} : A^{ℓ}(G) → H^{ℓ}(G) and ρ_{T} : A^{ℓ}(T ) → H^{ℓ}(T ) are the projections, then τ ∘ ρ_{G} = ρ_{T} ∘ π.

Theorem 5.1. If T′ is a degeneration of T with principal dependence (S, r), then Ω_{λ}(T′, T ) ∘ τ = τ ∘ Ω_{λ}(S, r). In other words, the following diagram commutes:

H^{ℓ}(G) ⟶^{τ} H^{ℓ}(T )
↓^{Ω_{λ}(S,r)}  ↓^{Ω_{λ}(T′,T )}
H^{ℓ}(G) ⟶^{τ} H^{ℓ}(T )

Proof. As noted in the introduction, the Gauss-Manin endomorphism Ω_{λ}(T′, T ) of H^{ℓ}(T ) is induced by the endomorphism Ω̃_{λ}(T′, T ) of H^{ℓ}(G), see [4, Thm. 7.3] and (1.2). In turn, Ω̃_{λ}(T′, T ) is the map in cohomology induced by the cochain endomorphism ω_{λ}^{•}(T′, T ) of the complex (A^{•}(G), e_{λ}), see Theorem 3.1. The map ω_{λ}^{•}(T′, T ) also induces a cochain endomorphism ω̄_{λ}^{•}(T′, T ) of (A^{•}(T ), a_{λ}), and the Gauss-Manin endomorphism Ω_{λ}(T′, T ) may be realized as the map in cohomology induced by the latter, see [5, Thm. 7.1]. In summary, we have the following commutative diagram.

(5.1)  [figure: A^{ℓ}(G); π; A^{ℓ}(T ); H^{ℓ}(G); τ; ρ_{G}; ω̄_{λ}^{ℓ}(T′, T ); H^{ℓ}(T ); ρ_{T}; A^{ℓ}(G); ω_{λ}^{ℓ}(T′, T ); π; A^{ℓ}(T ); H^{ℓ}(G); Ω̃_{λ}(T′, T ); τ; ρ_{G}; H^{ℓ}(T ); Ω_{λ}(T′, T ); ρ_{T}]

To establish the theorem, it suffices to show that the endomorphisms ω_{λ}^{•}(T′, T ) and ω_{λ}^{•}(S, r) of A^{•}(G) induce the same endomorphism of A^{•}(T ).

The Orlik-Solomon ideal I^{•}(T ) gives rise to a subcomplex I_{R}^{•}(T ) = I^{•}(T )⊗R of the Aomoto complex A_{R}^{•}(G), with quotient A_{R}^{•}(T ), the Aomoto complex of type T . Since ω_{λ}^{•}(T′, T ) and ω_{λ}^{•}(S, r) are specializations at λ of the corresponding endomorphims of the Aomoto complex A_{R}^{•}(G), it is enough to show that ω^{•}(T′, T ) and ω^{•}(S, r) induce the same endomorphism of A_{R}^{•}(T ).

By Theorem 3.2, there are dependence pairs (S_{i}, r_{i}), 1 ≤ i ≤ k, such that Dep(T )^{∗} contains Dep(T (S_{i}, r_{i}))^{∗} and Dep(T′)^{∗} = ⋃_{i=0}^{k} Dep(T (S_{i}, r_{i}))^{∗}, where (S_{0}, r_{0}) = (S, r) is the pair of principal dependence. It follows that there are constants c_{i} so that ω^{•}(T′) = ω^{•}(S, r) + ∑_{i=1}^{k} c_{i} · ω^{•}(S_{i}, r_{i}).

If Dep(T (S_{i}, r_{i}))^{∗} ⊂ Dep(T )^{∗}, it follows from Theorem 4.3 that the image of ω^{•}(S_{i}, r_{i}) : A_{R}^{•}(G) → A_{R}^{•}(G) is contained in I_{R}^{•}(T ). Consequently, the endomorphisms ω̄^{•}(T′) and ω̄^{•}(S, r) of the Aomoto complex A_{R}^{•}(T ) induced by ω^{•}(T′) and ω^{•}(S, r) are equal.

Finally, ω^{•}(T′) = ω^{•}(T′, T ) + ω^{•}(T ), see (3.3). It follows from the definitions that the image of ω^{•}(T ) is also contained in I_{R}^{•}(T ). Hence, the endomorphisms ω̄^{•}(T′) and ω̄^{•}(T′, T ) of A_{R}^{•}(T ) induced by ω^{•}(T′) and ω^{•}(T′, T ) are equal. □

Theorem 4.6 and Theorem 5.1 yield the result stated in the introduction.

Theorem 5.2. Let T′ be a degeneration of T with principal dependence (S, r), and λ a collection of T -nonresonant weights satisfying λ_{S} ≠ 0. Then the Gauss-Manin endomorphism Ω_{λ}(T′, T ) is diagonalizable, with spectrum contained in {0, λ_{S}}.

Proof. By Theorem 4.6, the endomorphism Ω_{λ}(S, r) of H^{ℓ}(G) is diagonalizable, with eigenvalues 0 and λ_{S}. By Theorem 5.1, we have Ω_{λ}(T′, T )∘τ = τ ∘Ω_{λ}(S, r). Checking that ker(τ) ⊂ H^{ℓ}(G) is an invariant subspace for Ω_{λ}(S, r), the result follows from Lemma 4.4. □

Remark 5.3. The Gauss-Manin endomorphism Ω_{λ}(T′, T ) of H^{ℓ}(T ) is determined by the endomorphism Ω_{λ}(S, r) of H^{ℓ}(G) and the projection τ : H^{ℓ}(G) → H^{ℓ}(T ) via the equality Ω_{λ}(T′, T ) ∘ τ = τ ∘ Ω_{λ}(S, r). Together with the explicit description of the eigenstructure of Ω_{λ}(S, r) provided by Theorems 4.3 and 4.6, this yields an algorithm for finding the (geometric) multiplicities of the eigenvalues of Ω_{λ}(T′, T ).

The Gauss-Manin connection ∇ = ∑ Θ_{T′} ⊗ Ω_{λ}(T′, T ) on the vector bundle H → B(T ) with fiber H^{ℓ}(T ) corresponds to a monodromy representation Ψ : π_{1}(B(T )) → Aut_{ℂ}(H^{ℓ}(T )). For a degeneration T′ of T , let γ_{T′} ∈ π_{1}(B(T )) be a simple loop in B(T ) around a generic point in B(T′). Then the automorphism Ψ(γ_{T′}) is conjugate to exp(−2π i Ω_{λ}(T′, T )), see for instance [3, Prop. 4.1]. Theorem 5.2 yields:

Corollary 5.4. Let T′ be a degeneration of T with principal dependence (S, r), and λ a collection of T -nonresonant weights satisfying λ_{S} ≠ 0. Then the automorphism Ψ(γ_{T′}) is diagonalizable, with spectrum contained in {1, exp(−2π i λ_{S})}.

We conclude with several examples which illustrate these results.

5.5. Codimension zero. Recall that G denotes the combinatorial type of a general position arrangement of n hyperplanes in ℂ^{ℓ}, and that n ≥ ℓ. Weights λ = (λ_{1}, …, λ_{n}) are G-nonresonant if λ_{j} ≠ 0 for each j. If n = ℓ, then H^{•}(G) = 0, so we assume that n > ℓ. Then dim H^{ℓ}(G) = \binom{n−1}{ℓ}. The moduli space B(G) has codimension zero in (ℂℙ^{ℓ})^{n}, and consists of all matrices b for which every (ℓ + 1) × (ℓ + 1) minor is nonzero, see (3.1). For general position arrangements, the Gauss-Manin connection was determined by Aomoto and Kita [2]. The corresponding connection 1-form is given by ∇ = ∑ Θ_{T} ⊗ Ω_{λ}(T , G), where the sum is over all ℓ + 1 element subsets S of [n + 1], T = T (S, ℓ + 1), and Θ_{T} is a logarithmic 1-form on (ℂℙ^{ℓ})^{n} with a simple pole along the divisor defined by the vanishing of the (ℓ + 1) × (ℓ + 1) minor of b with rows indexed by S. Theorem 4.6 gives:

Proposition 5.6. Let S be an ℓ + 1 element subset of [n], let T = T (S, ℓ + 1), and λ a collection of G-nonresonant weights satisfying λ_{S} ≠ 0. Then the Gauss-Manin endomorphism Ω_{λ}(T , G) is diagonalizable, with eigenvalues 0 and λ_{S}. The dimension of the λ_{S}-eigenspace is 1, and the dimension of the 0-eigenspace is \binom{n−1}{ℓ} − 1.

5.7. Codimension one. If T is a combinatorial type for which the cardinality of Dep(T )_{ℓ+1} is 1, then the moduli space B(T ) is of codimension one in (ℂℙ^{ℓ})^{n}. Write Dep(T )_{ℓ+1} = {K}. As shown by Terao [15], noted in the proof of Theorem 3.2, and illustrated in Example 3.4, the combinatorial type T admits three types of degeneration T′ = T (S, r). The principal dependencies of these degenerations are as follows.

I: (S, ℓ), where |S| = ℓ + 1 and |S ∩ K| ≤ ℓ − 1; II: (S, ℓ − 1), where S = K_{p}, for each p, 1 ≤ p ≤ ℓ + 1; III: (S, ℓ), where S = (m, K), for each m ∈ [n + 1] \ K.

For the combinatorial type T and T -nonresonant weights λ, the Gauss-Manin connection was determined by Terao [15]. The corresponding connection 1-form is given by ∇ = ∑ Θ_{T′} ⊗ Ω_{λ}(T′, T ), where T′ ranges over the three types of degeneration of T noted above. In [15], Terao also found the eigenvalues of the endomorphism Ω_{λ}(T′, T ) and their algebraic multiplicities. If λ satisfies λ_{S} ≠ 0 for each of the principal dependence sets S recorded above, Terao’s result concerning the eigenstructure of the endomorphism Ω_{λ}(T′, T ) may be strengthened as follows.

Proposition 5.8. Let T be a combinatorial type of codimension one, let T′ = T (S, r) be a degeneration of T , and λ a collection of T -nonresonant weights satisfying λ_{S} ≠ 0. Then the Gauss-Manin endomorphism Ω_{λ}(T′, T ) is diagonalizable, with eigenvalues 0 and λ_{S}.

1. If T′ is a degeneration of type I, the dimension of the λ_{S}-eigenspace is 1, and the dimension of the 0-eigenspace is dim H^{ℓ}(T ) − 1 = \binom{n−1}{ℓ} − 2.

2. If T′ is a degeneration of type II, the dimension of the λ_{S}-eigenspace is n−ℓ−1, and the dimension of the 0-eigenspace is \binom{n−1}{ℓ} − n + ℓ.

3. If T′ is a degeneration of type III, the dimension of the λ_{S}-eigenspace is ℓ, and the dimension of the 0-eigenspace is \binom{n−1}{ℓ} − ℓ − 1.

Proof. By Theorem 5.2, the endomorphism Ω_{λ}(T′, T ) is diagonalizable, with spectrum contained in {0, λ_{S}}.

Without loss, assume that Dep(T )_{ℓ+1} = {K}, where K = [ℓ + 1]. Then the nbc basis of A^{ℓ}(T ) consists of monomials a_{I}, where I ⊂ [n], |I| = ℓ, and I ≠ [2, ℓ + 1]. Write F = [2, ℓ + 1]. The projection π : A^{ℓ}(G) → A^{ℓ}(T ) is given by

π(e_{I}) = \begin{cases}a_{I} & if I ≠ F, \\ a_{1}∂a_{F} & if I = F. \end{cases}

The βnbc basis for H^{ℓ}(T ) consists of monomials ξ_{I}, where I ⊂ [2, n], |I| = ℓ, and I ≠ F. The projection ρ = ρ_{T} : A^{ℓ}(T ) → H^{ℓ}(T ) is given by

ρ(a_{I}) = \begin{cases}(λ_{i_{1}} ⋯ λ_{i_{ℓ}})^{−1}ξ_{I} & if 1 ∉ I, \\ −(λ_{i_{1}} ⋯ λ_{i_{ℓ}})^{−1} ∑_{j∉I} ξ_{j}ξ_{I_{1}} & if 1 ∈ I, I ⊄ K, \\ −(λ_{K}λ_{i_{1}} ⋯ λ_{i_{ℓ}})^{−1} ∑_{j∉K}[λ_{I}ξ_{j}ξ_{I_{1}} + ξ_{j}ξ_{p}∂ξ_{I_{1}}] & if 1 ∈ I, I = K \ \{p\}. \end{cases}

If T′ is a degeneration of type I with principal dependence (S, ℓ), then |S∩K| ≤ ℓ−1 and we can assume that S ∩ K ⊂ [3, ℓ + 1]. By Theorem 4.3, the endomorphism ω_{λ}^{ℓ}(S, ℓ) of A^{ℓ}(G) is diagonalizable, with eigenvalues 0 and λ_{S}. The 0-eigenspace is spanned by {e_{J}e_{L} | |J| ≤ ℓ − 1} ⋃{η_{S}e_{J} | |J| = ℓ − 1}, where J ⊂ S and L ⊂ [n]\S, and the λ_{S}-eigenspace is spanned by ∂e_{S}. By Theorem 5.1, the endomorphism Ω_{λ}(T′, T ) of H^{ℓ}(T ) satisfies Ω_{λ}(T′, T ) ∘ ρ ∘ π = ρ ∘ π ∘ ω_{λ}^{ℓ}(S, ℓ), see (5.1). Write S = (s_{1}, …, s_{ℓ+1}). Calculations with the projections π and ρ yield

ρ ∘ π(∂e_{S}) = (λ_{s_{1}} ⋯ λ_{s_{ℓ+1}})^{−1}∂ξ_{S},
ρ ∘ π(e_{J}e_{L}) = (λ_{i_{1}} ⋯ λ_{i_{ℓ}})^{−1}ξ_{J}ξ_{L}, where I = (J, L) and 1 ∉ L,
ρ ∘ π(η_{S}e_{J}) = λ_{s_{1}}λ_{s_{p}}(λ_{s_{1}} ⋯ λ_{s_{ℓ+1}})^{−1}(ξ_{S_{p}} ± ξ_{S_{1}}), where J = (s_{2}, …, ŝ_{p}, …, s_{ℓ+1}).

Checking that

{∂ξ_{S})} ⋃{ξ_{J}ξ_{L} | J ⊂ S, |J| ≤ ℓ − 1, L ⊂ [2, n] \ S} ⋃{ξ_{S_{p}} ± ξ_{S_{1}} | 2 ≤ p ≤ ℓ + 1}

forms a basis for H^{ℓ}(T ), we conclude that the dimensions of the eigenspaces are as asserted for a degeneration of type I.

If T′ is a degeneration of type II with principal dependence (S, ℓ − 1), we can assume that S = K_{1} = [2, ℓ + 1]. By Theorem 4.3, the endomorphism ω_{λ}^{ℓ}(S, ℓ − 1) of A^{ℓ}(G) is diagonalizable, with eigenvalues 0 and λ_{S}. The 0-eigenspace is spanned by {e_{J}e_{L} | |J| ≤ ℓ − 2} ⋃{η_{S}e_{J}e_{q} | |J| = ℓ − 2}, where J ⊂ S, L ⊂ [n] \ S, q ∉ S, and the λ_{S}-eigenspace is spanned by {e_{S}} ⋃{(∂e_{S})e_{q} | q ∉ S}. Note that the λ_{S}- eigenspace of ω_{λ}^{ℓ}(S, ℓ−1) has dimension n−ℓ+1. Note also that the λ_{S}-eigenvectors ∂e_{K} = e_{S} − e_{1}∂e_{S} and e_{λ}∂e_{S} are annihilated by the projection ρ ∘ π. On the other hand, it is readily checked that

(5.2)  {ρ ∘ π((∂e_{S})e_{q}) | ℓ + 2 ≤ q ≤ n}

is a linearly independent set of (n − ℓ − 1) λ_{S}-eigenvectors for Ω_{λ}(T′, T ) in H^{ℓ}(T ). Additionally, one can check that the set

(5.3) {ρ ∘ π(e_{J}e_{L}) | J ⊂ S, |J| ≤ ℓ − 2} ⋃{ρ ∘ π(η_{S}e_{J}e_{q}) | J ⊂ [3, ℓ + 1], |J| = ℓ − 2}

where L ⊂ [n] \ S and q ∉ S, is a linearly independent set of 0-eigenvectors for Ω_{λ}(T′, T ) in H^{ℓ}(T ). Checking that the dimension of the subspace spanned by the vectors (5.3) is dim H^{ℓ}(T ) − (n − ℓ − 1), since eigenvectors associated to distinct eigenvalues are linearly independent, the vectors (5.2) and (5.3) form a basis for H^{ℓ}(T ). Hence, the dimensions of the eigenspaces are as asserted for a degeneration of type II.

If T′ is a degeneration of type III with principal dependence (S, ℓ), we can assume that S = K ⋃{q} for some q ∈ [ℓ+2, n]. By Theorem 4.3, the endomorphism ω_{λ}^{ℓ}(S, ℓ) of A^{ℓ}(G) is diagonalizable, with eigenvalues 0 and λ_{S}. The 0-eigenspace is spanned by {e_{J}e_{L} | |J| ≤ ℓ − 1} ⋃{η_{S}e_{J} | |J| = ℓ − 2}, where J ⊂ S, L ⊂ [n] \ S, and the λ_{S}-eigenspace is spanned by {∂e_{J} | J ⊂ S, |J| = ℓ + 1}. Note that the λ_{S}-eigenspace of ω_{λ}^{ℓ}(S, ℓ) has dimension ℓ + 1. Note also that the λ_{S}-eigenvector ∂e_{K} is annihilated by the projection ρ ∘ π. Recall that F = [2, ℓ + 1]. Let S_{q} denote the subspace of H^{ℓ}(T ) spanned by {ξ_{I} | I ⊂ F ⋃{q}}, and let p_{q} : H^{ℓ}(T ) → S_{q} be the natural projection. For J ⊂ F, |J| = ℓ − 1, a calculation reveals that p_{q} ∘ ρ ∘ π(η_{K}e_{J}e_{q}) = λ_{S}(λ_{2} ⋯ λ_{ℓ+1}λ_{q})^{−1}ξ_{J}ξ_{q}. Consequently, the set {ρ ∘ π(η_{K}e_{J}e_{q}) | J ⊂ F, |J| = ℓ − 1} is a linearly independent set of ℓ λ_{S}-eigenvectors for Ω_{λ}(T′, T ) in H^{ℓ}(T ). Check that the set {ρ ∘ π(e_{J}e_{L}) | J ⊂ S_{1}, |J| ≤ ℓ − 1, L ⊂ [n] \ S} is a linearly independent set of dim H^{ℓ}(T ) − ℓ 0-eigenvectors for Ω_{λ}(T′, T ) in H^{ℓ}(T ). It follows that the dimensions of the eigenspaces are as asserted for a degeneration of type III. □

5.9. Further examples. We present three examples of higher codimension.

Example 5.10. Let S be the combinatorial type of the Selberg arrangement A in ℂ^{2} with defining polynomial Q(A) = u_{1}u_{2}(u_{1} −1)(u_{2} −1)(u_{1} −u_{2}) depicted in Figure 2. See [1, 14, 10] for detailed studies of the Gauss-Manin connections arising in the context of Selberg arrangements.

[figure: 5; 4; 3; 1; 2; A; 345; 1; 2; A′]

Figure 2. A Selberg arrangement and one degeneration

Here Dep(S)^{∗} = {126, 346, 135, 245}. Weights λ are S-nonresonant if

λ_{j} (1 ≤ j ≤ 6), λ_{1} + λ_{2} + λ_{6}, λ_{1} + λ_{3} + λ_{5}, λ_{2} + λ_{4} + λ_{5}, λ_{3} + λ_{4} + λ_{6} ∉ ℤ_{≥0}.

For S-nonresonant weights, the βnbc basis for H^{2}(S) is {Ξ_{2,4}, Ξ_{2,5}}, where Ξ_{2,j} = (λ_{2}a_{2}+λ_{4}a_{4}+λ_{5}a_{5})λ_{j}a_{j}, see Definition 4.5. Recall that λ_{J} = ∑_{j∈J} λ_{j}. The projection map τ : H^{2}(G) → H^{2}(S) is given by

τ(ξ_{i,j}) = \begin{cases}−Ξ_{2,4} − Ξ_{2,5} & if (i, j) = (2, 3), \\ (λ_{2,4}Ξ_{2,4} + λ_{4}Ξ_{2,5})/λ_{2,4,5} & if (i, j) = (2, 4), \\ (λ_{5}Ξ_{2,4} + λ_{2,5}Ξ_{2,5})/λ_{2,4,5} & if (i, j) = (2, 5), \\ 0 & if (i, j) = (3, 4), \\ (−λ_{5}Ξ_{2,4} − λ_{3,5}Ξ_{2,5})/λ_{1,3,5} & if (i, j) = (3, 5), \\ (−λ_{5}Ξ_{2,4} + λ_{4}Ξ_{2,5})/λ_{2,4,5} & if (i, j) = (4, 5). \end{cases}

The arrangement A′ in Figure 2 represents one degeneration type S′ of S. Here Dep(S′, S)^{∗} = {34, 35, 45, 134, 145, 234, 235, 345, 356, 456}. The sets 34, 35, 45, and 345 have r = 1, and the others r = 2. The principal dependence is (S, r), where S = 345 and r = 1. For S-nonresonant weights with λ_{S} ≠ 0, Ω_{λ}(S′, S) is diagonalizable, with spectrum contained in {0, λ_{S}} by Theorem 5.2. The projection τ annihilates the 0-eigenspace of Ω_{λ}(S, r), and restricts to a surjection E(λ_{S}) ↠ H^{2}(S), where E(λ_{S}) is the λ_{S}-eigenspace of Ω_{λ}(S, r), see Example 4.7. It follows that Ω_{λ}(S′, S) has eigenvalues λ_{S}, λ_{S}. Note that 0 is not an eigenvalue of Ω_{λ}(S′, S) in this instance.

Although the eigenvalues are determined by the principal dependence (S, r), the same principal dependence may occur for degenerations of different types. Thus the multiplicities of the eigenvalues depend on the combinatorial types as well.

Example 5.11. Consider the arrangement Ā of type T obtained from the arrangement A in Example 5.10 by rotating line 1 by a (small) angle about the triple point 135, see Figure 3. Here, lines 1 and 2 meet in affine space, so 126 is no longer dependent. This change implies that dim A^{2}(T ) = 7 and dim H^{2}(T ) = 3.

[figure: 5; 4; 3; 1; 2; Ā; 345; 1; 2; Ā′]

Figure 3. A line arrangement and one degeneration

Weights λ are T -nonresonant if

λ_{j} (1 ≤ j ≤ 6), λ_{1} + λ_{3} + λ_{5}, λ_{2} + λ_{4} + λ_{5}, λ_{3} + λ_{4} + λ_{6} ∉ ℤ_{≥0}.

For T -nonresonant weights, the βnbc basis for H^{2}(T ) is {Ξ_{2,3}, Ξ_{2,4}, Ξ_{2,5}}, where Ξ_{2,3} = λ_{2}λ_{3}a_{2,3} and Ξ_{2,j} = (λ_{2}a_{2} + λ_{4}a_{4} + λ_{5}a_{5})λ_{j}a_{j} for j = 4, 5. The projection map τ : H^{2}(G) → H^{2}(T ) is given by

τ(ξ_{i,j}) = \begin{cases}Ξ_{2,3} & if (i, j) = (2, 3), \\ (λ_{2,4}Ξ_{2,4} + λ_{4}Ξ_{2,5})/λ_{2,4,5} & if (i, j) = (2, 4), \\ (λ_{5}Ξ_{2,4} + λ_{2,5}Ξ_{2,5})/λ_{2,4,5} & if (i, j) = (2, 5), \\ 0 & if (i, j) = (3, 4), \\ (λ_{5}Ξ_{2,3} − λ_{3}Ξ_{2,5})/λ_{1,3,5} & if (i, j) = (3, 5), \\ (−λ_{5}Ξ_{2,4} + λ_{4}Ξ_{2,5})/λ_{2,4,5} & if (i, j) = (4, 5). \end{cases}

The combinatorial type T has a degeneration of type T′ similar to S′, represented by the arrangement Ā′ in Figure 3. As in Example 5.10, the principal dependence is (S, r), where S = 345 and r = 1. For T -nonresonant weights with λ_{S} ≠ 0, the spectrum of Ω_{λ}(T′, T ) is contained in {0, λ_{S}}. Calculations with the projection τ and the eigenspace decomposition of the endomorphism Ω_{λ}(S, r) of H^{2}(G) given in Example 4.7 reveal that Ω_{λ}(T′, T ) has eigenvalues λ_{S}, λ_{S}, 0.

Example 5.12. The combinatorial type S in Example 5.10 is a degeneration of the type T in Example 5.11. The principal dependence of this degeneration is (S, r), where S = 126 and r = 2. For T -nonresonant weights with λ_{S} ≠ 0, the spectrum of Ω_{λ}(S, T ) is contained in {0, λ_{S}}. A calculation shows that the eigenvalues are λ_{S}, 0, 0. It is interesting to note that λ_{S} = λ_{1,2,6} = −λ_{3,4,5}.

Acknowledgment. This work was completed during the program “Hyperplane Arrangements and Applications” at the Mathematical Sciences Research Institute in Berkeley, California. We thank MSRI for its support and hospitality, and for providing a stimulating mathematical environment.

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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803

E-mail address: cohen@math.lsu.edu
URL: http://www.math.lsu.edu/~cohen

Department of Mathematics, University of Wisconsin, Madison, WI 53706

E-mail address: orlik@math.wisc.edu