BOUNDARY MANIFOLDS OF PROJECTIVE HYPERSURFACES

DANIEL C. COHEN^{1} AND ALEXANDER I. SUCIU^{2}

Abstract. We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology ring.

1. Introduction

1.1. Boundary manifolds. There are many ways to understand the topology of a homogeneous polynomial f: ℂ^{ℓ+1} → ℂ. The most direct approach is to study the hypersurface V in ℂℙ^{ℓ} defined as the zero locus of f. Another approach is to view the complement, X = ℂℙ^{ℓ} \V, as the primary object of study. And perhaps the most thorough is to study the Milnor fibration f: ℂ^{ℓ+1} \ {f(x) = 0} → ℂ^{∗}. Of course, the different approaches are interrelated. For example, if the degree of f is n, then the Milnor fiber F = f^{−1}(1) is a cyclic n-fold cover of X. Consequently, knowledge of the cohomology groups of X with coefficients in certain local systems yields the cohomology groups of F.

In this paper, we take a different (yet still related) tack. We consider the boundary manifold, M, defined as the boundary of a regular neighborhood N of the subvariety V ⊂ ℂℙ^{ℓ}, see Durfee [9]. Clearly, X ≃ ℂℙ^{ℓ} \ N^{∘}, and M is the boundary of ℂℙ^{ℓ} \ N^{∘}. While the complement X has the homotopy type of a CW-complex of dimension at most ℓ, the boundary manifold M is a smooth, compact manifold of dimension 2ℓ−1.

There are many questions one can ask about the topology of M, for instance, concerning its fundamental group, and how it relates to the fundamental group of X. In the case where V is the union of an arrangement of lines in ℂℙ^{2}, work in this direction was done by Jiang-Yau [17], Westlund [31], and Hironaka [15]. Here, we resolve the asphericity question for the boundary manifold of an arbitrary hyperplane arrangement (see Propositions 2.9 and 4.5), leaving a more detailed study of the fundamental group and related invariants to future work.

2000 Mathematics Subject Classification. Primary 14J70, 32S22. Secondary 16W50, 32S35, 55M30.

Key words and phrases. Complex hypersurface, hyperplane arrangement, boundary manifold, cohomology ring, trivial extension, zero-divisor cup length, topological complexity, resonance variety, pfaffian.

1Partially supported by National Security Agency grant MDA904-00-1-0038 and a Faculty Research Grant from Louisiana State University.

2Partially supported by National Science Foundation grant DMS-0311142 and an RSDF grant from Northeastern University.

For a general hypersurface V, our main goal in this paper is to compute the cohomology ring of the boundary manifold M. We show that, under fairly mild hypotheses, the cohomology ring of the complement X functorially determines the cohomology ring of M, and derive a number of consequences. For instance, when the hypersurface V = ⋃_{H∈A} H is determined by an arrangement of hyperplanes A, these (Hodge theoretic) hypotheses are satisfied, and the cohomology of X = X(A) is thoroughly understood, thanks to classical results of Brieskorn and Orlik-Solomon. Our results then yield an explicit description of the cohomology ring of the boundary manifold M = M(A).

1.2. Cohomology ring of the boundary. Given a finite-dimensional graded algebra A over a ring R, we construct a new algebra, D(A). This is a particular case of a more general construction due to Reiten [27], which associates to a ring A and an A-bimodule B the “trivial extension” ring A ⋉ B := A ⊕ B, with multiplication (a, b)(a′, b′) = (aa′, a·b′+b·a′). Applying this construction to a graded (commutative) algebra A = ⨁_{k=0}^{ℓ} A^{k} and the A-bimodule B = Ā = ⨁_{k=ℓ−1}^{2ℓ−1} Hom(A^{2ℓ−k−1}, R) yields a graded (commutative) algebra D(A) = A ⋉ Ā, which we refer to as the double of A.

Now if V ⊂ ℂℙ^{ℓ} is a projective hypersurface, then the cohomology groups (with complex coefficients) of V, and those of the complement X = ℂℙ^{ℓ} \ V admit mixed Hodge structures. For each k ≥ 0, there is an increasing weight filtration {W_{m}}_{m≤2k} of the k-th cohomology group, such that each quotient W_{m}/W_{m−1} has pure Hodge structure of weight m. Our main results, proved in Section 3, may be summarized as follows.

Theorem. Let V be a hypersurface in ℂℙ^{ℓ}, with complement X and boundary manifold M. If either V is irreducible, or the weight filtration on the top cohomology group of X satisfies W_{ℓ+1}(H^{ℓ}(X; ℂ)) = 0, then the cohomology ring of the boundary manifold is isomorphic to the double of the cohomology ring of the complement:

(1.1)  H^{∗}(M; ℂ) ≅ D(H^{∗}(X; ℂ)).

If H^{ℓ}(X; ℂ) satisfies the above weight condition and the integral cohomology of X is torsion-free, our results can be used to show that the splitting (1.1) holds over the integers, H^{∗}(M; ℤ) ≅ D(H^{∗}(X; ℤ)). On the other hand, this splitting can fail with integral coefficients when V is irreducible (see Example 2.4). With complex coefficients, the splitting (1.1) can fail if neither of the hypotheses stated in the theorem holds (see below).

1.3. Arrangements and curves. When applied to a complex hyperplane arrangement, our result yields an analog for the boundary manifold of a well known theorem of Orlik and Solomon [23] concerning the cohomology ring of the complement. Suppose A is an arrangement of hyperplanes in ℂℙ^{ℓ}, and let M(A) be its boundary manifold. Then the integral cohomology ring H^{∗}(M(A); ℤ) is determined by the intersection poset L(A) and the ambient dimension ℓ, see Corollary 4.2.

For an algebraic curve V ⊂ ℂℙ^{2} (in particular, an arrangement of lines in ℂℙ^{2}), the associated boundary manifold M is a Waldhausen graph manifold. We show in Theorem 3.5 that the “doubling” formula (1.1) holds for a reducible curve V if and only if all its components are rational curves.

Cohomology rings of graph manifolds (with ℤ_{2} coefficients) have been the object of substantial recent study, see Aaslepp, et.al. [1]. For those graph manifolds which arise as boundary manifolds of arrangements of rational curves in ℂℙ^{2}, our methods, together with Cogolludo’s computation of the cohomology ring of the complement of such an arrangement in [4], provide an efficient alternative.

1.4. LS category and topological complexity. Let X^{I} be the space of continuous paths from the unit interval to X, and let π: X^{I} → X × X be the map sending a path to its endpoints. In [12], Farber defines the topological complexity of X, denoted by tc(X), to be the smallest integer k such that X ×X can be covered by k open sets, over each of which π has a section. This numerical invariant, which depends only on the homotopy type of X, is related to the Lusternik-Schnirelmann category by the inequalities cat(X) ≤ tc(X) ≤ 2 cat(X) − 1. Computing the topological complexity of X is crucial to solving the motion planning problem for the space X, see [12].

The topological complexity tc(X) admits a cohomological lower bound in terms of the zero-divisor length of H^{∗}(X; 𝕜), similar to the well known cup-length lower bound for cat(X). In the case when X = X(A) is the complement of a hyperplane arrangement, explicit computations of tc(X) were carried out by Farber and Yuzvinsky [13]. In Section 5, we compute the topological complexity of the boundary manifold M = M(A) for various classes of hyperplane arrangements, using our description of the cohomology ring of M and results from [12]. In particular, we show that the difference tc(M) − cat(M) can be made arbitrarily large, see Corollary 5.7.

1.5. Resonance. We conclude with a comparison of certain ring-theoretic invariants of the cohomology ring of the complement to those of the cohomology ring of the boundary manifold.

Suppose A is a finite-dimensional, graded, connected algebra over an algebraically closed field 𝕜 of characteristic 0. For each a ∈ A^{1}, multiplication by a defines a chain complex (A, a). The resonance varieties of A are the jumping loci for the cohomology of these complexes: R_{d}^{k}(A) = {a ∈ A^{1} | dim_{𝕜} H^{k}(A, a) ≥ d}.

In Section 6, we study the resonance varieties of the trivial extension, D(A) = A⋉Ā. As an application, we obtain information about the structure of the resonance varieties of the boundary manifold of a hyperplane arrangement A. Let A = H^{∗}(X(A); 𝕜) be the Orlik-Solomon algebra. It is well known that the components of the resonance varieties R_{d}^{k}(X(A)) = R_{d}^{k}(A) are linear subspaces of A^{1} = 𝕜^{n}. The behavior of the resonance varieties R_{d}^{k}(M(A)) = R_{d}^{k}(D(A)) is dramatically different. Indeed, we produce examples of arrangements for which the resonance varieties of the boundary manifold contain singular, irreducible components of arbitrarily high degree, see Corollary 6.6.

2. The boundary manifold

In this section, we introduce our main character, the boundary manifold of an (algebraic) hypersurface in complex projective space. We then compute its homology groups in terms of those of the complement to the hypersurface, and make a remark on the homotopy groups.

2.1. Thickenings. According to C.T.C. Wall [30], a thickening of a finite, k-dimen-sional CW-complex Y is a compact, m-dimensional manifold with boundary W^{m}, which is simply homotopy equivalent to Y. Such a thickening always exists, as soon as m ≥ 2k + 1: Embed Y as a sub-polyhedron in ℝ^{m}, and take W to be a smooth, regular neighborhood of Y.

Let M = ∂W be the boundary of the thickening W. In general, the homotopy type of the boundary manifold M is not determined by the homotopy type of Y. For example, both ℂℙ^{2} × D^{m−4} and the normal disk bundle of ℂℙ^{2} ⊂ S^{m} are thickenings of ℂℙ^{2}, but their boundary manifolds are not homotopy equivalent, see Lambrechts [20]. Nevertheless, if M is orientable, and m ≥ 2(k + 1), then the cohomology ring H^{∗}(M; ℤ) is completely determined by H^{∗}(Y; ℤ), by Poincaré duality and degree considerations.

2.2. Projective hypersurfaces. Let V be a hypersurface in ℂℙ^{ℓ}, given as the zero locus of a homogeneous polynomial f = f(x), where x = (x_{0}, …, x_{ℓ}) are homogeneous coordinates on ℂℙ^{ℓ}. A (closed) regular neighborhood, N, of V in ℂℙ^{ℓ} can be constructed either by triangulation, or by levels sets. In the first approach, triangulate ℂℙ^{ℓ} with V as a subcomplex, and take N to be the closed star of V in the second barycentric subdivision. In the second, define ϕ: ℂℙ^{ℓ} → ℝ by ϕ(x) = |f(x)|^{2} / ||x||^{2d}, where d = deg f, and take N = ϕ^{−1}([0, δ]), for sufficiently small δ > 0. As shown by Durfee [9], these constructions yield isotopic neighborhoods, independent of the choices made.

Clearly, N is a thickening of V. Hence, we may define the boundary manifold of V to be

(2.1)  M = ∂N.

This is a compact, orientable, smooth manifold of dimension 2ℓ − 1. If ℓ = 1, then V consists of, say, n points on the sphere, and so M is a disjoint union of n circles. If ℓ > 1, then M is connected. Here is a simple illustration.

Example 2.1. Let V be a pencil of n + 1 hyperplanes in ℂℙ^{ℓ}, ℓ ≥ 2, defined by the polynomial f = x_{0}^{n+1} − x_{1}^{n+1}. In this case, X may be realized as the complement of n parallel hyperplanes in ℂ^{ℓ}, and so it is homotopy equivalent to the n-fold wedge ⋁^{n} S^{1}. On the other hand, ℂℙ^{ℓ} \ N = (D^{2} \ {n disks}) × D^{2(ℓ−1)}; hence M is diffeomorphic to the n-fold connected sum #^{n}S^{1} × S^{2(ℓ−1)}.

Note that the complement X = ℂℙ^{ℓ} \ V is homotopy equivalent to the interior of the manifold with boundary ℂℙ^{ℓ} \ N^{∘}, and that M = ∂(ℂℙ^{ℓ} \ N^{∘}). Also observe that, while N is a thickening of V, the cohomology ring of M = ∂N is not a priori determined by that of V.

2.3. Cohomology groups. We now analyze in detail the cohomology groups of M. We start by relating these cohomology groups to those of X. Throughout this section, we use integral coefficients, unless otherwise noted.

Proposition 2.2. Let V be a hypersurface in ℂℙ^{ℓ}, with complement X and boundary manifold M. For each 0 ≤ k ≤ 2ℓ − 1, there is an exact sequence

(2.2)  \xymatrix{0 \ar[r] & H^{k}(X) \ar[r] & H^{k}(M) \ar[r] & H^{k+1}(X, M) \ar[r] & 0} .

Moreover, H^{k+1}(X, M) ≅ H_{2ℓ−k−1}(X), and the sequence splits, except possibly when k = ℓ.

Proof. Let i: X → M and j: V → ℂℙ^{ℓ} be the inclusion maps. Consider the following commuting diagram, with rows long exact sequences of pairs, and vertical isomorphisms given by the homotopy equivalence V ↪ N̄ and excision, respectively:

(2.3)  [figure: H^{k}(ℂℙ^{ℓ}, V); H^{k}(ℂℙ^{ℓ}); j^{∗}; H^{k}(V); H^{k+1}(ℂℙ^{ℓ}, V); H^{k}(ℂℙ^{ℓ}, N); ≅; ≅; H^{k}(ℂℙ^{ℓ}); =; H^{k}(N); ≅; H^{k+1}(ℂℙ^{ℓ}, N); ≅; ≅; H^{k}(X, M); H^{k}(X); i^{∗}; H^{k}(M); H^{k+1}(X, M)]

By Lefschetz Duality, H^{k}(ℂℙ^{ℓ} \ N^{∘}, M) ≅ H_{2ℓ−k}(ℂℙ^{ℓ} \ N^{∘}) for each k ≥ 0. Since X ≃ ℂℙ^{ℓ} \ N^{∘}, we obtain H^{k}(X, M) ≅ H_{2ℓ−k}(X).

By the Lefschetz hyperplane section theorem, the map j^{∗}: H^{k}(ℂℙ^{ℓ}) → H^{k}(V) is an isomorphism for k ≤ ℓ − 2 and a monomorphism for k = ℓ − 1. Chasing the diagram, we find that sequence (2.2) is exact, for each k ≤ ℓ − 2.

Now, it is well known that X is a Stein space, and thus has the homotopy type of a CW-complex of dimension at most ℓ. In particular, H^{k}(X) = 0 for k > ℓ, and H_{ℓ}(X) is finitely generated and torsion-free. Furthermore, the boundary map H^{k}(ℂℙ^{ℓ}, V; ℚ) → H^{k}(ℂℙ^{ℓ}; ℚ) is the zero map; see [7, p. 146]. By Lefschetz duality, H^{ℓ}(X, M) ≅ H_{ℓ}(X). Hence the map H^{ℓ}(X, M) → H^{ℓ}(X) is the zero map. We conclude that sequence (2.2) is exact for k ≥ ℓ − 1, as well.

For k < ℓ − 1 or k > ℓ, one of the side terms in (2.2) vanishes, so obviously the sequence splits. For k = ℓ−1, we know H_{ℓ}(X) is torsion-free, so (2.2) splits again. □

Corollary 2.3. The Betti numbers of M are given by b_{k}(M) = b_{k}(X) + b_{2ℓ−k−1}(X). Hence, the Poincaré polynomials of M and X are related by:

(2.4)  P (M, t) = P (X, t) + t^{2ℓ−1} · P (X, t^{−1}).

Proposition (2.2) determines the cohomology groups of M in terms of the (co)homology groups of X, except possibly the torsion in H^{ℓ}(M). By the Universal Coefficient Theorem, this torsion fits into the short exact sequence

(2.5)  \xymatrix{0 \ar[r] & Tors(H_{ℓ−1}(X)) \ar[r] & Tors(H^{ℓ}(M)) \ar[r] & Tors(H_{ℓ−1}(X)) \ar[r] & 0} .

This sequence may or may not split. As we shall see in examples below, both possibilities can occur.

Example 2.4. Let V be a smooth algebraic hypersurface in ℂℙ^{ℓ} of degree d. In this case, N can be taken to be a tubular neighborhood of V, diffeomorphic to the unit normal disk bundle ν. Hence M is the total space of the S^{1}-bundle over V with Euler number e = c_{1}(ν)[V].

In particular, if ℓ = 2, then V is a curve of genus g = \binom{d−1}{2}, with e = d^{2}. Hence, by the Gysin sequence, H^{2}(M) = ℤ_{d^{2}}. On the other hand, H_{1}(X) = ℤ_{d}. Thus, in this instance, (2.5) is a non-split exact sequence, of the form 0 → ℤ_{d} → ℤ_{d^{2}} → ℤ_{d} → 0.

2.4. Affine hypersurfaces and Milnor fibrations. Let V_{0} ⊂ ℂ^{ℓ} be an affine hypersurface, defined by the vanishing of a polynomial f_{0} = f_{0}(x_{1}, …, x_{ℓ}) of degree n. Let V be the projective closure of V_{0}, defined by the vanishing of the homogeneous polynomial f(x_{0}, x_{1}, …, x_{ℓ}) = x_{0}^{n+1} · f_{0}(x_{1}/x_{0}, …, x_{ℓ}/x_{0}). Clearly, ℂℙ^{ℓ} \ V = ℂ^{ℓ} \ V_{0}.

If f_{0} itself is homogeneous, then f(x_{0}, x_{1}, …, x_{ℓ}) = x_{0}·f_{0}(x_{1}, …, x_{ℓ}). Moreover, we can take the regular neighborhood N of V to be the union of a regular neighborhood of V_{0}, say N_{0}, with a tubular neighborhood of the hyperplane at infinity (after rounding corners). Thus, ℂℙ^{ℓ} \ N^{∘} is diffeomorphic to D^{2ℓ} \ (D^{2ℓ} ∩ N_{0}^{∘}), and so

M = (S^{2ℓ−1} \ (S^{2ℓ−1} ∩ N_{0})) ⋃ D^{2ℓ} ∩ ∂N_{0} .

As shown in [22], each of the two sides of the above decomposition is diffeomorphic to the total space of the Milnor fibration, F → Y → S^{1}, determined by the homogeneous polynomial f_{0}. Thus, M is the double of the manifold with boundary Y:

(2.6)  M = ∂(Y × I) = Y ∪_{∂Y} Y.

Furthermore, M fibers over the circle, with fiber the double of F.

Notice that, in this situation, the exact sequence (2.2) always splits. Indeed, the inclusion Y → X is a homotopy equivalence, which factors through the inclusions Y → M and i: M → X. Thus, i^{∗}: H^{∗}(X) → H^{∗}(M) is a split injection.

Example 2.5. Let f = x_{0}x_{1} ⋯ x_{ℓ} be the polynomial defining the Boolean arrangement in ℂℙ^{ℓ}. Then M = S^{ℓ−1} × T^{ℓ}, where T^{ℓ} is the ℓ-torus; see [7, Example 2.29].

Example 2.6. Let f = x_{0}(x_{1}^{n} − x_{2}^{n}) be the polynomial defining a near pencil of n + 1 lines in ℂℙ^{2}. In this case, Y admits a fibration over the circle (different from the Milnor fibration!), with fiber D^{2} \{n − 1 disks}, and monodromy a Dehn twist about the boundary D^{2}. It follows that M = S^{1} × Σ_{n−1}, where Σ_{g} denotes a surface of genus g.

Example 2.7. More generally, let f = x_{0}(x_{1}^{n_{1}} − y_{1}^{n_{1}}) ⋯ (x_{k}^{n_{k}} − y_{k}^{n_{k}}), with n_{i} ≥ 2. Then M = T^{k} × (#^{m}T^{k} × S^{2k−1}), where m = ∏_{i=1}^{k}(n_{i} − 1).

Example 2.8. Let f = x_{0}(x_{1}^{2} + ⋯ + x_{ℓ}^{2}). In this case, the Milnor fiber F of f_{0} = x_{1}^{2} + ⋯ + x_{ℓ}^{2} is diffeomorphic to the unit disk bundle of S^{ℓ−1}. Thus, M fibers over S^{1} with fiber E, where S^{ℓ−1} → E → S^{ℓ−1} is the bundle with Euler number 1 − (−1)^{ℓ}.

Now assume ℓ is odd and ℓ > 1. A computation with the Wang sequence for the bundle F → Y → S^{1} shows that H_{ℓ−1}(X) = ℤ_{2}; see [8, Example 3.2]. Hence, (2.5) is a split exact sequence, of the form 0 → ℤ_{2} → ℤ_{2} ⊕ ℤ_{2} → ℤ_{2} → 0.

2.5. On asphericity of the boundary. If V is a hypersurface in ℂℙ^{ℓ}, the inclusion map M → X is an (ℓ − 1)-equivalence, see for instance [7, Proposition 2.31]; in particular, π_{i}(M) ≅ π_{i}(X), for i < ℓ − 1. A natural question arises: Is M aspherical? In other words, do all the higher homotopy groups of M vanish?

If ℓ = 2, the manifold M^{3} is a graph manifold in the sense of Waldhausen. With a few exceptions (such as lens spaces), manifolds of this type are aspherical. In higher dimensions, though, this never happens.

Proposition 2.9. Let M be the boundary manifold of a hypersurface in ℂℙ^{ℓ}. If ℓ ≥ 3, then M is not aspherical.

Proof. Let π = π_{1}(M) be the fundamental group of M. Since the inclusion i: M → X is an (ℓ − 1)-equivalence, and since ℓ ≥ 3, the induced map i_{∗}: π_{1}(M) → π_{1}(X) is an isomorphism. Let g: X → K(π, 1) be a classifying map for the universal cover X̃ → X. By definition, g_{∗}: π_{1}(X) → π is an isomorphism. Hence, the composite map g ∘ i: M → K(π, 1) is a classifying map for M̃ → M.

Now suppose M is aspherical. Then the map g ∘ i: M → K(π, 1) must be a homotopy equivalence, since it induces an isomorphism on fundamental groups. Consequently, (g ∘ i)^{∗}: H^{2ℓ−1}(π) → H^{2ℓ−1}(M) = ℤ is an isomorphism. On the other hand, i^{∗}: H^{2ℓ−1}(X) → H^{2ℓ−1}(M) is the zero map, since the CW-complex X has dimension at most ℓ. This contradiction finishes the proof. □

3. The cohomology ring of the boundary manifold

Let V ⊂ ℂℙ^{ℓ} be a projective hypersurface, with complement X = ℂℙ^{ℓ} \ V, and associated boundary manifold M. In this section, we determine the structure of the cohomology ring H^{∗}(M; ℂ) under certain conditions. These conditions are given below in terms of the mixed Hodge structure on H^{∗}(X; ℂ), respectively H^{∗}(V; ℂ). First, we discuss the relevant algebraic structure.

3.1. The double of a graded ring. If A is a ring and B is an A-bimodule, the trivial extension of A by B, written A ⋉ B, is the additive group A ⊕ B, with multiplication (a, b)(a′, b′) = (aa′, a · b′ + b · a′), see [27]. Note that A ≅ {(a, 0)} is a subring of the trivial extension, and that B ≅ {(0, b)} is a square-zero ideal.

Now let A = ⨁_{k=0}^{ℓ} A^{k} be a finite-dimensional graded ring over a base ring R. We will assume R is a commutative ring with 1, and all the graded pieces A^{k} are finitely generated free R-modules. Define the double D(A) of A to be the trivial extension of A by the graded A-bimodule Ā = ⨁_{k=ℓ−1}^{2ℓ−1} Ā^{k}, where Ā^{k} = Hom(A^{2ℓ−k−1}, R), and the A-bimodule structure is given by a·b(x) = b(xa) and b·a(x) = b(ax) for a, x ∈ A and b ∈ Ā. If A is a graded commutative ring, it is readily checked that D(A) = A ⋉ Ā is a graded commutative ring as well.

Let μ: A ⊗ A → A, μ(a, a′) = aa′, denote the multiplication map of the ring A. Then the multiplication map D(μ): D(A) ⊗ D(A) → D(A) of the double restricts to μ on A ⊗ A and vanishes on Ā ⊗ Ā, while on A ⊗ Ā it vanishes, except for

(3.1)  D(μ)(a_{j}^{k}, ā_{p}^{r}) = ∑_{i} μ_{i,j,p} ā_{i}^{r−k}, if μ(a_{i}^{r−k}, a_{j}^{k}) = ∑_{p} μ_{i,j,p} a_{p}^{r},

where {a_{j}^{k}} is a (fixed) homogeneous basis for A^{k} and {ā_{j}^{k}} is the dual basis for Ā^{2ℓ−k−1} = Hom(A^{k}, ℂ). The proof of the next result is straightforward.

Proposition 3.1. The doubling construction is functorial. In particular, if A_{1} and A_{2} are isomorphic as graded rings, then D(A_{1}) and D(A_{2}) are isomorphic as graded rings.

Let b_{k}(A) = rank A^{k} be the Betti numbers of A, and let Hilb(A, t) = ∑_{k=0}^{ℓ} b_{k}(A)·t^{k} be its Hilbert series. Then:

(3.2)  Hilb(D(A), t) = Hilb(A, t) + t^{2ℓ−1} · Hilb(A, t^{−1}).

In particular, if A is connected (i.e., b_{0}(A) = 1), then D(A) is an Artin-Gorenstein ring.

Recall from Proposition 2.3 that the cohomology of the boundary manifold M is additively given by H^{q}(M) ≅ H^{q}(X) ⊕ H^{q+1}(X, M). Let R be a coefficient ring.

Theorem 3.2. Assume H^{∗}(X; R) is a free R-module. If H^{∗}(X, M; R) is a squarezero subring of H^{∗}(M; R), then H^{∗}(M; R) ≅ D(H^{∗}(X; R)) as graded rings.

Proof. Recall that the inclusion i: M → X induces an injection i^{∗}: H^{∗}(X) → H^{∗}(M) in cohomology. Let A = i^{∗}(H^{∗}(X; R)), and note that A is a subring of H^{∗}(M; R). Comparing formulas (2.4) and (3.2), and using the R-freeness assumption for H^{∗}(X; R), we see that H^{∗}(M; R) and D(A) = A⋉Ā are additively isomorphic. So it suffices to show that the cup-product structure in H^{∗}(M; R) coincides with the multiplicative structure in D(A). This is clearly the case for the restriction to the common subring A.

For simplicity, let us suppress the coefficient ring R from the notation. Fix a generator ω ∈ H^{2ℓ−1}(M), and note that ω ∉ A. For each q, 0 ≤ q ≤ ℓ, let {a_{1}^{q}, …, a_{b_{q}}^{q}} be a basis for A_{q} ≅ H^{q}(X), where b_{q} = b_{q}(A). By Poincaré duality, there are linearly independent elements ā_{1}^{q}, …, ā_{b_{q}}^{q} in Hq̄(M) so that a_{i}^{q}∪ā_{j}^{q} = δ_{i,j}ω, where q̄ = 2ℓ−q−1 and δ_{i,j} is the Kronecker index. Since A is a subring of H^{∗}(M) and ω ∉ A, the dual classes ā_{i}^{q} are also not in A. Identifying H^{q}(M) = H^{q}(X) ⊕ H^{q+1}(X, M), it follows that {ā_{1}^{q}, …, ā_{b_{q}}^{q}} forms a basis for H^{q̄+1}(X, M) ⊂ Hq̄(M). Consequently, H^{q}(M)

has basis {a_{1}^{q}, …, a_{b_{q}}^{q}, āq̄ ; 1, …, āq̄ ; bq̄}.

By hypothesis, we have ā_{i}^{p} ∪ ā_{j}^{q} = 0 for all p, q and i, j. It remains to consider the cup-product a_{j}^{p} ∪ ā_{k}^{q} ∈ H^{p+q̄}(M). If p = 0, then a_{j}^{p} ∪ ā_{k}^{q} = 1 ∪ ā_{k}^{q} = ā_{k}^{q}. If p > q, then a_{j}^{p} ∪ ā_{k}^{q} = 0. So assume that 0 < p ≤ q ≤ ℓ, which implies that p + q̄ ≥ ℓ.

If p + q̄ > ℓ, then a_{j}^{p} ∪ ā_{k}^{q} = ∑_{i=1}^{b_{q−p}} c_{i,j,k}ā_{i}^{q−p} for some constants c_{i,j,k}. Write the multiplication in A ≅ H^{∗}(X) as a_{i}^{r} · a_{j}^{p} = ∑_{l=1}^{b_{r+p}} μ_{i,j,l}a_{l}^{r+p}, and note that μ_{j,i,l} = (−1)^{rp}μ_{i,j,l} in this instance. For a fixed i, cupping with a_{i}^{q−p} yields a_{i}^{q−p} ∪ a_{j}^{p} ∪ ā_{k}^{q} = c_{i,j,k}ω. Since

a_{i}^{q−p} ∪ a_{j}^{p} ∪ ā_{k}^{q} = (∑_{l=1}^{b_{q−p}} μ_{i,j,l}a_{l}^{q}) ∪ ā_{k}^{q} = μ_{i,j,k}ω,

we must have c_{i,j,k} = μ_{i,j,k}, and so a_{j}^{p} ∪ ā_{k}^{q} = ∑_{i=1}^{b_{q−p}} μ_{i,j,k}ā_{i}^{q−p}.

We are left with the case p + q̄ = ℓ, that is, p = 1 and q = ℓ. We then have a_{j}^{1} ∪ ā_{k}^{ℓ} = ∑_{i=1}^{b_{ℓ−1}} c_{i,j,k}ā_{i}^{ℓ−1} + ∑_{i=1}^{b_{ℓ}} d_{i,j,k}a_{i}^{ℓ} for some constants c_{i,j,k} and d_{i,j,k}. Since 0 = ā_{i}^{ℓ} ∪ a_{j}^{1} ∪ ā_{k}^{ℓ} = ±d_{i,j,k}ω, we have d_{i,j,k} = 0. Then, a calculation as above yields c_{i,j,k} = μ_{i,j,k}, where a_{i}^{ℓ−1} · a_{j}^{1} = ∑_{k=1}^{b_{ℓ}} μ_{i,j,k}a_{k}^{ℓ}. Thus, a_{j}^{1} ∪ ā_{k}^{ℓ} = ∑_{i=1}^{b_{ℓ−1}} μ_{i,j,k}ā_{i}^{ℓ−1}.

Notice that these calculations show that the square-zero subring H^{∗}(X, M) is, in fact, an ideal in H^{∗}(M). Using these calculations, and formula (3.1), it is readily checked that the cup-product structure in H^{∗}(M) coincides with the multiplicative structure in D(H^{∗}(X)). □

The freeness assumption from Theorem 3.2 holds, for example, when R = ℤ and H^{∗}(X) is torsion-free, or when R = 𝕜 is a field. This assumption is necessary, as illustrated by the smooth plane curve of degree d > 1 from Example 2.4. Indeed, for such a curve, H^{2}(M; ℤ) = ℤ_{d^{2}} does not split as a direct sum, and so H^{∗}(M; ℤ) ≅ D(H^{∗}(X; ℤ)), even though H^{∗}(X, M; ℤ) is a square-zero subring of H^{∗}(M; ℤ), by degree considerations.

3.2. Hodge structures. Now we pursue conditions which insure that the hypotheses of Theorem 3.2 hold. These conditions will be given in terms of mixed Hodge structures. For the rest of this section, we shall take coefficients in the ring R = ℂ.

If V is a smooth projective variety, then, by a classical theorem of Hodge, each cohomology group H^{m}(V) admits a pure Hodge structure of weight m. That is, for H = H^{m}(V), there is a direct sum decomposition

(3.3)  H = ⨁_{p+q=m} H^{p,q}, where \overline{H^{p,q}} = H^{q,p} (complex conjugation).

If X is a quasi-projective variety, then, by a well known theorem of Deligne, each cohomology group of X admits a mixed Hodge structure. That is, for each k, there is an increasing weight filtration

(3.4)  0 = W_{−1} ⊂ W_{0} ⊂ ⋯ ⊂ W_{2k} = H^{k}(X),

such that each quotient W_{m}/W_{m−1} of the subspaces W_{m} = W_{m}(H^{k}(X)) of H^{k}(X) admits a pure Hodge structure of weight m as in (3.3).

The following properties of the weight filtration will be of use. See [26] for further details.

(1) If X is projective, then W_{k} = H^{k}(X) for each k.

(2) If X is smooth, then 0 = W_{k−1} ⊂ H^{k}(X) for each k.

(3) For any smooth compactification ι: X → X̄ of X, W_{k} = ι^{∗}(H^{k}(X̄)) for each k.

(4) The weight filtration is functorial. For an algebraic map f: X → Y, the induced homomorphism f^{∗} strictly preserves the filtration: If x ∈ W_{m}(H^{k}(X)) is in the image of f^{∗}, there is an element y ∈ W_{m}(H^{k}(Y)) with f^{∗}(y) = x.

It follows from work of Durfee and Hain [11] that the cohomology of the boundary manifold M of a projective hypersurface V admits a mixed Hodge structure. Furthermore, the cup-product of H^{∗}(M) is a morphism of mixed Hodge structures, and the top cohomology H^{2ℓ−1}(M) is of weight 2ℓ (and type (ℓ, ℓ)).

Theorem 3.3. Let V be a hypersurface in ℂℙ^{ℓ} with complement X, and associated boundary manifold M. If V is irreducible, then H^{∗}(M; ℂ) ≅ D(H^{∗}(X); ℂ) as graded algebras.

Proof. If ℓ = 1, then V is a point in ℂℙ^{1}. In this instance, X is contractible, M is a circle, and it is readily checked that H^{∗}(M) ≅ D(H^{∗}(X)).

So we may assume that ℓ ≥ 2. By Theorem 3.2, it suffices to show that H^{∗}(X, M) is a square-zero subalgebra of H^{∗}(M). For this, it is enough to show that u∪v = 0 for u ∈ H^{r+1}(X, M) ⊂ H^{r}(M) and v ∈ H^{s+1}(X, M) ⊂ H^{s}(M), where ℓ − 1 ≤ r, s ≤ ℓ.

Recall that, for k ≤ 2ℓ−2, the inclusion j: V → ℂℙ^{ℓ} induces a monomorphism in kth cohomology. From diagram (2.3), we see that H^{k+1}(X, M) is isomorphic to H_{0}^{k}(V), the primitive cohomology of V, given by H_{0}^{k}(V) = coker[j^{∗}: H^{k}(ℂℙ^{ℓ}) → H^{k}(V)].

It is known that the connecting homomorphism in the long exact sequence of the pair is weight-preserving, see [26]. This fact, and the properties recorded above, imply that all cohomology classes in H^{k+1}(X, M) ≅ H_{0}^{k}(V) (for k ≤ 2ℓ − 2) are of weight at most k.

Now take u ∈ H^{r+1}(X, M) ⊂ H^{r}(M) and v ∈ H^{s+1}(X, M) ⊂ H^{s}(M) as above. If r = s = ℓ, then clearly u ∪ v = 0. If, say, r = ℓ − 1 and s = ℓ, then u is of weight at most ℓ − 1 and v is of weight at most ℓ. Hence, u ∪ v is of weight at most 2ℓ − 1 in H^{2ℓ−1}(M). But W_{2ℓ−1}(H^{2ℓ−1}(M)) = 0 by the results of Durfee and Hain noted above. So we must have u ∪ v = 0.

Finally, if r = s = ℓ − 1, then u ∪ v is of weight at most 2ℓ − 2 in H^{2ℓ−2}(M). Since V is irreducible, H^{1}(X) = 0, the map j^{∗}: H^{2ℓ−2}(ℂℙ^{ℓ}) → H^{2ℓ−2}(V) is an isomorphism, and H^{2ℓ−1}(X, M) ≅ H_{0}^{2ℓ−2}(V) = 0. If ℓ = 2, then all non-trivial classes in H^{2}(M) = H^{2}(X) are of weight at least 3 by Poincaré duality, since all classes in H^{1}(M) ≅ H_{0}^{1}(V) = H^{1}(V) are of weight at most 1. If ℓ ≥ 3, then H^{2ℓ−2}(M) = H^{2ℓ−2}(X) = 0 since X has the homotopy type of an ℓ-dimensional complex. It follows that u ∪ v = 0 in either case. □

Theorem 3.4. Let V be a hypersurface in ℂℙ^{ℓ} with complement X, and associated boundary manifold M. If W_{ℓ+1}(H^{ℓ}(X; ℂ)) = 0, then H^{∗}(M; ℂ) ≅ D(H^{∗}(X; ℂ)).

Proof. If ℓ = 1, then V is a union of, say, n + 1 points in ℂℙ^{1}. In this instance, X is homotopic to a bouquet of n circles, M is a disjoint union of n + 1 circles, and it is readily checked that H^{∗}(M) ≅ D(H^{∗}(X)).

If ℓ ≥ 2, by Theorem 3.2, it suffices to show that H^{∗}(X, M) is a square-zero subalgebra of H^{∗}(M). For this, as above, it is enough to show that u ∪ v = 0 for u ∈ H^{r+1}(X, M) ⊂ H^{r}(M) and v ∈ H^{s+1}(X, M) ⊂ H^{s}(M), where (r, s) = (ℓ−1, ℓ) or (r, s) = (ℓ−1, ℓ−1). By Poincaré duality, there are elements a, b ∈ H^{∗}(X) ⊂ H^{∗}(M) so that a ∪ u = b ∪ v = ω ∈ H^{2ℓ−1}(M).

If (r, s) = (ℓ − 1, ℓ), then a ∈ H^{ℓ}(X) and b ∈ H^{ℓ−1}(X). Then, since X is smooth, W_{ℓ−2}(H^{ℓ−1}(X)) = 0, and b is of weight at least ℓ − 1. Since W_{ℓ+1}(H^{ℓ}(X)) = 0 by hypothesis, a is of weight at least ℓ + 2. Since ω is of weight 2ℓ, is follows that u is of weight at most ℓ − 2 and v is of weight at most ℓ + 1. Consequently, u ∪ v is of weight at most 2ℓ − 1 in H^{2ℓ−1}(M), which is pure of weight 2ℓ. Hence u ∪ v = 0. If (r, s) = (ℓ − 1, ℓ − 1), then a, b ∈ H^{ℓ}(X) are both of weight at least ℓ + 2, and a similar argument shows that u ∪ v = 0. □

3.3. Plane algebraic curves. For an arbitrary projective hypersurface, the cohomology ring of the boundary manifold (with ℂ coefficients) need not admit the structure of a double. We illustrate this phenomenon in dimension two.

Theorem 3.5. Let V = V_{1} ∪ ⋯ ∪ V_{k} be a reducible algebraic curve in ℂℙ^{2}, with complement X and boundary manifold M. Then H^{∗}(M; ℂ) ≅ D(H^{∗}(X; ℂ)) if and only if all the irreducible components V_{j} are rational curves.

Proof. If an irreducible component V_{j} of V is a rational curve, then the normalization of V_{j} has genus 0. It follows that all nontrivial cohomology classes in H_{0}^{1}(V_{j}) = H^{1}(V_{j}) are of weight 0. Using this, an inductive argument with the Mayer-Vietoris sequence reveals that the same holds for H_{0}^{1}(V) = H^{1}(V). It follows that H^{2}(X) ≅ H_{0}^{1}(V) is pure of weight 4, see [7, p. 246]. So H^{∗}(M) ≅ D(H^{∗}(X)) by Theorem 3.4.

Conversely, if an irreducible component V_{j} of V is not a rational curve, then the degree of V_{j} is necessarily at least three. In this situation, H^{1}(V) = H_{0}^{1}(V) contains nontrivial classes of weights 0 and 1 (see [10]). It follows that H^{2}(X) contains classes of weights 3 and 4 (see [7]). (Note that the weight condition of Theorem 3.4 fails.) In this instance, it is readily checked that the cup-product H_{0}^{1}(V) ⊗ H_{0}^{1}(V) → H_{0}^{2}(V) is nontrivial. Hence, H_{0}^{∗}(V) ⊂ H^{∗}(M) is not a square-zero subalgebra, compare Theorem 3.2, and H^{∗}(M) ≅ D̸(H^{∗}(X)). □

Suppose V is an arrangement of rational curves in ℂℙ^{2}, with complement X, and boundary manifold M. A presentation for the cohomology ring H^{∗}(X; ℂ) was given in [4, Theorem 0.4]. Our Theorem 3.5 can now be used to compute the cohomology ring H^{∗}(M; ℂ).

4. Hyperplane arrangements

Let A be an arrangement of hyperplanes in ℂℙ^{ℓ}. For each hyperplane H of A, let f_{H} be a linear form with H = {f_{H} = 0}. Then f = Q(A) = ∏_{H∈A} f_{H} is a defining polynomial for A, the hypersurface V = V (A) is given by V = f^{−1}(0) = ⋃_{H∈A} H, and the complement of the arrangement is X = X(A) = ℂℙ^{ℓ} \ V.

4.1. Boundary manifold of an arrangement. Let M = M(A) be the boundary manifold of the hypersurface V = V (A). The next theorem expresses the (integral) cohomology ring of M in terms of the Orlik-Solomon algebra A = A(A) = H^{∗}(X(A); ℤ) of the arrangement A.

Theorem 4.1. Let A be an arrangement of hyperplanes in ℂℙ^{ℓ} with complement X and associated boundary manifold M. Then H^{∗}(M; ℤ) ≅ D(H^{∗}(X; ℤ)).

Proof. For any hyperplane arrangement A, the cohomology H^{k}(X, ℂ) is pure of weight 2k, that is, the weight filtration takes the form 0 = W_{2k−1} ⊂ W_{2k} = H^{k}(X; ℂ), for every k, see Shapiro [29], and also Kim [18]. Hence, by Theorem 3.4, we have H^{∗}(M; ℂ) ≅ D(H^{∗}(X; ℂ)).

Let A = H^{∗}(X; ℤ) be the integral Orlik-Solomon algebra of A. It is well known that A = ⨁_{k=0}^{ℓ} A^{k} is torsion-free. Let D(A) = A ⋉ Ā be the integral double of A, the trivial extension of A by Ā = ⨁_{k=ℓ−1}^{2ℓ−1} Hom_{ℤ}(A^{2ℓ−k−1}, ℤ), with A-bimodule structure as given in §3.1. Since A = H^{∗}(X; ℤ) is torsion-free, H^{∗}(M; ℤ) is also torsion-free, see Proposition 2.3. Since H^{∗}(M; ℂ) ≅ D(H^{∗}(X; ℂ)), it follows that H^{∗}(M; ℤ) ≅ D(A). □

Let L(A) be the intersection poset of the arrangement A, the set of all nonempty intersections of elements of A, ordered by reverse inclusion. By the Orlik-Solomon theorem (see [24, 32]), the integral cohomology ring of X(A) is determined by L(A). Our next result shows that the cohomology of M(A) is determined by L(A) and the ambient dimension.

Corollary 4.2. If A_{1} and A_{2} are hyperplane arrangements in ℂℙ^{ℓ} with L(A_{1}) ≅ L(A_{2}), then H^{∗}(M(A_{1}); ℤ) ≅ H^{∗}(M(A_{2}); ℤ).

Proof. By the Orlik-Solomon theorem, A(A_{1}) ≅ A(A_{2}). Proposition 3.1 implies that the (integral) doubles are isomorphic. Thus, by Theorem 4.1, H^{∗}(M(A_{1}); ℤ) ≅ H^{∗}(M(A_{2}); ℤ). □

4.2. Computing cup products. We now exhibit an explicit basis for the cohomology of the boundary manifold of an arrangement, and compute cup products in that basis. Write A = {H_{0}, H_{1}, …, H_{n}}, and designate H_{0} as the hyperplane at infinity in ℂℙ^{ℓ}. Let A′ = {H_{1}, …, H_{n}} be the corresponding affine arrangement in ℂ^{ℓ} = ℂℙ^{ℓ} \ H_{0}. Notice that A is the projective closure of A′.

The rank of the affine arrangement A′ is the maximal number of linearly independent hyperplanes in A′. If A′ ⊂ ℂ^{ℓ} has rank ℓ, then A′ is said to be essential. Observe that the projective arrangement A ⊂ ℂℙ^{ℓ} is essential if it contains ℓ + 1 independent hyperplanes. For an arrangement of rank r, it is well known that the Betti numbers, b_{k}(X), of the complement are nonzero for all k, 0 ≤ k ≤ r. See [24] as a general reference.

Order the hyperplanes of A′ = {H_{1}, …, H_{n}} by their indices. A circuit is an inclusion-minimal dependent set of hyperplanes (in A′), and a broken circuit is a set S for which there exists j < min(S) so that {H_{j}} ∪ {H_{i} | i ∈ S} is a circuit. Let nbc = nbc(A′) denote the collection of subsets I ⊂ [n] for which ⋂_{i∈I} H_{i} ≠ ∅ and I contains no broken circuits. If the rank of A′ is r, then all elements of nbc are of cardinality at most r. Note also that ∅ ∈ nbc.

Clearly, the complement of A in ℂℙ^{ℓ} is diffeomorphic to the complement of A′ in ℂ^{ℓ}. The integral cohomology of X = X(A) = X(A′) is isomorphic to the Orlik-Solomon algebra A = A(A′), a quotient of an exterior algebra on n generators in degree 1. A basis for A is indexed by the set nbc; denote this basis for A by {a_{I} | I ∈ nbc}. If |I| = k, then a_{I} ∈ A^{k}. In particular, the unit in A is 1 = a_{∅} ∈ A^{0}. Express the cup-product in A = H^{∗}(X) by

(4.1)  a_{I}a_{J} = ∑_{K∈nbc} μ_{I,J,K}a_{K}.

Denote the images of the generators a_{I} of A = H^{∗}(X) in H^{∗}(M) by the same symbols. By Poincaré duality, there are elements ā_{I} ∈ H^{∗}(M) so that a_{I} ā_{J} = δ_{I,J}ω, where ω is a (fixed) generator of H^{2ℓ−1}(M) ≅ ℤ. In particular, ā_{∅} = ω. Since H^{∗}(M) = D(A), using (3.1), we obtain the following.

Corollary 4.3. The set {a_{I}, ā_{I} | I ∈ nbc} forms a basis for H^{∗}(M), and the cupproduct in H^{∗}(M) is given by

a_{I}a_{J} = ∑_{K∈nbc} μ_{I,J,K}a_{K}, a_{J} ā_{K} = ∑_{I∈nbc} μ_{I,J,K} ā_{I}, ā_{I} ā_{J} = 0.

Example 4.4. Let A be a near-pencil in ℂℙ^{2}, with defining polynomial Q(A) = x_{0}(x_{1}^{n} − x_{2}^{n}). As noted in Example 2.6, the boundary manifold M is diffeomorphic to S^{1} × Σ_{n−1}.

The complement X of A has Poincaré polynomial P (X, t) = 1 + nt + (n − 1)t^{2}. A basis for the Orlik-Solomon algebra A = H^{∗}(X) is {1 = a_{∅}, a_{1}, …, a_{n}, a_{1,2}, …, a_{1,n}}. The cup-product in A is given by a_{1}a_{j} = a_{1,j} and a_{i}a_{j} = a_{1,j} − a_{1,i} for i > 1.

The boundary M has Poincaré polynomial P (M, t) = 1+(2n−1)t+(2n−1)t^{2} +t^{3}. A basis for the cohomology ring D(A) = H^{∗}(M) is given by the above basis for the Orlik-Solomon algebra, together with the dual classes {ā_{1,2}, …, ā_{1,n}, ā_{1}, …, ā_{n}, ā_{∅} = ω}. By Corollary 4.3, the cup-product in D(A) is given by the multiplication in A recorded above, ā_{I} ā_{J} = 0 for all I and J, a_{j}ā_{k} = a_{1,j}ā_{1,k} = δ_{j,k}ω, and a_{j}ā_{1,k} = −ā_{k} + δ_{j,k}(ā_{1} + ⋯ + ā_{n}).

Now, H^{∗}(Σ_{n−1} × S^{1}) = H^{∗}(Σ_{n−1}) ⊗ H^{∗}(S^{1}) is generated by a_{j} ⊗ 1, b_{j} ⊗ 1, 1 ≤ j ≤ n − 1, Γ ⊗ 1, and 1 ⊗ z, where a_{j}, b_{j}, Γ generate H^{∗}(Σ_{n−1}) and satisfy a_{j}b_{k} = δ_{j,k}Γ, and z generates H^{∗}(S^{1}). An explicit isomorphism H^{∗}(S^{1} × Σ_{n−1} × S^{1}) → D(A) is defined by

a_{j} ⊗ 1 ↦ e_{j+1} − e_{1}, b_{j} ⊗ 1 ↦ f_{1,j+1}, 1 ⊗ z ↦ e_{1}, Γ ⊗ 1 ↦ f_{1} + ⋯ + f_{n}.

4.3. The K(π, 1) problem. A hyperplane arrangement A is said to be a K(π, 1)- arrangement if the complement X = X(A) is aspherical, i.e., its universal cover is contractible. Classical examples include the braid arrangement (Fadell-Neuwirth), certain reflection arrangements (Brieskorn) and simplicial arrangements (Deligne).

The boundary manifold of an arrangement in ℂℙ^{1} is a disjoint union of circles. For ℓ ≥ 3, Proposition 2.9 shows that the boundary manifold of an arrangement in ℂℙ^{ℓ} is never aspherical. In the remaining case, ℓ = 2, we have the following result.

Proposition 4.5. Let A be a line arrangement in ℂℙ^{2}. The boundary manifold M = M(A) is aspherical if and only if A is essential.

Proof. If A is not essential, then A is a pencil of lines in ℂℙ^{2}, and so, by Example 2.1, M is a connected sum of S^{1} × S^{2}’s. Thus, π_{2}(M) ≠ 0.

If A is essential, it follows from work of Jiang and Yau [17] that M is an irreducible, sufficiently large Waldhausen graph manifold. Hence, M is aspherical. (In fact, by [28], M admits a metric of non-positive curvature.) □

5. Topological complexity

In this section, we relate the topological complexity of the boundary manifold of a hyperplane arrangement to that of the complement. We start by relating the zero-divisor length of a graded algebra to that of its double.

5.1. Cup length and zero-divisor length. Let A = ⨁_{k=0}^{ℓ} A^{k} be a finite-dimen-sional graded algebra over a field 𝕜 (as usual, we assume all graded pieces are finitedimensional). Define the cup length of A, denoted cl(A), to be the largest integer q for which there exist homogeneous elements a_{1}, …, a_{q} ∈ A^{>0} such that a_{1} ⋯ a_{q} ≠ 0.

The tensor product A ⊗ A has a natural graded algebra structure, with multiplication given by (u_{1} ⊗v_{1})·(u_{2} ⊗v_{2}) = (−1)^{|v_{1}|·|u_{2}|}u_{1}u_{2} ⊗v_{1}v_{2}. Multiplication in A defines an algebra homomorphism μ: A ⊗ A → A. Let J(A) be the kernel of this map. The zero-divisor length of A, denoted by zcl(A), is the length of the longest non-trivial product in this ideal.

Lemma 5.1. The ideal J(A) = ker(μ: A⊗A → A) is generated by the set of elements {ζ_{a} := a ⊗ 1 − 1 ⊗ a | a ∈ A}.

Proof. Let z = ∑_{i=1}^{k} a_{i} ⊗ b_{i} be an element of J(A). Then ∑_{i=1}^{k} a_{i}b_{i} = 0 in A, and it is readily checked that z − ∑_{i=1}^{k} ζ_{a_{i}}(1 ⊗ b_{i}) = ∑_{i=1}^{k} 1 ⊗ a_{i}b_{i} = 1 ⊗ (∑_{i=1}^{k} a_{i}b_{i}) = 0 in A ⊗ A. □

These two notions of length behave quite nicely with respect to the doubling operation for graded algebras.

Proposition 5.2. Let A be a connected, finite-dimensional graded algebra, with double D(A) = A ⋉ Ā. Then, cl(D(A)) = cl(A) + 1 and zcl(D(A)) = zcl(A) + 2.

Proof. Suppose that cl(A) = q, and let a = a_{1} ⋯ a_{q} be an element in A of length q. Then a · ā is a nonzero element in D(A), of length q + 1. Thus cl(D(A)) ≥ cl(A) + 1. The equality cl(D(A)) = cl(A) + 1 then follows from the fact that Ā is a square-zero ideal in D(A).

Next, suppose that zcl(A) = q, and let z = z_{1} ⋯ z_{q} be an element in J(A) of length q. Recall the basis {a_{j}^{k}} of A from §3.1, and write z = ∑ c_{j_{1},j_{2}}^{k_{1},k_{2}}a_{j_{1}}^{k_{1}} ⊗ a_{j_{2}}^{k_{2}}. Let m be maximal so that i_{1} + i_{2} = m and there is a nonzero coefficient c_{r_{1},r_{2}}^{i_{1},i_{2}} in this sum. Then, one can check that

z(ā_{r_{1}}^{i_{1}} ⊗ 1)(1 ⊗ ā_{r_{2}}^{i_{2}}) = ±c_{r_{1},r_{2}}^{i_{1},i_{2}}ω ⊗ ω + z′,

where ω = ¯1 generates D(A)^{2ℓ−1} and z′ is a linear combination of elements a_{j_{1}}^{k_{1}}ā_{r_{1}}^{i_{1}} ⊗ a_{j_{2}}^{k_{2}}ā_{r_{2}}^{i_{2}} in D(A) of bidegree different from (2ℓ − 1, 2ℓ − 1). So ẑ = z(ā_{r_{1}}^{i_{1}} ⊗ 1)(1 ⊗ ā_{r_{2}}^{i_{2}}) is a nonzero element in J(D(A)), of length at least q + 2. Thus zcl(D(A)) ≥ zcl(A) + 2.

To show that zcl(D(A)) = zcl(A)+2, it suffices to check that ẑζ_{α} = ẑ(α⊗1−1⊗α) = 0 for α ∈ D(A). We may assume that α is an element of the basis {a_{j}^{k}, ā_{j}^{k}} for D(A).

If α = ā_{j}^{k} ∈ Ā, then ẑζ_{α} = 0 since Ā is a square-zero ideal in D(A). If α = a_{j}^{k} ∈ A and ẑζ_{α} ≠ 0, then zζ_{α} is a nonzero element of length q + 1 in J(A), contradicting the assumption that zcl(A) = q. □

5.2. LS category and topological complexity. Let p: Y → X be a fibration. The sectional category of p, denoted secat(p), is the smallest integer q such that X can be covered by q open subsets, over each of which p has a section. A cohomological lower bound is given by:

(5.1)  secat(p) > cl(ker(p^{∗}: H^{∗}(Y; 𝕜) → H^{∗}(X; 𝕜))),

see James [16] as a classical reference. If p: P X → X is the path-fibration of a pointed space X, then secat(p) = cat(X), the Lusternik-Schnirelmann category of X. The category of X depends only on the homotopy-type of X. Since P X is contractible, the inequality (5.1) reduces to cat(X) > cl(X) := cl(H^{∗}(X; 𝕜)). If X is a finite simplicial complex, then cat(X) ≤ dim(X) + 1. Moreover, cat(X × Y) ≤ cat(X) + cat(Y) − 1.

Now let X^{I} be the space of all continuous paths from I = [0, 1] to X, with the compact-open topology, and let π: X^{I} → X × X be the fibration given by π(γ) = (γ(0), γ(1)). The topological complexity of X, introduced by Farber in [12] and denoted by tc(X), may be realized as the sectional category of π. Again, tc(X) = secat(π) depends only on the homotopy type of X. Using the fact that X^{I} ≃ X, and the Künneth formula, (5.1) reduces to tc(X) > zcl(X) := zcl(H^{∗}(X; 𝕜)). If X is a finite simplicial complex, then cat(X) ≤ tc(X) ≤ 2 cat(X) − 1; in particular, tc(X) ≤ 2 dim(X) + 1. Moreover, tc(X × Y) ≤ tc(X) + tc(Y) − 1.

As noted in [12], topological complexity is not determined by the LS category. For example, cat(S^{n}) = 2 for all n ≥ 1, whereas tc(S^{n}) = 2 for n odd and tc(S^{n}) = 3 for n even; also, cat(T^{n}) = tc(T^{n}) = n + 1, but cat(Σ_{g}) = 3 and tc(Σ_{g}) = 5 for g ≥ 2.

In [13], Farber and Yuzvinsky study the invariants tc(X) and zcl(X) in the case when X is the complement of a (central, essential) hyperplane arrangement in ℂ^{ℓ}. They show that tc(X) ≤ 2ℓ, and that this upper bound is attained for some classes of arrangements, including generic arrangements of sufficiently large cardinality and the reflection arrangements of types A, B, and D.

5.3. Topological complexity of the boundary manifold. Using Theorem 4.1 and Proposition 5.2, we see that the cup and zero-divisor lengths of the boundary manifold of an arrangement are determined in a simple fashion by the respective lengths of the complement.

Corollary 5.3. Let A be an arrangement of hyperplanes in ℂℙ^{ℓ}, with complement X and boundary manifold M. Then:

cl(M) = cl(X) + 1 and zcl(M) = zcl(X) + 2.

Moreover, if A is essential, then cl(M) = ℓ + 1.

The relationship between the LS category and topological complexity of the boundary manifold on one hand, and the complement on the other hand, is more subtle, as the following example indicates.

cat(X) 1 2  2  3  3  3
tc(X) 2 3  4  4  5  6
cat(M) 2 3  3  4  4  4
tc(M) 2 4  5  5  6  7
f  x_{0} x_{0}x_{1} x_{0}^{3} − x_{1}^{3} x_{0}x_{1}x_{2} x_{0}(x_{1}^{3} − x_{2}^{3}) (x_{0}^{2} − x_{1}^{2})(x_{0}^{2} − x_{2}^{2})(x_{1}^{2} − x_{2}^{2})

Table 1. Possible values of LS category and topological complexity for the complement X and boundary manifold M of a line arrangement in ℂℙ^{2}.

Example 5.4. Let A be the Boolean arrangement ℂℙ^{ℓ}. Then X ≃ T^{ℓ} and M = T^{ℓ} × S^{ℓ−1}. An easy computation shows that cat(M) = cat(X) + 1 = ℓ + 2; on the other hand, tc(M) = tc(X) + 2 = ℓ + 3 if ℓ is even, but tc(M) = tc(X) + 3 = ℓ + 4 if ℓ is odd.

For projective line arrangements, we can narrow down the possible values of the category and topological complexity of the boundary manifold.

Proposition 5.5. Let A be a line arrangement in ℂℙ^{2}, with boundary manifold M. If A is not essential, then cat(M) = 2 or 3 and tc(M) = 4, 5, or 6. If A is essential, then cat(M) = 4 and tc(M) = 5, 6, or 7.

Proof. As shown in [14], the LS category of a closed 3-manifold M depends only on π_{1}(M): it is 2, 3, or 4, according to whether π_{1}(M) is trivial, a non-trivial free group, or not a free group.

Suppose A is a pencil of n+1 lines. If n = 0, then M = S^{3}, so cat(M) = tc(M) = 2. If n = 1, then M = S^{1} × S^{2}, so cat(M) = 3 and tc(M) = 4. If n > 1, then M = #^{n}S^{1} × S^{2}, and so cat(M) = 3, and tc(M) = 5.

On the other hand, if A is essential, then, as noted in Proposition 4.5, M is aspherical. In particular, cd(π_{1}(M)) = 3, and so π_{1}(M) cannot be free. Hence, cat(M) = 4, and the bounds on tc(M) follow at once. □

All the various possibilities listed in Proposition 5.5 do occur. For example, if A is a near-pencil of n+1 ≥ 4 lines, then M = S^{1}×Σ_{n−1}, and so cat(M) = 4 and tc(M) = 6. We summarize in Table 1 the possible values for the LS category and topological complexity of both the complement and the boundary manifold of an arrangement in ℂℙ^{2}, together with sample representatives for the defining polynomials.

In high dimensions, a complete understanding of the possible values for cat(M) and tc(M) is not at hand. Nevertheless, we have the following class of arrangements (mentioned in Example 2.7), where precise formulas can be given.

Proposition 5.6. Let A be the hyperplane arrangement in ℂℙ^{2k} defined by the polynomial f = x_{0} ∏_{i=1}^{k}(x_{i}^{n_{i}} − y_{i}^{n_{i}}), with n_{i} ≥ 3. If X is the complement and M is the boundary manifold, then:

cat(X) = 2k + 1,  tc(X) = 3k + 1,
cat(M) = 2k + 2,  tc(M) = 3k + 3.

Proof. We have X ≃ T^{k} × ∏_{i=1}^{k} ⋁^{n_{i}−1} S^{1}, while M = T^{k} × (#^{m}T^{k} × S^{2k−1}), where m = ∏_{i=1}^{k}(n_{i} − 1). A computation shows that cl(X) = 2k and zcl(X) = 3k. Hence, by Corollary 5.3, cl(M) = 2k + 1 and zcl(M) = 3k + 2.

Let W = T^{k} × S^{2k−1}. Note that cl(W) = k + 1, while cat(W) ≤ cat(T^{k}) + cat(S^{2k−1}) − 1 = k + 2; hence cat(W) = k + 2. In fact, if we consider W with its standard CW decomposition, we can take W = ⋃_{i=0}^{k+1} U_{i}, with U_{0} a small ball around the 0-cell e^{0}, U_{i} the union of the (open) i and i + 2k − 1 cells, for 1 ≤ i ≤ k, and U_{k+1} the top cell e^{3k−1}; plainly, each U_{i} is contractible in W.

Now, #^{m}W is obtained by attaching a top cell to the wedge of m copies of W \U_{k+1} at the basepoint e^{0}; thus, we may find a decomposition #^{m}W = ⋃_{i=0}^{k+1} V_{i} as before, with V_{i} contractible in #^{m}W. It follows that cat(#^{m}W) = k + 2, and so tc(#^{m}W) ≤ 2k + 3. Thus, tc(M) ≤ tc(T^{k}) + tc(#^{m}W) − 1 ≤ 3k + 3, and we are done. □

As a consequence, we see that the difference between the topological complexity and the LS category of the boundary manifold of an arrangement can be arbitrarily large.

Corollary 5.7. For each k ≥ 1, there is an arrangement A with boundary manifold M = M(A) for which tc(M) − cat(M) = k.

6. Resonance

In this section, we study the resonance varieties of the trivial extension of a graded algebra. As an application, we obtain information about the structure of the resonance varieties of the boundary manifold of a hyperplane arrangement. Throughout, 𝕜 will denote an algebraically closed field of characteristic 0.

6.1. Resonance varieties. Let A = ⨁_{k=0}^{ℓ} A^{k} be a graded, graded-commutative, connected algebra over 𝕜. Assume each graded piece A^{k} is finite-dimensional. For each a ∈ A^{1}, we have a · a = 0; thus, multiplication by a defines a chain complex

(6.1)  (A, a): \xymatrix{0 \ar[r] & A^{0} \ar[r]^{a} & A^{1} \ar[r]^{a} & A^{2} \ar[r]^{a} & ⋯ \ar[r]^{a} & A^{ℓ} \ar[r] & 0} .

By definition, the resonance varieties of A are the jumping loci for the cohomology of these complexes:

(6.2)  R_{d}^{k}(A) = {a ∈ A^{1} | dim_{𝕜} H^{k}(A, a) ≥ d},

for 0 ≤ k ≤ ℓ and 0 ≤ d ≤ b_{k} = b_{k}(A). Notice that A^{1} = R_{0}^{k}(A) ⊃ R_{1}^{k}(A) ⊃ ⋯ ⊃ R_{b_{k}}^{k}(A) ⊃ {0}. The sets R_{d}^{k}(A) are algebraic subvarieties of the affine space A^{1} = 𝕜^{n}, and are isomorphism-type invariants of the graded algebra A. They have been the subject of considerable recent interest, particularly in the context of hyperplane arrangements, see for instance [6, 21, 32], and references therein.

An element a ∈ A^{1} is said to be nonresonant if the dimensions of the cohomology groups H^{∗}(A, a) are minimal. If A is the Orlik-Solomon algebra of an arrangement of rank ℓ, and a ∈ A^{1} is nonresonant, then H^{k}(A, a) = 0 for k ≠ ℓ, see for instance [32].

6.2. Resonance varieties of a doubled algebra. We now compare the resonance varieties of A to those of the doubled algebra D(A) = A ⋉ Ā, under the assumption that ℓ ≥ 3. Notice that for such ℓ, we have D(A)^{1} = A^{1}.

Theorem 6.1. If A is a graded, connected 𝕜-algebra and ℓ ≥ 3, then the resonance varieties of D(A) are given by

R_{d}^{k}(D(A)) = \begin{cases}R_{d}^{k}(A) & if k ≤ ℓ − 2, \\ ⋃_{p+q=d}(R_{p}^{ℓ−1}(A) ∩ R_{q}^{ℓ}(A)) & if k = ℓ − 1 or k = ℓ, \\ R_{d}^{2ℓ−k−1}(A) & if k ≥ ℓ + 1. \end{cases}

Proof. Fix a basis {a_{i}^{p}} for A, and let {a_{i}^{p}, ā_{j}^{q}} be the corresponding basis for the double D(A) as in §3.1. Let m_{k} = m_{k}(a) and D(m_{k}) = D(m_{k}(a)) denote the matrices of the maps A^{k−1} → A^{k} and D(A)^{k−1} → D(A)^{k} given by multiplication by a ∈ A^{1} = D(A)^{1} in the bases specified above. An exercise in linear algebra reveals that

(6.3)  R_{d}^{k}(A) = {a ∈ A^{1} | dim_{𝕜} A^{k} − rank m_{k} − rank m_{k+1} ≥ d} .

Similarly,

(6.4) R_{d}^{k}(D(A)) = {a ∈ D(A)^{1} | dim_{𝕜} D(A)^{k} − rank D(m_{k}) − rank D(m_{k+1}) ≥ d} .

The chain complex (D(A), a) has terms D(A)^{k} = A^{k} for k ≤ ℓ − 2, D(A)^{ℓ−1} = A^{ℓ−1} ⊕ Ā^{ℓ}, D(A)^{ℓ} = Ā^{ℓ−1} ⊕ A^{ℓ}, and D(A)^{k} = Ā^{2ℓ−k−1} for k ≥ ℓ + 1. Using the definition of the multiplication in D(A), one can check that the boundary maps of this chain complex have matrices D(m_{k}) = m_{k} for k ≤ ℓ − 2,

D(m_{ℓ−1}) = [m_{ℓ−1} 0] , D(m_{ℓ}) = [\begin{matrix}0 & m_{ℓ} \\ ±mℓ^{⊤} & 0 \end{matrix}] , D(m_{ℓ+1}) = [\begin{matrix}±mℓ−1^{⊤} \\ 0 \end{matrix}] ,

and D(m_{k}) = ±m_{2ℓ−k} for k ≥ ℓ + 2. Calculating ranks of these matrices, and using the above descriptions of the resonance varieties R_{d}^{k}(A) and R_{d}^{k}(D(A)) yields the result. □

If ℓ = 2, then D(A)^{1} = A^{1} ⊕ Ā^{2}. If (a, b) · (a, b) = 0 for all (a, b) ∈ D(A)^{1}, then (D(A), (a, b)) is a chain complex for each (a, b) as in (6.1), and the resonance varieties of D(A) are

R_{d}^{k}(D(A)) = {(a, b) ∈ D(A)^{1} | dim_{𝕜} H^{k}(D(A), (a, b)) ≥ d}.

In this situation, the boundary maps of the chain complex (D(A), (a, b)) have matrices

D(m_{1}) = [m_{1} m̄_{1}] , D(m_{2}) = [\begin{matrix}ϕ & m_{2} \\ −m2^{⊤} & 0 \end{matrix}] , D(m_{3}) = [\begin{matrix}m1^{⊤} \\ & m̄1^{⊤} \end{matrix}] ,

where, as above, m_{k} = m_{k}(a) is the matrix of multiplication by a, A^{k−1} → A^{k}, m̄_{1} = m̄_{1}(b) is the matrix of multiplication by b, Ā^{2} → Ā^{1}, and ϕ = ϕ(b) is the matrix of multiplication by b, A^{1} → Ā^{1}. Since A and D(A) are graded commutative, the matrix ϕ is skew-symmetric. The structure of these matrices, D(m_{3}) = D(m_{1})^{⊤} and D(m_{2})^{⊤} = −D(m_{2}), follows from the multiplication in D(A), see (3.1).

6.3. Aomoto complexes. The chain complex (6.1) may be realized as the specialization at a of the Aomoto complex of the algebra A. Let R_{A} = Sym(A_{1}) be the symmetric algebra on the 𝕜-dual of A^{1}, and let x = {x_{1}, …, x_{n}} be the basis for A_{1} dual to the basis {a_{1}^{1}, …, a_{n}^{1}} for A^{1}. Then R_{A} becomes identified with the polynomial ring R = 𝕜[x]. The Aomoto complex of A is the chain complex

(6.5)  \xymatrix{A^{0} ⊗_{𝕜} R \ar[r]^{d^{1}} & A^{1} ⊗_{𝕜} R \ar[r]^{d^{2}} & A^{2} ⊗_{𝕜} R \ar[r]^{d^{3}} & ⋯ \ar[r]^{d^{ℓ}} & A^{ℓ} ⊗_{𝕜} R} ,

where the boundary maps are multiplication by ∑_{j=1}^{n} a_{j}^{1} ⊗ x_{j}. Notice that the multiplication map μ: A^{1} ⊗ A^{p−1} → A^{p} can be recovered from the boundary map d^{p}. Denote the matrix of d^{1} by d_{x}, and that of d^{2} by Δ = Δ_{A}. If the multiplication A^{1} ⊗A^{1} → A^{2} is given by a_{i}^{1}a_{j}^{1} = ∑_{k=1}^{m} μ_{i,j,k}a_{k}^{2}, the latter is an n×m matrix of linear forms over R, with entries

(6.6)  Δ_{j,k} = ∑_{i=1}^{n} μ_{i,j,k}x_{i}.

The (transpose of the) matrix Δ_{A} is the (linearized) Alexander matrix of the algebra A, which appears in various guises in, for instance, [5, 6, 21, 25].

The Aomoto complex of the double D(A) may be constructed analogously. In light of Theorem 6.1, we focus on the case ℓ = 2. Here, D(A)^{1} = A^{1} ⊕ Ā^{2}, with basis {a_{i}^{1}, ā_{j}^{2}}, where 1 ≤ i ≤ n and say 1 ≤ j ≤ m. Identify the ring R_{D(A)} = Sym(A_{1}⊕Ā_{2}) with the polynomial ring 𝕜[x, y]. Then, the Aomoto complex of D(A) is the chain complex

(6.7)  \xymatrix{D(A)^{0} ⊗_{𝕜} S \ar[r]^{D^{1}} & D(A)^{1} ⊗_{𝕜} S \ar[r]^{D^{2}} & D(A)^{2} ⊗_{𝕜} S \ar[r]^{D^{3}} & D(A)^{3} ⊗_{𝕜} S} ,

where the boundary maps are multiplication by ∑_{i=1}^{n} a_{i}^{1} ⊗ x_{i} + ∑_{j=1}^{m} ā_{j}^{2} ⊗ y_{j}. Denote the matrix of D^{1} by (d_{x} d_{y}), and that of D^{2} by Δ_{D(A)}. Then it follows from (3.1) that the matrix of D^{3} is (d_{x} d_{y})^{⊤}, and that

(6.8)  Δ_{D(A)} = (\begin{matrix}Φ & Δ_{A} \\ −ΔA^{⊤} & 0 \end{matrix}) ,

where Φ is the n × n matrix with entries Φ_{i,j} = ∑_{k=1}^{m} μ_{i,j,k}y_{k}. Notice that Δ_{D(A)} is a skew-symmetric matrix of linear forms, and that d_{x}Φ = d_{y}ΔA^{⊤}.

If A = ⨁_{k=0}^{ℓ} A^{k} and ℓ ≥ 3, the relationship between the Aomoto complexes of A and D(A) is implicit in the proof of Theorem 6.1. We relate these complexes in the case ℓ = 2. Consider the Aomoto complex of A and its dual

\xymatrix{A^{0} ⊗_{𝕜} S \ar[r]^{d_{x}} & A^{1} ⊗_{𝕜} S \ar[r]^{Δ_{A}} & A^{2} ⊗_{𝕜} S & and & Ā^{2} ⊗_{𝕜} S \ar[r]^{−Δ_{A}^{⊤}} & Ā^{1} ⊗_{𝕜} S \ar[r]^{−d_{x}^{⊤}} & Ā^{0} ⊗_{𝕜} S,}

where we have extended scalars and changed the signs for reasons which will become apparent.

Lemma 6.2. The maps {d_{y}, −Φ, dy^{⊤}} provide a chain map

\xymatrix{A^{0} ⊗_{𝕜} S \ar[r]^{d_{x}} \ar[d]^{d_{y}} & A^{1} ⊗_{𝕜} S \ar[r]^{Δ_{A}} \ar[d]^{−Φ} & A^{2} ⊗_{𝕜} S \ar[d]^{d_{y}^{⊤}} \\ Ā^{2} ⊗_{𝕜} S \ar[r]^{−Δ_{A}^{⊤}} & Ā^{1} ⊗_{𝕜} S \ar[r]^{−d_{x}^{⊤}} & Ā^{0} ⊗_{𝕜} S.}

Furthermore, the Aomoto complex of D(A) is the mapping cone of this chain map.

An alternate way to compute the resonance varieties R_{d}^{1}(A) is by taking the zero locus of the determinantal ideals of the linearized Alexander matrix of A. If Ψ is a p × q matrix (p ≤ q) with polynomial entries, define R_{d}(Ψ) = V (E_{p−d}(Ψ)) where E_{r}(Ψ) is the ideal of r × r minors. Proceeding as in the proof of Theorem 3.9 from [21] (see also [6]), we find that R_{d}^{1}(A) = R_{d}(Δ_{A}). Similarly, R_{d}^{1}(D(A)) = R_{d}(Δ_{D(A)}).

6.4. Resonance of arrangements. For a space X with the homotopy type of a finite CW-complex, define the resonance varieties of X by R_{d}^{k}(X) = R_{d}^{k}(H^{∗}(X; 𝕜)).

Let A be an arrangement of hyperplanes, with complement X, boundary manifold M, and Orlik-Solomon algebra A = H^{∗}(X; 𝕜). If A is an arrangement in ℂℙ^{ℓ} with ℓ ≥ 3, then it follows from Theorem 6.1 that the resonance varieties of the complement, R_{d}^{k}(X) = R_{d}^{k}(A), completely determine the resonance varieties, R_{d}^{k}(M) = R_{d}^{k}(D(A)), of the boundary manifold. So assume that A ⊂ ℂℙ^{2} is a line arrangement.

The complex (A, a) of (6.1) may be realized as the specialization A^{•} ⊗_{𝕜} R|_{x↦a} of the Aomoto complex of A. Since D(A)^{1} = A^{1} ⊕ A^{2}, the resonance varieties of the boundary manifold are given by

R_{d}^{k}(M) = R_{d}^{k}(D(A)) = {(a, b) ∈ A^{1} ⊕ A^{2} | dim H^{k}(D(A), (a, b)) ≥ d}.

The complex (D(A), (a, b)) may be realized as the specialization D(A)^{•} ⊗_{𝕜} S|_{(x,y)↦(a,b)} of the Aomoto complex of D(A). The properties of the boundary maps of the complex (6.7) noted above imply that the resonance varieties of M satisfy R_{d}^{k}(M) = R_{d}^{3−k}(M).

Recall that, for nonresonant a ∈ A^{1}, we have H^{k}(A, a) = 0 for k = 0, 1. Write b_{k} = b_{k}(A) = dim_{𝕜} A^{k}, and β = 1 − b_{1} + b_{2}. Note that β = dim_{𝕜} H^{2}(A, a).

Proposition 6.3. Let A ⊂ ℂℙ^{2} be a line arrangement with Orlik-Solomon algebra A and double D(A).

(1) If a ∈ A^{1} is nonresonant for A, then for any b, (a, b) ∈ D(A)^{1} is nonresonant for D(A). Furthermore, H^{0}(D(A), (a, b)) = H^{3}(D(A), (a, b)) = 0 and H^{1}(D(A), (a, b)) = H^{2}(D(A), (a, b)) = 𝕜^{β}.

(2) If a ∈ R_{d}^{1}(A) is nonzero, then for any b, (a, b) ∈ R_{d+β}^{1}(D(A)).

(3) If b ≠ 0, then (0, b) ∈ R_{d}^{1}(D(A)), where d = b_{2} − 1 + dim_{𝕜} ( ker Φ|_{y↦b} ).

Proof. Given (a, b) ∈ D(A)^{1}, by Lemma 6.2, there is a corresponding short exact sequence of chain complexes \xymatrix{0 \ar[r] & (Ā^{#}, a)^{−1} \ar[r] & (D(A), (a, b)) \ar[r] & (A, a) \ar[r] & 0} :

[figure: Ā^{2}; −Δ_{A}^{⊤}; Ā^{1}; −d_{x}; Ā^{0}; D(A)^{0}; _{(_{d_{x} d_{y}})}; D(A)^{1}; _{(\begin{matrix}Φ & Δ_{A} \\ −ΔA^{⊤} & 0 \end{matrix})}; D(A)^{2}; _{(\begin{matrix}dx^{⊤} \\ dy^{⊤} \end{matrix})}; D(A)^{3}; A^{0}; d_{x}; A^{1}; Δ_{A}; A^{2}]

where (Ā^{#}, a) denotes the specialization at a of the dual of the Aomoto complex of A. Passing to cohomology yields a long exact sequence (6.9)

[figure: 0; H^{0}(D(A)); H^{0}(A); H^{0}(Ā^{#}); H^{1}(D(A)); H^{1}(A); H^{1}(Ā^{#}); H^{2}(D(A)); H^{2}(A); H^{2}(Ā^{#}); H^{3}(D(A)); 0]

where, for instance H^{k}(A) = H^{k}(A, a). Using the fact that H^{k}(Ā^{#}) ≅ H^{2−k}(A), calculations with this long exact sequence may be used to establish all three assertions. □

As a consequence of Proposition 6.3, we obtain the following.

Corollary 6.4. The resonance varieties of the doubled algebra D(A) satisfy (1) R_{d}^{1}(D(A)) = D^{1}(A) for d ≤ β.

(2) R_{d}^{1}(A) × A^{2} ⊆ R_{d+β}^{1}(D(A)).

(3) {0} × R_{d}(Φ) ⊆ R_{d+b_{2}(A)}^{1}(D(A)).

All irreducible components of R_{d}^{1}(X) = R_{d}^{1}(A) are linear, see [6]. From items (1) and (2) in the above Corollary, it is clear that the resonance varieties of M contain linear components as well. However, item (3) leaves open the possibility that R_{d}^{1}(M) = R_{d}^{1}(D(A)) contains non-linear components, for d ≥ b_{2}(A). This does indeed occur, as shown next.

6.5. General position arrangements. Let A_{n} be a projective line arrangement consisting of n + 1 lines in general position. We identify the resonance varieties of the boundary manifold M_{n}^{3} = M(A_{n}).

The Orlik-Solomon algebra, A = E/𝔪^{3}, is the rank 2 truncation of the exterior algebra generated by e_{1}, …, e_{n}, where 𝔪 = (e_{1}, …, e_{n}). Note that A has Betti numbers b_{1} = n, b_{2} = \binom{n}{2}, and that β = 1 − b_{1} + b_{2} = \binom{n−1}{2}.

For this arrangement, the Alexander matrix Δ_{A} is the transpose of the matrix of the Koszul differential δ_{2} : E^{2} ⊗ S → E^{1} ⊗ S. The submatrix Φ of the Alexander matrix Δ_{D(A)} recorded in (6.8) is the generic n × n skew-symmetric matrix of linear forms Φ_{n}, with entries (Φ_{n})_{i,j} = y_{i,j} above the diagonal.

Identity D(A)^{1} = E^{1} × E^{2}. Note that R_{d}^{1}(A) = R_{d}^{1}(E) = {0} for d > 0. An analysis of the long exact sequence (6.9) in light of this observation yields the following sharpening of Corollary 6.4 for general position arrangements.

Proposition 6.5. The resonance varieties of the boundary manifold M_{n} of a general position arrangement of n + 1 lines in ℂℙ^{2} are given by:

R_{d}^{1}(M_{n}) = \begin{cases}E^{1} × E^{2} & if d ≤ \binom{n−1}{2}, \\ \{0\} × E^{2} & if \binom{n−1}{2} < d < \binom{n}{2}, \\ \{0\} × R_{d−\binom{n}{2}}(Φ_{n}) & if \binom{n}{2} ≤ d < \binom{n}{2} + n, \\ \{0\} × \{0\} & if d = \binom{n}{2} + n. \end{cases}

If Ψ is a skew-symmetric matrix of size n with polynomial entries, define the Pfaffian varieties of Ψ by

(6.10)  P_{d}(Ψ) = V (Pf_{2(⌊n/2⌋−d)}(Ψ)),

where Pf_{2r}(Ψ) is the ideal of 2r × 2r Pfaffians of Ψ. For n even, the ideal Pf_{n}(Ψ) is principal, generated by Pfaff(Ψ), the maximal Pfaffian of Ψ. Well known properties of Pfaffians (see, for instance [3, Cor. 2.6]) may be used to establish the following relationship between the resonance and Pfaffian varieties of Ψ:

(6.11)  V (E_{2r−1}(Ψ)) = V (E_{2r}(Ψ)) = V (Pf_{2r}(Ψ)).

In other words, for n even, we have R_{2d+1}^{1}(Ψ) = R_{2d}^{1}(Ψ) = P_{d}(Ψ), while for n odd, we have R_{2d}^{1}(Ψ) = R_{2d−1}^{1}(Ψ) = P_{d−1}(Ψ).

For n = 2k, the Pfaffian of the generic skew-symmetric matrix Φ_{n} is given by

(6.12)  Pfaff(Φ_{n}) = ∑_{m} σ(m)ω(m),

where the sum is over all perfect matchings m = {{i_{1}, j_{1}}, {i_{2}, j_{2}}, …, {i_{k}, j_{k}}}, partitions of [2k] into blocks of size two with i_{p} < j_{p}), and where σ(m) is the sign of

the corresponding permutation ( \begin{matrix}1 & 2 & 3 & 4 & ... & 2k−1 & 2k \\ i_{1} & j_{1} & i_{2} & j_{2} & ... & i_{k} & j_{k} \end{matrix} ), and ω(m) = y_{i_{1},j_{1}}y_{i_{2},j_{2}} ⋯ y_{i_{k},j_{k}},

see for instance [2]. Note that Pfaff(Φ_{n}) is a polynomial of degree k = n/2 in the variables y_{i,j}.

For arbitrary n, it is known that the Pfaffian variety P_{d}(Φ_{n}) is irreducible, with singular locus P_{d+1}(Φ_{n}), see [3, 19]. These facts, together with Proposition 6.5 and (6.11), yield the following.

Corollary 6.6. Let M_{n} be the boundary manifold of the general position arrangement A_{n}. For n ≥ 4 and \binom{n}{2} < d < \binom{n}{2}+n−2, the resonance variety R_{d}^{1}(M_{n}) is a singular, irreducible variety.

Acknowledgment. A significant portion of this work was done while the authors attended the Fall 2004 program “Hyperplane Arrangements and Applications” at the Mathematical Sciences Research Institute in Berkeley, California. We thank MSRI for its support and hospitality during this stay.

References

1. K. Aaslepp, M. Drawe, C. Hayat-Legrand, C. Sczesny, H. Zieschang, On the cohomology of Seifert and graph manifolds, Topology Appl. 127 (2003), 3–32. MR 2003j:57038

2. R. Brualdi, H. Ryser, Combinatorial matrix theory, Encyclopedia of Mathematics and its Applications, vol. 39, Cambridge Univ. Press, Cambridge, 1991. MR 93a:05087

3. D. Buchsbaum, D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 0453723

4. J. Cogolludo, Topological invariants of the complement to arrangements of rational plane curves, Mem. Amer. Math. Soc. 159 (2002). MR 2003k:32037

5. D. Cohen, A. Suciu, Alexander invariants of complex hyperplane arrangements, Trans. Amer. Math. Soc. 351 (1999), 4043–4067. MR 99m:52019

6. , Characteristic varieties of arrangements, Math. Proc. Cambridge Phil. Soc. 127 (1999), no. 1, 33–53. MR 2000m:32036

7. A. Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 94b:32058

8. A. Dimca, A. Némethi, Hypersurface complements, Alexander modules and monodromy, in: Real and complex singularities, 19–43, Contemp. Math., vol. 354, Amer. Math. Soc., Providence, RI, 2004. MR 2087802

9. A. Durfee, Neighborhoods of algebraic sets, Trans. Amer. Math. Soc. 276 (1983), no. 2, 517–530. MR 84m:32013

10. , A naive guide to mixed Hodge theory, Singularities, Part 1 (Arcata, Calif., 1981), 313–320, Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983. MR 85b:32044

11. A. Durfee, R. Hain, Mixed Hodge structures on the homotopy of links, Math. Ann. 280 (1988), no. 1, 69–83. MR 89c:14012

12. M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211–221. MR 2004c:6813

13. M. Farber, S. Yuzvinsky, Topological robotics: Subspace arrangements and collision free motion planning, in: Geometry, topology, and mathematical physics, pp. 145–156, Amer. Math. Soc. Transl. Ser. 2, vol. 212, Amer. Math. Soc., Providence, RI, 2004. MR 2070052

14. J.C. Gómez-Larrañaga, F. González-Acuña, Lusternik-Schnirelmann category of 3-manifolds, Topology 31 (1992), no. 4, 791–800. MR 93j:55005

15. E. Hironaka, Boundary manifolds of line arrangements, Math. Annalen 319 (2001), 17–32. MR 2001m: 32061

16. I.M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331–348. MR 80i:55001

17. T. Jiang, S.S.-T. Yau, Intersection lattices and topological structures of complements of arrangements in ℂℙ^{2}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 357–381. MR 99g:52020

18. M. Kim, Weights in cohomology groups arising from hyperplane arrangements, Proc. Amer. Math. Soc. 120 (1994), 697–703. MR 94b:52022

19. H. Kleppe, D. Laksov, The algebraic structure and deformation of Pfaffian schemes, J. Algebra 64 (1980), 1, 167–189. MR 82b:14027

20. P. Lambrechts, Cochain model for thickenings and its application to rational LS-category, Manuscripta Math. 103 (2000), 143–160. MR 2001m:55009

21. D. Matei, A. Suciu, Cohomology rings and nilpotent quotients of real and complex arrangements, in: Arrangements–Tokyo 1998, Adv. Stud. Pure Math., vol. 27, Math. Soc. Japan, Tokyo, 2000, pp. 185–215. MR 2002b:32045

22. J.W. Milnor, Singular points of complex hypersurfaces, Annals of Math. Studies, vol. 61, Princeton Univ. Press, Princeton, NJ, 1968. MR 0239612

23. P. Orlik, L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189. MR 81e:32015 24. P. Orlik, H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss., vol. 300, Springer-

Verlag, Berlin, 1992; MR 94e:52014.

25. S. Papadima, A. Suciu, Chen Lie algebras, Int. Math. Res. Not. 2004:21 (2004), 1057–1086. MR 2004m:17043.

26. C. Peters, J. Steenbrink, Mixed Hodge structures, book in preparation, Ergebnisse der Mathematik, Springer, Berlin.

27. I. Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417–420. MR 0296067

28. V. Schroeder, Rigidity of nonpositively curved graphmanifolds, Math. Ann. 274 (1986), 19–26. MR 87h:53054

29. B.Z. Shapiro, The mixed Hodge structure of the complement to an arbitrary arrangement of affine complex hyperplanes is pure, Proc. Amer. Math. Soc. 117 (1993), no. 4, 931–933. MR 93e:32045

30. C.T.C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR 0192509

31. E. Westlund, The boundary manifold of an arrangement, Ph.D. thesis, University of Wisconsin, Madison, WI, 1997.

32. S. Yuzvinsky, Orlik-Solomon algebras in algebra and topology, Russian Math. Surveys 56 (2001), 293–364. MR 2002i:14047

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, Northeastern University, Boston, MA 02115

  E-mail address: a.suciu@neu.edu
  URL: http://www.math.neu.edu/~suciu