aa r X i v : . [ m a t h . AG ] J u l CHEN RANKS AND RESONANCE
DANIEL C. COHEN AND HENRY K. SCHENCK
Abstract.
The Chen groups of a group G are the lower central series quo-tients of the maximal metabelian quotient of G . Under certain conditions,we relate the ranks of the Chen groups to the first resonance variety of G , ajump locus for the cohomology of G . In the case where G is the fundamentalgroup of the complement of a complex hyperplane arrangement, our resultspositively resolve Suciu’s Chen ranks conjecture. We obtain explicit formulasfor the Chen ranks of a number of groups of broad interest, including pureArtin groups associated to Coxeter groups, and the group of basis-conjugatingautomorphisms of a finitely generated free group. Introduction
Let G be a group, with commutator subgroup G ′ = [ G, G ], and second commu-tator subgroup G ′′ = [ G ′ , G ′ ]. The Chen groups of G are the lower central seriesquotients gr k ( G/G ′′ ) of G/G ′′ . These groups were introduced by K.T. Chen in [4],so as to provide accessible approximations of the lower central series quotients ofa link group. For example, if G = F n is the free group of rank n (the fundamen-tal group of the n -component unlink), the Chen groups are free abelian, and theirranks, θ k ( G ) = rank gr k ( G/G ′′ ), are given by(1.1) θ k ( F n ) = ( k − · (cid:18) k + n − k (cid:19) , for k ≥ . While apparently weaker invariants than the lower central series quotients of G itself, the Chen groups sometimes yield more subtle information. For instance, if G = P n is the Artin pure braid group, the ranks of the Chen groups distinguish G from a direct product of free groups, while the ranks of the lower central seriesquotients fail to do so, see [8]. In this paper, we study the Chen ranks θ k ( G ) fora class of groups which includes all arrangement groups (fundamental groups ofcomplements of complex hyperplane arrangements), and potentially fundamentalgroups of more general smooth quasi-projective varieties. We relate these Chenranks to the first resonance variety of the cohomology ring of G . For arrangementgroups, our results positively resolve Suciu’s Chen ranks conjecture, stated in [38].Let A = L ℓk =0 A k be a finite-dimensional, graded, graded-commutative, con-nected algebra over an algebraically closed field k of characteristic 0. For each a ∈ A , we have a = 0, so right-multiplication by a defines a cochain complex(1.2) ( A, a ) : 0 / / A a / / A a / / A a / / · · · a / / A k a / / · · · . Mathematics Subject Classification.
Primary 20F14, 14M12; Secondary 20F36, 52C35.
Key words and phrases.
Chen group, resonance variety, hyperplane arrangement.Cohen supported by NSF 1105439, NSA H98230-11-1-0142.Schenck supported by NSF 1068754, NSA H98230-11-1-0170.
In the context of arrangements, with A the cohomology ring of the complement, thecomplex ( A, a ) was introduced by Aomoto [1], and subsequently used by Esnault-Schechtman-Viehweg [15] and Schechtman-Terao-Varchenko [35] in the study oflocal system cohomology. In this context, if a ∈ A is generic, the cohomology of( A, a ) vanishes, except possibly in the top dimension, see Yuzvinsky [40].In general, the resonance varieties of A , or of G in the case where A = H ∗ ( G ; k ),are the cohomology jump loci of the complex ( A, a ), R kd ( A ) = { a ∈ A | dim H k ( A, a ) ≥ d } , homogeneous algebraic subvarieties of A . These varieties, introduced by Falk [16]in the context of arrangements, are isomorphism-type invariants of the algebra A .They have been the subject of considerable recent interest in a variety of areas, see,for instance, Dimca-Papadima-Suciu [13], Yuzvinsky [41], and references therein.We will focus on the first resonance variety of the group G , R ( G ) = R ( H ∗ ( G ; k )) = { a ∈ A | H ( A, a ) = 0 } . Assume that G is finitely presented. The group G is 1-formal (over the field k ) ifand only if the Malcev Lie algebra of G is is isomorphic, as a filtered Lie algebra, tothe completion with respect to degree of a quadratic Lie algebra (see [32] for details),and is said to be a commutator-relators group if it admits a presentation G = F/R ,where F is a finitely generated free group and R is the normal closure of a finitesubset of [ F, F ]. For any finitely generated 1-formal group, Dimca-Papadima-Suciu[13] show that all irreducible components of the resonance variety R d ( H ∗ ( G ; k )) arelinear subspaces of H ( G ; k ). For a finitely presented, commutator-relators group,the resonance variety R ( G ) may be realized as the variety defined by the annihi-lator of the linearized Alexander invariant B of G , a module over the polynomialring S = Sym( H ( G ; k )), R ( G ) = V (ann( B )), see Section 2 below. We can thusview R ( G ) as a scheme.Let A = H ∗ ( G ; k ), and let µ : A ∧ A → A be the cup product map, µ ( a ∧ b ) = a ∪ b . A non-zero subspace U ⊆ A is said to be p -isotropic with respect to thecup product map if the restriction of µ to U ∧ U has rank p . For instance, A is0-isotropic if G is a free group, while A is 1-isotropic if G is the fundamental groupof a closed, orientable surface. Call subspaces U and V of A projectively disjointif they meet only at the origin, U ∩ V = { } . In the formulas below, we use theconvention (cid:0) nm (cid:1) = 0 if n < m . Our main result is as follows. Theorem A.
Let G be a finitely presented, -formal, commutator-relators group.Assume that the components of R ( G ) are (i) -isotropic, (ii) projectively disjoint,and (iii) reduced (viewing R ( G ) as a scheme). Then, for k ≫ , θ k ( G ) = ( k − X m ≥ h m (cid:18) m + k − k (cid:19) , where h m denotes the number of m -dimensional components of R ( G ) . Example 1.1.
Examples illustrating the necessity of the hypotheses in the theoreminclude the following. Let [ u, v ] = uvu − v − denote the commutator of u, v ∈ G .(a) The Heisenberg group G = h g , g | [ g , [ g , g ]] , [ g , [ g , g ]] i is not 1-formal.Here, R ( G ) = H ( G ; k ) is 0-isotropic since the cup product is trivial. But θ k ( G ) = θ k ( F ). Since G is nilpotent, the Chen groups of G are trivial. See [13]. HEN RANKS AND RESONANCE 3 (b) The fundamental group G of a closed, orientable surface of genus g ≥ R ( G ) = H ( G ; k ) is not 0-isotropic, and θ k ( G ) = θ k ( F g ). See [30].(c) Let G = G Γ be the right-angled Artin group corresponding to the graph Γ withvertex set V = { , , , , } and edge set E = { , , , , } . The resonancevariety R ( G ) is the union of two 3-dimensional subspaces in H ( G ; k ) which arenot projectively disjoint, and θ k ( G ) = 2 θ k ( F ). See [31].(d) Let G = h g , g , g , g | [ g , g ] , [ g , g ] , [ g , g ] , [ g , g ][ g , g ] i . As a variety, R ( G ) is a 2-dimensional subspace of H ( G ; k ). But R ( G ) is not reduced, and θ k ( G ) = θ k ( F ). We do not know if this group is 1-formal. See Example 3.3.Groups which satisfy the hypotheses of Theorem A include all arrangementgroups. More generally, let X be a smooth quasi-projective variety, and assumethat G = π ( X ) is a commutator-relators group. If X admits a smooth com-pactification with trivial first Betti number, work of Deligne [12] and Morgan [27]implies that G is 1-formal. In this instance, Dimca-Papadima-Suciu [13] show thatthe irreducible components of R ( G ) are all projectively disjoint and 0-isotropic.Thus, Theorem A applies when the components of R ( G ) are reduced. Anotherinteresting example is provided by the “group of loops.”Let F n be the free group of rank n . The basis-conjugating automorphism group,or pure symmetric automorphism group, is the group PΣ n of all automorphisms of F n which send each generator x i to a conjugate of itself. Results of Dahm [11] andGoldsmith [20] imply that this group may also be realized as the “group of loops,”the group of motions of a collection of n unknotted, unlinked oriented circles in3-space, where each circle returns to its original position. Theorem B.
For k ≫ , the ranks of the Chen groups of the basis-conjugatingautomorphism group PΣ n are θ k ( PΣ n ) = ( k − (cid:18) n (cid:19) + ( k − (cid:18) n (cid:19) . Our interest in the relationship between Chen ranks and resonance stems fromthe theory of hyperplane arrangements. Let A = { H , . . . , H n } be an arrange-ment in C ℓ , with complement M = C ℓ r S ni =1 H i . It is well known that thefundamental group G = π ( M ) is a commutator-relators group, and is 1-formal.Furthermore, in low dimensions, the cohomology of G is isomorphic to that of M , H ≤ ( G ; k ) ∼ = H ≤ ( M ; k ), see Matei-Suciu [25]. Consequently, the first resonancevarieties of G and of the Orlik-Solomon algebra A = H ∗ ( M ; k ) coincide. Falk andYuzvinsky observed that the irreducible components of R ( G ) are 0-isotropic (see[16] and [18]), and Libgober-Yuzvinsky [23] showed that these components are pro-jectively disjoint. In Section 5 below, we show that the components of R ( G ) arereduced. Thus, Theorem A yields a combinatorial formula for the Chen ranks ofthe arrangement group G in terms of the Orlik-Solomon algebra of A . In [38], Suciuconjectured that this formula, the Chen ranks conjecture, holds.Theorem A facilitates the explicit calculation of the Chen ranks of a number ofarrangement groups of broad interest. For example, let W be a finite reflectiongroup, and A W the arrangement of reflecting hyperplanes. The fundamental group P W of the complement of A W is the pure braid group associated to W . If W = A n is the type A Coxeter group, then P W = P A n = P n +1 is the classical Artin pure DANIEL C. COHEN AND HENRY K. SCHENCK braid group on n +1 strings, whose Chen ranks were determined in [8]. We completethe picture for the remaining infinite families. Theorem C.
Let
P A n , P B n and P D n be the pure braid groups associated to theCoxeter groups A n , B n and D n . For k ≫ , the ranks of the Chen groups of thesepure braid groups are θ k ( P A n ) = ( k − (cid:18) n + 24 (cid:19) ,θ k ( P B n ) = ( k − (cid:20) (cid:18) n (cid:19) + 9 (cid:18) n (cid:19)(cid:21) + ( k − (cid:18) n (cid:19) ,θ k ( P D n ) = ( k − (cid:20) (cid:18) n (cid:19) + 9 (cid:18) n (cid:19)(cid:21) . Preliminaries
Alexander invariant.
Let G be a finitely presented group, with abelian-ization G/G ′ and a : G → G/G ′ the natural projection. Let Z G be the integralgroup ring of G , and let J G = ker( ǫ ) be the kernel of the augmentation map ǫ : Z G → Z , given by ǫ (cid:0)P m g g ) = P m g . Classically associated to the group G are the Z ( G/G ′ )-modules A G = Z ( G/G ′ ) ⊗ Z G J G and B G = G ′ /G ′′ . The Alexan-der module A G is induced from J G by the extension of the abelianization map a to group rings. The action of G/G ′ on the Alexander invariant B G is givenon cosets of G ′′ by gG ′ · hG ′′ = ghg − G ′′ for g ∈ G and h ∈ G ′ . These mod-ules, and the augmentation ideal of Z ( G/G ′ ), comprise the Crowell exact sequence0 → B G → A G → J G/G ′ → J = J G/G ′ , and consider the J -adic filtration { J k B G } k ≥ of the Alexanderinvariant. Let gr( B G ) = L k ≥ J k B G /J k +1 B G be the associated graded moduleover the ring gr( Z ( G/G ′ )) = L k ≥ J k /J k +1 . A basic observation of Massey [24]shows that gr k ( G ′ /G ′′ ) = gr k − ( B G )for k ≥
2, where the associated graded on the left is taken with respect to the lowercentral series filtration. Consequently, the Chen ranks of G are given by X k ≥ θ k +2 ( G ) t k = Hilb( B G ⊗ k , t ) , for any field k of characteristic zero.Now assume that G = h g , . . . , g n | r , . . . , r m i admits a commutator-relatorspresentation with n generators, and let p : F n → G be the natural projection,where F n = h g , . . . , g n i . Set t i = a ◦ p ( g i ). The choice of basis t , . . . , t n for G/G ′ ∼ = Z n identifies the group ring Z ( G/G ′ ) with the Laurent polynomial ringΛ = Z [ t ± , . . . , t ± n ]. With this identification, the augmentation ideal is given by J = ( t − , . . . , t n − A G and Alexander invariant B G maybe obtained using the Fox calculus [19]. For 1 ≤ i ≤ n , let ∂ i = ∂∂g i : F n → Z F n bethe Fox free derivatives. The Alexander module has presentationΛ m D −−→ Λ n −→ A G −→ , where D = (cid:0) a ◦ p ◦ ∂ i ( r j ) (cid:1) is the Alexander matrix of (abelianized) Fox derivatives. HEN RANKS AND RESONANCE 5
Let ( C ∗ , d ∗ ) be the standard Koszul resolution of Z over Λ, where C = Λ, C = Λ n with basis e , . . . , e n , and C k = Λ( nk ). The differentials are given by d k ( e I ) = P kr =1 ( − r + k ( t i r − e I r { i r } , where e I = e i ∧ · · · ∧ e i k if I = { i , . . . , i k } .By the fundamental formula of Fox calculus [19], we have d ◦ D = 0. This yieldsa chain map(2.1) Λ m D −−−−→ Λ n d −−−−→ Λ α y id y id y · · · d −−−−→ Λ( n ) d −−−−→ Λ( n ) d −−−−→ Λ n d −−−−→ ΛBasic homological algebra insures the existence of the map α satisfying d ◦ α = D . Analysis of the mapping cone of this chain map as in [24, 9] then yields apresentation for the Alexander invariant B G = ker( d ) / im( D ):Λ( n ) + m ∆= d + α −−−−−−−→ Λ( n ) −→ B G −→ . Linearized Alexander invariant.
The ring Λ = Z [ t ± , . . . , t ± n ] may beviewed as a subring of the formal power series ring P = Z [[ x , . . . , x n ]] via theMagnus embedding, defined by ψ ( t i ) = 1 + x i . Note that the augmentation ideal J = ( t − , . . . , t n −
1) is sent to the ideal m = ( x , . . . , x n ). Passing to associatedgraded rings, with respect to the filtrations by powers of J and m respectively,the homomorphism gr( ψ ) identifies gr(Λ) with the polynomial ring S = gr( P ) = Z [ x , . . . , x n ].For q ≥
0, let ψ ( q ) : Λ → P/ m q +1 denote the q -th truncation of ψ . Since G is a commutator-relators group, all entries of the Alexander matrix D are in theaugmentation ideal J . It follows that ψ (0) ( D ) is the zero matrix. Consequently, allentries of the linearized Alexander matrix ψ (1) ( D ) are in m / m = gr ( P ), so maybe viewed as linear forms in the variables of S .For the n -generator, commutator-relators group G , the polynomial ring S = Z [ x , . . . , x n ] may be identified with the symmetric algebra on H ( G ) = G/G ′ = Z n .Let E = V H ( G ) be the exterior algebra (over Z ), and let ( E ⊗ S, ∂ ∗ ) be the Koszulcomplex of S . Observe that ∂ k = ψ (1) ( d k ), in particular, ∂ is the matrix of variablesof S , with image the ideal m = ( x , . . . , x n ). Note also that ∂ ◦ ψ (1) ( D ) = 0. Thelinearized Alexander invariant of G is the S -module B Z = ker( ∂ ) / im( ψ (1) ( D )). Theorem 2.1 (Papadima-Suciu [30] Cor. 9.7) . Let G be a -formal, commutator-relators group with associated linearized Alexander invariant B Z , and let k be afield of characteristic zero. Then X k ≥ θ k +2 ( G ) t k = Hilb( B Z ⊗ k , t ) . A presentation for the linearized Alexander invariant may be obtained by aprocedure analogous to that used to obtain one for the Alexander invariant itself.Linearizing the equality d ◦ α = D , we obtain ∂ ◦ α = ψ (1) ( D ), where α = ψ (0) ( α ), the entries of which are in S/ m . Thus, α is induced by a map Z m → Z ( n ), DANIEL C. COHEN AND HENRY K. SCHENCK which we denote by the same symbol. These considerations yield a chain map(2.2) S m D (1) −−−−→ S n ∂ −−−−→ S α y id y id y · · · ∂ −−−−→ S ( n ) ∂ −−−−→ S ( n ) ∂ −−−−→ S n ∂ −−−−→ S where D (1) = ψ (1) ( D ). Analysis of the mapping cone of this chain map then yieldsa presentation for the linearized Alexander invariant:(2.3) S ( n ) + m ∆ lin = ∂ + α −−−−−−−−−→ S ( n ) −→ B Z −→ . Resonance.
In the case where G is an arrangement group, it is known [10]that the annihilator of the linearized Alexander invariant defines the first resonancevariety. We show that this holds for an arbitrary commutator-relators group. Let k be a field of characteristic zero, and consider B := B Z ⊗ k , the linearized Alexanderinvariant with coefficients in k . Abusing notation, let S = k [ x , . . . , x n ]. Theorem 2.2.
Let G be an n generator, commutator-relators group. The resonancevariety R ( G ) is the variety defined by the annihilator of the linearized Alexanderinvariant B , R ( G ) = V (ann B ) .Proof. Let X be the 2-dimensional CW-complex corresponding to a commutator-relators presentation of G with n generators, and let R ( X ) ⊂ k n be the firstresonance variety of the cohomology ring H ∗ ( X ; k ). We first show that R ( G ) = R ( X ).A K ( G,
1) space may be obtained from X by attaching cells of dimension atleast 3. The resulting inclusion map X ֒ → K ( G,
1) induces an isomorphism be-tween H ( G ; k ) and H ( X ; k ). Identify these cohomology groups. Dualizing theHopf exact sequences reveals that H ( G ; k ) ֒ → H ( X ; k ). Consequently, if a, b ∈ H ( G ; k ) = H ( X ; k ), then a ∪ b = 0 in H ( G k ) if and only if a ∪ b = 0 in H ( X ; k ).It follows that R ( G ) = R ( X ).Consider the chain map (2.2), now with S = k [ x , . . . , x n ]. For f a map of free S -modules and a ∈ k n , denote the evaluation of f at a by f ( a ). The resonancevariety R ( G ) = R ( X ) may be realized as the variety in H ( G ; k ) = k n definedby the vanishing of the ( n − × ( n −
1) minors of the linearized Alexander matrix D (1) , see [25, 39]. In other words, R ( G ) = { a ∈ k n | rank D (1) ( a ) < n − } . An exercise with the mapping cone of (2.2) reveals that rank D (1) ( a ) < n − lin ( a ) < (cid:0) n (cid:1) . This implies that R ( G ) = V (ann B ). (cid:3) Over the field k , the presentation (2.3) of the linearized Alexander invariantmay be simplified. Recall that X is the presentation 2-complex corresponding toan n -generator, commutator-relators presentation of G . Let H = H ( X ; Z ) = H ( G ; Z ) = Z n . Specializing (2.2) at x i = 0 (resp., (2.1) at t i = 1) yields α : H ( X ; k ) → H ( H ; k ), which is dual to the cup product map µ : H ∧ H → H ( X ; k ), see [25]. The cohomology ring E = H ∗ ( H ; k ) is an exterior algebra.Let I ⊂ E be the ideal generated by ker( µ : E ∧ E → H ( X ; k )). Writing A = E/I , for a ∈ A = E , we obtain a short exact sequence of cochain complexes HEN RANKS AND RESONANCE 7 → ( I, a ) → ( E, a ) → ( A, a ) →
0. If a = 0, ( E, a ) is acyclic, and H ( A, a ) = 0 ifand only if ker( I a −−→ I ) = 0. Thus, R ( G ) = { a ∈ A | ker( I a −−→ I ) = 0 } . If e , . . . , e n ∈ E generate the exterior algebra E , let δ k : E k − ⊗ S → E k ⊗ S denote multiplication by P ni =1 x i e i , dual to the Koszul differential ∂ k : E k ⊗ S → E k − ⊗ S . Tensoring the exact sequence 0 → I → E → A → S , we obtaina commutative diagram of cochain complexes, with exact columns(2.4) · · · δ A ←−−−− A ⊗ S δ A ←−−−− A ⊗ S δ A ←−−−− A ⊗ S δ A ←−−−− A ⊗ S x µ ⊗ id x id x id x · · · δ ←−−−− E ⊗ S δ ←−−−− E ⊗ S δ ←−−−− E ⊗ S δ ←−−−− E ⊗ S x x · · · δ I ←−−−− I ⊗ S δ I ←−−−− I ⊗ S where δ kI denotes the restriction, and δ kA denotes the induced map on the quotient.Referring to the diagram (2.2), we have A ⊗ S ∼ = S m / ker( α : S m → S ( n )). Themap D (1) = ψ (1) ( D ) is trivial on this kernel, let ¯ D (1) denote the induced map onthe quotient. Then the linearized Alexander invariant may be realized as B =ker( ∂ ) / im( ¯ D (1) ), and the map δ A is dual to ¯ D (1) . Dualizing the diagram (2.4), weobtain an exact sequence of chain complexes0 → ( A ⊗ S, ∂ A ) → ( E ⊗ S, ∂ E ) → ( I ⊗ S, ∂ I ) → , where ∂ • is dual to δ • . Since ( E ⊗ S, ∂ E ) is acyclic, we have B = H ( A ⊗ S, ∂ A ) = H ( I ⊗ S, ∂ I ), yielding the following presentation for B :(2.5) I ⊗ S ∂ I −−−→ I ⊗ S −→ B −→ . Chen ranks from resonance
In this section, we prove Theorem A. For a group G satisfying the hypotheses ofTheorem A, we determine the ranks of the Chen groups of G from the resonancevariety R ( G ),Let G be a finitely presented, 1-formal, commutator-relators group. These as-sumptions on G insure (i) that the ranks of the Chen groups are given by theHilbert series of the linearized Alexander invariant B of G (with coefficients in k ): X k ≥ θ k ( G ) t k = Hilb( B , t ) , see Papadima-Suciu [30, Cor. 9.7]; and (ii) that all irreducible components of R ( G )are linear subspaces of H ( G ; k ), see Dimca-Papadima-Suciu [13, Thm. B]. Assumethat these components of R ( G ) are 0-isotropic, projectively disjoint, and reduced.Let L be an irreducible component of R ( G ), and let { l , . . . , l m } be a basis for L , where the l i are linearly independent elements of A = E = H ( G ; k ). Since L is 0-isotropic, we have l i ∧ l j = 0 in A for 1 ≤ i < j ≤ m . Consequently, the ideal(3.1) I L = h l i ∧ l j | ≤ i < j ≤ m i DANIEL C. COHEN AND HENRY K. SCHENCK is a subideal of the ideal I generated by ker( µ : E ∧ E → A ), where A = H ( X ; k ) and X is the presentation 2-complex corresponding to an n -generator,commutator-relators presentation of G as in the previous section. Correspondingto L , we have a “local” linearized Alexander invariant B L , presented in analogywith (2.5) by(3.2) I L ⊗ S ∂ L −−−→ I L ⊗ S −→ B L −→ , where ∂ Lk is dual to the map δ kI L : I k − L ⊗ S → I kL ⊗ S given by the restriction to I L of multiplication by P ni =1 x i e i .The inclusion I L ⊆ I yields a short exact sequence of cochain complexes of free S -modules(3.3) . . . ←−−−− I /I L ⊗ S ψ ←−−−− I /I L ⊗ S ψ ←−−−− I /I L ⊗ S x x x . . . ←−−−− I ⊗ S δ I ←−−−− I ⊗ S δ I ←−−−− I ⊗ S x x x . . . ←−−−− I L ⊗ S δ IL ←−−−− I L ⊗ S δ IL ←−−−− I L ⊗ S, where ψ is the induced map. Dualizing and passing to homology yields a surjection(3.4) π L : B = coker( ∂ I ) − ։ coker( ∂ L ) = B L . Let L , . . . , L k be the irreducible components of the resonance variety R ( G ),and define π : B → L ki =1 B L i by π = L ki =1 π L i . This yields an exact sequence(3.5) 0 −→ K −→ B π −−→ k M i =1 B L i −→ C −→ . Let m = h x , . . . , x n i be the maximal ideal in the polynomial ring S . We will showthat the modules K = ker π and C = coker π are supported only at m .For 1 ≤ j ≤ k , let q j = I ( L j ) be the prime ideal in S with V ( q j ) = L j . Sincelocalization is an exact functor, the sequence (3.5) remains exact after localizingat any ideal q such that V ( q ) ⊂ R ( G ). The assumption that the components of R ( G ) are projectively disjoint implies that ( B L i ) q j = 0 if i = j . This impliesthat C q j = 0 for 1 ≤ j ≤ k . For an embedded prime p for R ( G ), different from m , the same argument shows that C p = 0. Thus C can be supported only at themaximal ideal. In the case where G is an arrangement group, this yields the lowerbound on the Chen ranks found in [37]; however the proof there relies on a resultof Eisenbud-Popescu-Yuzvinsky [14] which is special to the case of arrangements.To establish Theorem A, it remains to show that the kernel K in (3.5) is sup-ported only at the maximal ideal. Let L = V ( q ) be an m -dimensional irreduciblecomponent of R ( G ). Given L ⊂ R ( G ), let J L be the ideal in the exterior algebra E with generating set { g ∈ I | l ∧ g ∈ I L ∀ l ∈ L } . If L has basis { l , . . . , l m } ,then from the definition (3.1) of I L , we have I L : h l , . . . , l m i = h l , . . . , l m i in E .Moreover, since J L = ( I L : h l , . . . , l m i ) ∩ I , we have(3.6) J L = h l , . . . , l m i ∩ I. Proposition 3.1.
The ideals I L and J L are equal if and only if ( B ) q ≃ ( B L ) q . HEN RANKS AND RESONANCE 9
Proof.
Choose a basis for E so that L = span { e , . . . , e m } and I L is consequentlygenerated by e i e j := e i ∧ e j , 1 ≤ i < j ≤ m .First, suppose I L = J L . Since h e , . . . , e m i∩ I = I L , we may choose (independent)elements { f , . . . , f r } in E whose images form a basis for the quotient I /I L , andwhose initial terms in( f i ) = g i are distinct elements of the ideal generated by e i e j , m < i < j ≤ n . Note that { e f , . . . , e f r } are independent in I , for if not, then r X i =1 c i e f i = e r X i =1 c i f i = 0 , so P ri =1 c i f i ∈ h e i . Since the initial terms in( f i ) = g i are distinct, P ri =1 c i f i = 0 isimpossible, and P ri =1 c i f i cannot be a nontrivial multiple of e for the same reason.Recall from (3.3) that ψ : I /I L ⊗ S → I /I L ⊗ S denotes the map induced by δ I : I ⊗ S → I ⊗ S given by multiplication by P ni =1 x i e i . Let M be the submatrixof (the matrix of) ψ with rows corresponding to e f , . . . , e f r ∈ I . Then M = x · id + M ′ , with M ′ ∈ k [ x , . . . , x n ] r × r , where id is the identity matrix of size r = dim I /I L . To see this, note that δ I ( f j ) = x e f j + n X i =2 x i e i f j . While there can be syzygies with e i f j = e f k + · · · , for such a relation we have i ≥ x i with i ≥ M ) = x r + h , with deg x ( h ) < r .In the localization S q , since x q = h x m +1 , . . . x n i , det( M ) is a unit. Thisimplies that the maximal Fitting ideal Fitt (( ψ ) q ) contains a unit. Consequently,after localizing, the cokernel of ψ vanishes. From the long exact sequence arisingfrom the dual of (3.3), this cokernel is the kernel of the map ( B ) q → ( B L ) q → π L of (3.4) induces an isomorphism ( B ) q ≃ ( B L ) q .Now suppose the irreducible component L of R ( G ) is such that I L = J L . Let g ∈ ( J L \ I L ), assume as above that L has basis { e , . . . , e m } , and write L ∗ =span { e m +1 , . . . , e n } . Note that g is indecomposable, for if g = l ∧ l with l ∈ L , l ∈ L ∗ , then the plane span { l , l } ⊆ R ( G ) intersects L in a line, contradictingthe assumption that the components of R ( G ) are projectively disjoint.Denote the ideal I L + g in the exterior algebra E by K L , and let δ qK L : K q − L ⊗ S → K qL ⊗ S be multiplication by ω = P ni =1 x i e i . Let B K L be the S -module presentedby the dual ∂ K L of δ K L ,(3.7) I K L ⊗ S ∂ KL −−−−→ I K L ⊗ S −→ B K L . We will construct a free resolution K • of B K L .First, consider the cochain complex C • = ( I • L ⊗ S, δ • L ) corresponding to the ideal I L in E . This complex is acyclic. To see this, let C • (0) denote the Koszul (cochain)complex V ∗ L ⊗ S , with differential given by multiplication by ω ′ = P mi =1 x i e i .Multiplication by x m +1 induces a chain map C • (0) → C • (0). Let C • (1) be thecorresponding mapping cone. Since C • (0) is acyclic, so is C • (1). Continuing inthis manner, multiplication by x m + j +1 induces a chain map C • ( j ) → C • ( j ), andthe resulting mapping cone C • ( j + 1) is acyclic. Thus, C • = C • ( n − m ) is acyclic.Let C • = ( I • L ⊗ S, ∂ L • ) be the chain complex dual to C • . Since C • is acyclic and free, the dual complex C • gives a free resolution of the module B L ,(3.8) · · · −→ I k ⊗ S ∂ Lk −−−→ · · · −→ I ⊗ S ∂ L −−−→ I ⊗ S −→ B L . Let b C • be the Koszul complex V • L ∗ ⊗ S , with differential given by multiplicationby − ω ′′ = − P ni = m +1 x i e i . Multiplication by ω ′ g = P mi =1 x i e i g induces a chain map b C • → C • . Let K • be the mapping cone of this chain map. Then we have a shortexact sequence of cochain complexes 0 → C • → K • → b C • →
0. Since C • and b C • are acyclic and free, K • is acyclic and free. It is readily checked that the map δ K L : I K L ⊗ S → I K L ⊗ S , f ωf = ω ′ f + ω ′′ f , coincides with the first differentialof K • . It follows that the chain complex K • dual to K • is a free resolution of B K L .Recall that q = h x m +1 , . . . , x n i . The above considerations yield a short exactsequence of chain complexes 0 → b C • → K • → C • →
0, where b C • is dual to b C • ,and is a free resolution of S/ q . We thus have a short exact sequence of S -modules(3.9) 0 −→ S/ q −→ B K L −→ B L −→ . Since the localization ( S/ q ) q is nontrivial, we have ( B K L ) q ( B L ) q . The surjection π L : B → B L of (3.4) factors through B K L , so the fact that ( B K L ) q ( B L ) q implies that ( B ) q ( B L ) q , which completes the proof. (cid:3) Proposition 3.1 implies that if I L = J L for each irreducible component L of R ( G ), then the modules K = ker π and C = coker π in the exact sequence (3.5) areof finite length, and hence the Hilbert polynomials of the modules B and L ki =1 B L i are equal. If dim L = m , a straightforward exercise using the resolution (3.8) showsthat the Hilbert polynomial of B L is ( k − (cid:0) m + k − k (cid:1) . Consequently, to completethe proof of Theorem A, it suffices to prove the following. Proposition 3.2.
The ideals I L and J L are equal if and only if the irreduciblecomponent L of R ( G ) is reduced.Proof. We continue with the notation established in the proof of Proposition 3.1.Let L = span { e , . . . , e m } be an m -dimensional irreducible component of R ( G ),and q = I ( L ) the prime ideal in S with V ( q ) = L . The component L of R ( G ) isreduced if in the primary decompositionann( B ) = \ Q i with p Q i = q i , the q -primary component is equal to q .Assume that I L = J L . For simplicity, bigrade E , viewing elements of L as ofbidegree (1 ,
0) and elements of L ∗ = span { e m +1 , . . . , e n } as of bidegree (0 , I L and I L in a grading where (1 , > (0 ,
1) (e.g., lex order).Then, the map ∂ L : I L ⊗ S → I L ⊗ S of (3.2) presenting the module B L has matrix[ ∂ L ] = (cid:2) d L ( x , . . . , x m ) x m +1 · id x m +2 · id · · · x n · id (cid:3) = (cid:2) d L X (cid:3) , where d L ( x , . . . , x m ) is the second Koszul differential on x , . . . , x m , id is the iden-tity matrix of size | I L | = (cid:0) m (cid:1) , and X = (cid:2) x m +1 · id · · · x n · id (cid:3) . Using this, onecan check that ann( B L ) = h x m +1 , . . . , x n i = q . Since ( B ) q ≃ ( B L ) q when I L = J L by Proposition 3.1, and localization commutes with taking annihilators, it followsthat L is reduced in this instance.For the other direction, we show that I L = J L implies that L is not reduced. Let g be an indecomposable element in J L \ I L , and let K L = I L + g . Changing basesin L and L ∗ = span { e m +1 , . . . , e n } , we may assume that g = e e m +1 + · · · + e k e m + k HEN RANKS AND RESONANCE 11 for some k , 2 ≤ k ≤ m . To see that L is not reduced, it suffices to exhibit anelement β in the module B K L which is not annihilated by q = h x m +1 , . . . , x n i .Choose ordered bases for K L and K L whose initial segments are the bases of I L and I L above, so that g appears last in the basis for K L , and g ∧ L ∗ is the finalsegment of the basis for K L . With respect to these ordered bases (in light of themapping cone construction in the proof of Proposition 3.1), the map ∂ K L : K L ⊗ S → K L ⊗ S of (3.7) presenting B K L has matrix[ ∂ K L ] = (cid:20) d L X y x (cid:21) , where x = [ x m +1 · · · x n ]. As ∂ K L is dual to the multiplication map δ K L : K L ⊗ S → K L ⊗ S , the last row of [ ∂ K L ] corresponds to δ K L ( g ) = ( P ni =1 x i e i ) ∧ g . Using this,and g = P ki =1 e i e m + i , one can check that the the entries of y are in k [ x , . . . , x m ].Let β be the class of e e ∈ K L ⊗ S in B K L , and assume that x m +1 β = 0, i.e., that x m +1 e e ∈ im( ∂ K L ). Then there exists u ∈ K L ⊗ S so that ∂ K L ( u ) = x m +1 e e .From the form of the matrix of ∂ K L above, we must have u = e e e m +1 + v for some v ∈ K L ⊗ S . Since the coefficient of e e e m +1 in δ K L ( g ) is − x , the transpose of thecolumn of [ ∂ K L ] corresponding to e e e m +1 is (cid:2) x m +1 · · · − x (cid:3) . In otherwords, ∂ K L ( e e e m +1 ) = x m +1 e e − x g . Thus, ∂ K L ( v ) = ∂ K L ( u − e e e m +1 ) = x g , and the assumption that x m +1 e e ∈ im( ∂ K L ) implies that x g ∈ im( ∂ K L ) aswell. This in turn implies that the kernel of the map B K L ։ B L is annihilated by x , a contradiction since this kernel is S/ q = S/ h x m +1 , . . . , x n i , see (3.9). (cid:3) Example 3.3.
For the group G in Example 1.1(d), I = h e e , e e + e e i , and R ( G ) = L = span { e , e } , so that I L = h e e i . Since J L = I = I L , the resonancecomponent L is not reduced. A calculation reveals that the Chen ranks of G aregiven by θ k ( G ) = 2( k − θ k ( F ) = k − k ≥ Basis-conjugating automorphism groups
As a first application of Theorem A, we compute the Chen ranks of the basis-conjugating automorphism group G = PΣ n , proving Theorem B.Let F n be the free group generated by x , . . . , x n . The basis-conjugating au-tomorphism group PΣ n is the group of all automorphisms of F n which send eachgenerator x i to a conjugate of itself. As noted in the introduction, this group maybe realized as the group of motions of a collection of n unknotted, unlinked orientedcircles in 3-space, where each circle returns to its original position. McCool [26]found the following presentation for the basis-conjugating automorphism group:(4.1) PΣ n = h β i,j , ≤ i = j ≤ n | [ β i,j , β k,l ] , [ β i,k , β j,k ] , [ β i,j , ( β i,k · β j,k )] i , where [ u, v ] = uvu − v − , the indices in the relations are distinct, and the generators β i,j are the automorphisms of F n defined by β i,j ( x k ) = ( x k if k = j , x − j x i x j if k = i .The integral cohomology of PΣ n was determined by Jensen-McCammond-Meier[22], resolving a conjecture of Brownstein-Lee [3]. We rephrase their result for afield k of characteristic zero. Let E be the exterior algebra over k generated by degree one elements e p,q , 1 ≤ p = q ≤ n , and let I be the two-sided ideal in E generated by e i,j e j,i , ≤ i < j ≤ n, ( e k,i − e j,i )( e k,j − e i,j ) , ≤ i, j, k ≤ n, i < j, k / ∈ { i, j } . Then the cohomology algebra of the basis-conjugating automorphism group PΣ n is isomorphic to the quotient of E by I , H ∗ ( PΣ n ; k ) ∼ = E/I . Using this descriptionof the cohomology, Cohen [5] computed the first resonance variety of PΣ n :(4.2) R ( PΣ n ) = [ ≤ i From work of Berceanu-Papadima [2], it is known that PΣ n is 1-formal. And it is clear from (4.1) that PΣ n is a commutator-relators group.Checking that the components C i,j , C i,j,k from (4.2) are all projectively disjoint,and are all 0-isotropic, the Chen ranks of PΣ n are given by Theorem A providedthat all these components are reduced.The symmetric group on n letters acts on PΣ n by permuting indices, σ ( β i,j ) = β σ ( i ) ,σ ( j ) , and hence on the cohomology and resonance variety of PΣ n . In light ofthis action, it suffices to show that the resonance components C , and C , , arereduced. We establish this using Proposition 3.1.Let L = C , , = span { e , − e , , e , − e , , e , − e , } , and let g ∈ I , where I is the ideal in the exterior algebra E defining the cohomology of PΣ n . Write g = X i Summarizing, we have h = X ≤ i The upper triangular McCool groups illustrate the necessity of thehypotheses of Theorem A. For each n ≥ 2, the upper triangular McCool group isthe subgroup PΣ + n of PΣ n generated by the elements β i,j with 1 ≤ i < j ≤ n ,subject to the relevant relations (4.1). Thus, PΣ + n is a commutator-relators group.Moreover, in [2], Berceanu-Papadima remark that PΣ + n is 1-formal.In [7], Cohen-Pakianathan-Vershinin-Wu determine the integral cohomology of PΣ + n , see also [6]. We rephrase their result for a field k of characteristic zero. Let E + be the exterior algebra over k generated by elements e p,q , 1 ≤ p < q ≤ n , ofdegree one, and let I + be the two-sided ideal in E + generated by e j,k ( e i,k − e i,j ),1 ≤ i < j < k ≤ n . Then, H ∗ ( PΣ + n ; k ) ∼ = E + /I + .Using the above description of the cohomology ring, one can check that theresonance variety R ( PΣ +4 ) has componentsspan { e , − e , , e , } , span { e , − e , , e , } , andspan { e , − e , , e , − e , , e , } . These components are projectively disjoint. However, the 3-dimensional component L = span { e , − e , , e , − e , , e , } is neither 0-isotropic nor reduced. For theformer, note that ( e , − e , )( e , − e , ) is nonzero in H ∗ ( PΣ +4 ; k ). For the latter, L has an embedded component span { e , } . Accordingly (see Proposition 3.2), theideals I L and J L are not equal. For instance, e , ( e , − e , ) − e , ( e , − e , ) isin J L , but not in I L .In light of the above observations, it is not surprising that the Chen ranks for-mula of Theorem A does not hold for the upper triangular McCool groups. Forexample, a computation reveals that, for k ≫ 0, the Chen ranks of PΣ +4 are givenby θ k ( PΣ +4 ) = 1 + θ k ( F ) + θ k ( F ), which differs from the value 2 θ k ( F ) + θ k ( F )naively predicted from the resonance variety R ( PΣ +4 ).5. Hyperplane arrangements Let A = { H , . . . , H n } be a hyperplane arrangement in C ℓ , with complement M ( A ) = C ℓ r S ni =1 H i . We assume that A is a central arrangement, i.e., that eachhyperplane of A passes through the origin. Let L ( A ) = { T H ∈B H | B ⊆ A} bethe intersection lattice of A , with rank function given by codimension. We referto elements of L ( A ) as flats. Recall that k is a field of characteristic zero. A wellknown theorem of Orlik-Solomon [28] yields a presentation for the cohomology ring A = H ∗ ( M ( A ); k ), the Orlik-Solomon algebra, in terms of the lattice L ( A ). Let G = π ( M ( A )) be the fundamental group of the complement. As noted in theintroduction, the arrangement group G is a 1-formal, commutator-relators group,and the resonance varieties R ( G ) and R ( A ) coincide. In this context, we denotethis variety by R ( A ), the first resonance variety of the arrangement A .Falk [16] initiated the study of resonance varieties in the context of arrangements.Among his main innovations was the concept of a neighborly partition. A partitionΠ of A is neighborly if, for any rank two flat Y ∈ L ( A ) and any block π of Π, | Y | − ≤ | Y ∩ π | = ⇒ Y ⊆ π, Partitions with a single block will be called trivial, others nontrivial. Flats containedin a single block of Π will be referred to as monochrome, others polychrome. Flatsof multiplicity two are necessarily monochrome.Falk showed that all components of R ( A ) arise from nontrivial neighborly par-titions of subarrangements of A , and conjectured that R ( A ) was a subspace ar-rangement. This was proved simultaneously by Cohen-Suciu in [10] and Libgober-Yuzvinsky in [23], and the latter also showed that the irreducible components of R ( A ) are projectively disjoint. As noted by Falk-Yuzvinsky [18] (see also [16]),these components are 0-isotropic. Arrangements which admit nontrivial neighborlypartitions, and corresponding resonance components, include all central arrange-ments in C , the rank 3 braid arrangement, the Pappus, Hessian, and type BCoxeter arrangements in C , etc., see [10, 16, 34, 38] among others.Let Π be a neighborly partition of a subarrangement A ′ of A . Following [16, 17,23], we explicitly describe the corresponding component L Π of R ( A ). Let E be theexterior algebra over k , with generators e , . . . , e n corresponding to the hyperplanesof A . The Orlik-Solomon algebra is given by H ∗ ( M ( A ); k ) = A = E/I , where I is the Orlik-Solomon ideal of A . This ideal is generated by boundaries of circuits, ∂e i · · · e i k , where { H i , . . . , H i k } is a minimally dependent set of hyperplanes in A and ∂ : E → E is defined by ∂ ∂e i = 1, and ∂ ( uv ) = ( ∂u ) v + ( − | u | u ( ∂v ), | u | denoting the degree of u , see [29, Ch. 3]. For u = P ni =1 u i e i ∈ E , write ∂ i u = u i ,and ∂ X u = P X ⊂ H i u i for a rank 2 flat X . Let poly(Π) denote the set of rank twoflats X which are polychrome with respect to Π. Then the resonance componentcorresponding to the neighborly partition Π of A ′ ⊂ A is given by(5.1) L Π = { u ∈ E | ∂u = 0 , ∂ X u = 0 ∀ X ∈ poly(Π) , ∂ i u = 0 ∀ H i / ∈ A ′ } . Since the irreducible components of R ( A ) are 0-isotropic and projectively dis-joint, to bring Theorem A to bear in the context of arrangement groups, it sufficesto show that these components are also reduced. To establish this, in addition toFalk’s description of components of R ( A ) in terms of neighborly partitions de-scribed above, we will also use the more recent Falk-Yuzvinsky [18] description ofresonance components in terms of multinets, which utilizes the Libgober-Yuzvinsky[23] treatment in terms of the Vinberg classification of generalized Cartan matrices.A multiarrangement is an arrangement A together with a multiplicity function ν : A → N which assigns a positive integer ν ( H ) to each hyperplane in A . A ( k, d )-multinet on a multiarrangement ( A , ν ) (of lines in CP ) is a pair (Π , X ), whereΠ = ( β | β | · · · | β k ) is a partition of A with k ≥ X is a set of rank 2flats of A , satisfying(i) For each block β i of the partition Π, the sum P H ∈ β i ν ( H ) = d is constant,independent of i ;(ii) For each H ∈ β i and H ′ ∈ β j with i = j , the flat H ∩ H ′ is in X ; HEN RANKS AND RESONANCE 15 (iii) For each flat X ∈ X , P H ∈ β i ,X ⊂ H ν ( H ) is constant, independent of i ;(iv) For each i and H, H ′ ∈ β i , there is a sequence H = H , H , . . . , H r = H ′ of hyperplanes of A such that H j − ∩ H j / ∈ X for 1 ≤ j ≤ r .As shown by Falk-Yuzvinsky [18], an ℓ -dimensional component of R ( A ) cor-responds to a multiplicity function and an ( ℓ + 1 , d )-multinet on a subarrange-ment of A , for some d . If (Π , X ) is a multinet on ( A ′ , ν ), where A ′ ⊂ A andΠ = ( β | β | · · · | β ℓ +1 ), the corresponding resonance component L = L Π has basis(5.2) { ν − ν ℓ +1 , ν − ν ℓ +1 , . . . , ν ℓ − ν ℓ +1 } , where ν i = X H j ∈ β i ν ( H j ) e j = X H j ∈ β i ν i,j e j ,see [18, Thms. 2.4, 2.5]. Note that the partition Π is neighborly, the elements of X are the polychrome flats of Π, the coefficients ν i,j = ν ( H j ) are positive integers,and ∂ν i = ∂ν j for 1 ≤ i, j ≤ ℓ + 1 by (i) above. Theorem 5.1. For any hyperplane arrangement A , the irreducible components ofthe resonance variety R ( A ) are reduced.Proof. Let L ⊂ R ( A ) be an irreducible component. By Propositions 3.1 and 3.2,it suffices to show that the ideals I L = V L and J L = h g ∈ I | xg ∈ I L ∀ x ∈ L i inthe exterior algebra E are equal.The component L corresponds to a neighborly partition Π of a subarrangement A ′ of A . If Π has ℓ + 1 blocks, then dim L = ℓ , see [18, 23], (5.2), and below.Write A ′ = { H , . . . , H m } , and let E ′ be the subalgebra of E with generatorscorresponding to the hyperplanes of A ′ . Choose generators ξ , . . . , ξ n for E sothat E ′ is generated by ξ , . . . , ξ m and L = span { ξ , . . . , ξ ℓ } . Then I L = h ξ i ξ j | ≤ i < j ≤ ℓ i . Let g ∈ J L be a generator. Write g = g + g + g , where g ∈ V span { ξ , . . . , ξ m } , g ∈ span { ξ , . . . , ξ m } ∧ span { ξ m +1 , . . . , ξ n } , and g ∈ V span { ξ m +1 , . . . , ξ n } . Then, the condition xg ∈ I L for all x ∈ L implies that g = g = 0. Thus, the generators of J L are elements of the subalgebra E ′ of E .Consequently, we can assume without loss that A ′ = A .For the remainder of the proof, we use the standard generators e , . . . , e n of E ,in correspondence with the hyperplanes of A . If the neighborly partition Π of A isgiven by Π = ( β | β | · · · | β ℓ +1 ), the resonance component L has basis given by(5.2), and dim L = ℓ . Recall from (3.6) that J L = h ν − ν ℓ +1 , . . . , ν ℓ − ν ℓ +1 i ∩ I .Consequently, if g ∈ J L is a generator (not necessarily in I L ), we can write g =( ν − ν ℓ +1 ) u + · · · + ( ν ℓ − ν ℓ +1 ) u ℓ . We will show that g ∈ I L by showing that foreach i , 1 ≤ i ≤ ℓ , u i satisfies the conditions of (5.1), that is, u i ∈ L = L Π . Notethat since A ′ = A , the last of the conditions in (5.1) is vacuous.Let X ∈ L ( A ) be a rank two flat which is polychrome with respect to Π, thatis, X ∈ X . Then X meets each block of Π, see [23, Rem. 3.10]. We can assumethat X is contained in the hyperplanes H , . . . , H ℓ +1 (and possibly others), andthat H j ∈ β j for 1 ≤ j ≤ ℓ + 1. Write the basis elements of L as ν i − ν ℓ +1 = y i + z i ,where y i ∈ span { e r | X ⊂ H r } and z i = ν i − ν ℓ +1 − y i ∈ span { e r | X H r } foreach i , 1 ≤ i ≤ ℓ . From our assumptions concerning the hyperplanes containing X and (5.2), we have y i = ν i,i e i − ν ℓ +1 ,ℓ +1 e ℓ +1 + X k>ℓ +1 ,X ⊂ H k ( ν i,k − ν ℓ +1 ,k ) e k . Since the multiplicities ν i,j = ν ( H j ) are positive integers, it is readily checked that { y , . . . , y ℓ } is linearly independent. We now show that g = ( ν − ν ℓ +1 ) u + · · · + ( ν ℓ − ν ℓ +1 ) u k is in I L . Since ν i − ν ℓ +1 ∈ L , we have ∂ ( ν i − ν ℓ +1 ) = 0. Similarly, since g is in the Orlik-Solomonideal I , we have ∂g = 0. Computing ∂g = − ( ν − ν ℓ +1 ) ∂u −· · ·− ( ν ℓ − ν ℓ +1 ) ∂u ℓ = 0,the fact that { ν − ν ℓ +1 , . . . , ν ℓ − ν ℓ +1 } is linearly independent implies that ∂u = · · · = ∂u ℓ = 0. Write u i = v i + w i , where v i ∈ span { e r | X ⊂ H r } and w i = u i − v i .Then g = ℓ X i =1 ( ν i − ν ℓ +1 ) u i = ℓ X i =1 ( y i + z i )( v i + w i ) = ℓ X i =1 ( y i v i + y i w i + z i v i + z i w i ) . Since g ∈ J L ⊂ I , I = L Y ∈ L ( A ) I Y , and w i , z i ∈ span { e r | X H r } for each i ,the element h = P ℓi =1 y i v i is in I X . Thus, ∂h = ∂y v + · · · + ∂y ℓ v ℓ − y ∂v − · · · − y ℓ ∂v ℓ = 0 . Since ν i − ν ℓ +1 ∈ L = L Π and X is a polychrome flat for Π, we have ∂y i = ∂ X ( ν i − ν ℓ +1 ) = 0 for each i . Consequently, ∂h = − y ∂v − · · · − y ℓ ∂v ℓ = 0, andsince { y , . . . , y ℓ } is linearly independent, we have ∂v = · · · = ∂v ℓ = 0, that is, ∂ X u i = 0 for each i . But the polychrome flat X was arbitrary, so u , . . . , u ℓ satisfy ∂ X u i = 0 for each i and every flat X which is polychrome with respect to Π. Thisimplies that u , . . . , u ℓ ∈ L , see (5.1). Therefore, g = P ℓi =1 ( ν i − ν ℓ +1 ) u i ∈ I L . (cid:3) Thus, Theorem A provides a formula for the ranks of the Chen groups of G interms of the resonance variety R ( A ). This formula, θ k ( G ) = P m ≥ h m θ k ( F m ) for k ≫ 0, where h m is the number of irreducible components of R ( A ) of dimension m , for the Chen ranks was conjectured by Suciu [38]. In [37], Schenck-Suciu provedthat θ k ( G ) ≥ P m ≥ h m θ k ( F m ) for k ≫ 0. Suciu’s original conjecture predictedequality in this Chen ranks formula for all k ≥ 4, but in [37] it is shown the valuefor which θ k ( G ) is given by a fixed polynomial in k depends on the Castelnuovo-Mumford regularity of the linearized Alexander invariant of G . Example 5.2. Let A be the Hessian arrangement in C , defined by the polynomial Q = xyz Q ≤ i,j ≤ ( x + ω i z + ω j z ), where ω = exp(2 πi/ A consists of the twelve lines in CP passing through the nine inflection points ofa smooth plane cubic curve. Four lines meet at each of the nine inflection points,yielding nine rank 2 flats in L ( A ) of cardinality 4, and associated 3-dimensional com-ponents of R ( A ). The arrangement A has 54 subarrangements lattice-isomorphicto the rank 3 braid arrangement. Each of these contributes a 2-dimensional compo-nent to R ( A ). The arrangement A itself admits a nontrivial neighborly partition,and has a corresponding 3-dimensional component of R ( A ). A calculation revealsthat these (10 3-dimensional and 54 2-dimensional) components constitute all irre-ducible components of R ( A ). Consequently, if G is the fundamental group of thecomplement of A , by Theorem A we have θ k ( G ) = 10( k − 1) + 54( k − 1) for k ≫ Coxeter arrangements In this section, we use Theorem A to determine the Chen ranks of the pure braidgroups associated to the Coxeter groups of types A, B, and D, proving Theorem C. Example 6.1. Let A n be the braid arrangement, the type A Coxeter arrangementin C n with hyperplanes ker( x i − x j ), 1 ≤ i < j ≤ n . The complement of A n is the configuration space of n ordered points in C , with fundamental group the HEN RANKS AND RESONANCE 17 Artin pure braid group G = P n . The resonance variety R ( A n ) has (cid:0) n +14 (cid:1) two-dimensional irreducible components, see [10, 33]. Theorem A yields θ k ( P n ) =( k − (cid:0) n +14 (cid:1) for k ≫ 0, as first calculated in [8].More generally, let Γ be a simple graph on vertex set { , . . . , n } , and let A Γ be the corresponding graphic arrangement in C n , consisting of the hyperplanesker( x i − x j ) for which { i, j } is an edge of Γ. The resonance variety R ( A Γ ) has κ + κ two-dimensional irreducible components, where κ m denotes the number ofcomplete subgraphs on m vertices in Γ, see [37]. If G is the fundamental group ofthe complement of A Γ , Theorem A yields θ k ( G ) = ( k − κ + κ ) for k ≫ 0, asfirst calculated in [37].6.1. Resonance of the Coxeter arrangement of type D. Let D n be the typeD Coxeter arrangement in C n , with n ( n − 1) hyperplanes H ± i,j = ker( x i ± x j ),1 ≤ i < j ≤ n . The rank 2 flats of D n are H − i,j ∩ H − i,k ∩ H − j,k , H − i,j ∩ H + i,k ∩ H + j,k , H − p,q ∩ H + p,q ,H + i,j ∩ H − i,k ∩ H + j,k , H + i,j ∩ H + i,k ∩ H − j,k , H ± p,q ∩ H ± r,s , where 1 ≤ i < j < k ≤ n , 1 ≤ p < q ≤ n , 1 ≤ r < s ≤ n , and { p, q } ∩ { r, s } = ∅ .Note that D n has 4 (cid:0) n (cid:1) rank two flats of multiplicity 3, and (cid:0) n (cid:1) + 12 (cid:0) n (cid:1) rank twoflats of multiplicity 2. We determine the structure of the variety R ( D n ) ⊂ k n ( n − .Recall that A n is the type A Coxeter arrangement in C n . Denote the hyperplanesof A n by H − i,j = ker( x i − x j ), 1 ≤ i < j ≤ n . Note that A is lattice-isomorphic to D . It is well known that A and A subarrangements give rise to two-dimensionalcomponents of the first resonance variety, see Falk [16]. It is also known that D subarrangements also yield two-dimensional components of the first resonancevariety, see Pereira-Yuzvinsky [34]. These facts yield a number of components of R ( D n ), which we record explicitly.Each triple 1 ≤ i < j < k ≤ n yields four A subarrangements of D n :1 { H − i,j , H − i,k , H − j,k } , { H + i,j , H − i,k , H + j,k } , { H − i,j , H + i,k , H + j,k } , { H + i,j , H + i,k , H − j,k } . Denote the corresponding components of R ( D n ) by U qi,j,k , 1 ≤ q ≤ 4. Eachsuch triple also yields the A subarrangement { H ± i,j , H ± i,k , H ± j,k } , with correspondingresonance component V i,j,k .Each 4-tuple 1 ≤ i < j < k < l ≤ n yields eight A subarrangements of D n :1 { H − i,j , H − i,k , H − i,l , H − j,k , H − j,l , H − k,l } { H − i,j , H + i,k , H + i,l , H + j,k , H + j,l , H − k,l } { H + i,j , H − i,k , H + i,l , H + j,k , H − j,l , H + k,l } { H + i,j , H + i,k , H − i,l , H − j,k , H + j,l , H + k,l } { H + i,j , H + i,k , H + i,l , H − j,k , H − j,l , H − k,l } { H − i,j , H − i,k , H + i,l , H + j,k , H + j,l , H − k,l } { H − i,j , H + i,k , H − i,l , H + j,k , H − j,l , H + k,l } { H + i,j , H − i,k , H − i,l , H − j,k , H + j,l , H + k,l } Denote the corresponding components of R ( D n ) by V qi,j,k,l , 1 ≤ q ≤ 8. Each such4-tuple also yields the subarrangement { H ± i,j , H ± i,k , H ± i,l , H ± j,k , H ± j,l , H ± k,l } isomorphicto D , with corresponding resonance component W i,j,k,l . Theorem 6.2. The first resonance variety of the arrangement D n is given by R ( D n ) = [ i The inclusion of the union in R ( D n ) follows from the preceding discussion,so it suffices to establish the opposite inclusion. For this, it is enough to show thata subarrangement of D n not isomorphic to A , A , or D does not contribute acomponent to R ( D n ). Let B be such a subarrangement. We show that B does notadmit a nontrivial neighborly partition.If B ⊂ D n is a subarrangement of cardinality at most 3 that is not isomorphicto A , then B is in general position, and admits no nontrivial neighborly partition.So we may assume that |B| ≥ B . If B contains 3 hyperplanes H ai,j , H bk,l , H cr,s ,where a, b, c ∈ { + , −} , with |{ i, j, k, l, r, s }| ≥ 5, we assert that Π must be trivial.If |{ i, j, k, l, r, s }| = 6, then the hyperplanes H ai,j , H bk,l , H cr,s are in general position,so must lie in the same block, say Π , of Π. Let H dp,q be any other hyperplane in B , where d ∈ { + , −} . Then H dp,q must be in general position with at least one of H ai,j , H bk,l , H cr,s , which implies that H dp,q ∈ Π . Thus, Π = Π is trivial.If H and H ′ are hyperplanes of an arrangement B such that the rank two flat H ∩ H ′ is not contained in any other hyperplane H ′′ of B , i.e., codim H ∩ H ′ ∩ H ′′ > H ⋔ H ′ in B .If |{ i, j, k, l, r, s }| = 5, permuting indices if necessary, we can assume that { i, j } = { , } , { k, l } = { , } , and { r, s } = { , } . Let Π be the block of Π containing H a , . Then, H a , ⋔ H b , and H a , ⋔ H c , in B , which implies that H b , , H c , ∈ Π .Let H dp,q be any other hyperplane in B . We must show that H dp,q ∈ Π . If { p, q } = { , } or 3 ≤ p , then H a , ⋔ H dp,q is a rank 2 flat of B , which implies that H dp,q ∈ Π . So we may assume that p ∈ { , } and q ≥ 3. If q ≥ 5, then H b , ⋔ H dp,q in B = ⇒ H dp,q ∈ Π . Similarly, if q = 3, then H c , ⋔ H dp,q in B = ⇒ H dp,q ∈ Π . Itremains to consider the instance p ∈ { , } and q = 4.There is a multiplicity 3, rank 2 flat H ep, ∩ H dp, ∩ H b , ∈ L ( D n ), for some e ∈ { + , −} . Note that H ep, ⋔ H b , in D n . If H ep, / ∈ B , then H dp, ⋔ H b , in B = ⇒ H dp, ∈ Π . If H ep, ∈ B , then H ep, ⋔ H b , in B = ⇒ H dp, ∈ Π . In thislast instance, the flat H ep, ∩ H dp, ∩ H b , ∈ L ( B ) must be monochrome since Π isneighborly, which forces H dp, ∈ Π .We are left with the case where any triple of hyperplanes H ai,j , H bk,l , H cr,s in B satisfies |{ i, j, k, l, r, s }| ≤ 4, which implies that all hyperplanes of B ⊂ D n involveat most 4 indices. Thus, B is a proper subarrangement of D (with |B| ≥ B ( D ∼ = A (for any of the 9 choices of A subarrangements of D ), it is readilychecked that B admits no nontrivial neighborly partition. So we may assume that A ( B ( D , for some choice of A ⊂ D .For such B , there are pairs of indices i < j and k < l so that |{ H ± i,j } ∩ B| = 1and |{ H ± k,l } ∩ B| = 2. If k < l is the only pair of indices with |{ H ± k,l } ∩ B| = 2, then B = A ∪ { H ± k,l } ( D is an arrangement of 7 hyperplanes containing a copy of A . Checking that the hyperplane H bk,l ∈ B r A is transverse to A , we see that B admits no nontrivial neighborly partition.Consequently, we may assume that there is more than one pair of indices k < l with |{ H ± k,l } ∩ B| = 2. At least one such a pair satisfies |{ i, j } ∩ { k, l }| = 1. Write H ai,j ∈ B and H ¯ ai,j / ∈ B , and assume that { H aj,k , H ¯ aj,k } ⊂ B , where { a, ¯ a } = { + , −} .(The other cases are similar, and left to the reader.) As before, let Π be the blockof Π containing H ai,j . HEN RANKS AND RESONANCE 19 Suppose a = +. Since A ( B , at least one of H + i,k , H − i,k is in B . In L ( D n ) thereare rank 2 flats H + i,j ∩ H + i,k ∩ H − j,k , H + i,j ∩ H − i,k ∩ H + j,k , H − i,j ∩ H − i,k ∩ H − j,k , H − i,j ∩ H + i,k ∩ H + j,k . Since H − i,j / ∈ B , the last two of these yield flats of multiplicity 2 in L ( B ) (or nothingin L ( B )). Consequently, if H bi,k ∈ B , then H bi,k , H + j,k , H − j,k all lie in the same blockof Π. Then H + i,j ∩ H bi,k ∩ H ¯ bj,k ∈ L ( B ), where { b, ¯ b } = { + , −} , and this flat mustbe monochrome. So H + i,j , H bi,k , H + j,k , H − j,k ∈ Π . From this, it follows easily thatΠ = Π, and hence Π is trivial.A similar argument shows that Π is trivial if a = − , completing the proof. (cid:3) Resonance of the Coxeter arrangement of type B. Let B n be the typeB Coxeter arrangement in C n , consisting of the n hyperplanes H i = ker( x i ),1 ≤ i ≤ n , and H ± i,j = ker( x i ± x j ), 1 ≤ i < j ≤ n . The rank 2 flats of B n are H − i,j ∩ H − i,k ∩ H − j,k , H − i,j ∩ H + i,k ∩ H + j,k , H i ∩ H ± j,k , H p ∩ H q ∩ H − p,q ∩ H + p,q ,H + i,j ∩ H − i,k ∩ H + j,k , H + i,j ∩ H + i,k ∩ H − j,k , H j ∩ H ± i,k , H k ∩ H ± i,j , H ± p,q ∩ H ± r,s , where 1 ≤ i < j < k ≤ n , 1 ≤ p < q ≤ n , 1 ≤ r < s ≤ n , and { p, q } ∩ { r, s } = ∅ . Note that B n has (cid:0) n (cid:1) rank two flats of multiplicity 4, 4 (cid:0) n (cid:1) rank two flats ofmultiplicity 3, and 6 (cid:0) n (cid:1) + 12 (cid:0) n (cid:1) rank two flats of multiplicity 2. We determine thestructure of the variety R ( B n ) ⊂ k n .Since D n ⊂ B n , there is an inclusion R ( D n ) ⊂ R ( B n ). As noted previously, A and A subarrangements give rise to two-dimensional components of the firstresonance variety. It is also known that B and B subarrangements yield resonancecomponents, of dimensions 3 and 2 respectively, see [16]. These facts yield a numberof components of R ( B n ), which we now specify.Each 2-tuple 1 ≤ i < j ≤ n yields a subarrangement B ( i, j ) of B n , isomorphicto B , and a corresponding rank two flat H i ∩ H j ∩ H − i,j ∩ H + i,j . Let L i,j be the three-dimensional component of R ( B n ) corresponding to this flat (resp., to B ( i, j )).Each 3-tuple 1 ≤ i < j < k ≤ n determines a subarrangement B ( i, j, k ) of B n , defined by x i x j x k ( x i − x j )( x i − x k )( x j − x k ), which is isomorphic to B . Thissubarrangement yields 12 two-dimensional components of R ( B n ), 11 correspondingto A subarrangements of B ( i, j, k ), and 1 component corresponding to B ( i, j, k )itself. The A subarrangements of B ( i, j, k ) are1 { H i , H j , H k , H − i,j , H − i,k , H − j,k } , { H i , H j , H k , H − i,j , H + i,k , H + j,k } , { H i , H j , H k , H + i,j , H − i,k , H + j,k } , { H i , H j , H k , H + i,j , H + i,k , H − j,k } , { H i , H − i,j , H + i,j , H − i,k , H + i,k , H − j,k } , { H i , H − i,j , H + i,j , H − i,k , H + i,k , H + j,k } , { H j , H − i,j , H + i,j , H − i,k , H − j,k , H + j,k } , { H j , H − i,j , H + i,j , H + i,k , H − j,k , H + j,k } , { H k , H − i,j , H − i,k , H + i,k , H − j,k , H + j,k } , { H k , H + i,j , H − i,k , H + i,k , H − j,k , H + j,k } , { H − i,j , H + i,j , H − i,k , H + i,k , H − j,k , H + j,k } . Note that only the last of these is contained in D n . For the 10 other A subar-rangements of B ( i, j, k ), let Y qi,j,k , 1 ≤ q ≤ 10, be the corresponding components of R ( B n ). Let Z i,j,k be the component of R ( B n ) corresponding to B ( i, j, k ) itself. Theorem 6.3. The first resonance variety of the arrangement B n is given by R ( B n ) = R ( D n ) ∪ [ i 2, it is readily checked that A 6∼ = A , B admits no nontrivial neighborly partition.Suppose A ′′ ⊃ { H ai,j , H bk,l , H cr,s } . If |{ i, j, k, l, r, s }| ≥ 5, these three hyperplanesmust lie in the same block Π of a neighborly partition Π of A . For each hyperplane H p ∈ A ′ ⊂ A , one of these three and H p forms a rank two flat of A of multiplicitytwo, which implies that H p ∈ Π as well. Then, arguing as in the proof of Theorem6.2 reveals that Π is trivial.We are left with the case where A = A ′ ∪ A ′′ , |A ′ | ≥ |A ′′ | ≥ 3, and allhyperplanes of A ′′ ⊂ D n involve at most 4 indices, say { i, j, k, l } . Assume first that H m ∈ A ′ for some m / ∈ { i, j, k, l } . Then L ( A ) contains the flats H m ∩ H ar,s foreach { r, s } ⊂ { i, j, k, l } for which H ar,s ∈ A ′′ . If Π is a neighborly partition of A ,it follows that H m and H ar,s lie in the same block Π of Π for all such { r, s } . If H t ∈ A ′ for t ∈ { i, j, k, l } , then since m / ∈ { i, j, k, l } , H m ∩ H t is a multiplicity twoflat of A , which implies that H t ∈ Π as well and Π is trivial.Consequently, we can assume that all hyperplanes of A = A ′ ∪ A ′′ involve onlythe indices { i, j, k, l } , so A is a subarrangement of the B arrangement involvingthese indices. We consider the various possibilities for |A ′ | .If |A ′ | = 4, then H i , H j , H k , H l ∈ A . Suppose, without loss, that H ak,l ∈ A .Then H i ⋔ H ak,l and H j ⋔ H ak,l are rank two flats of A , so H i , H j , H ak,l must lie inthe same block Π of a neighborly partition Π of A . If H bi,q or H cj,q are in A , for q ∈ { k, l } , then H j ⋔ H bi,q , H i ⋔ H ci,q ∈ L ( A ), which implies H bi,q , H cj,q ∈ Π . Therank two flat H i ∩ H q ∩ H − i,q ∩ H + i,q ∈ L ( B n ) yields a rank two flat in A . Since H i , H − i,q , H + i,q ∈ Π (if either of the latter two hyperplanes are in A ), this flat in A must be monochrome. Hence, H k , H l ∈ Π , and Π is trivial.If |A ′ | = 3, we can assume that H i , H j , H k , H ak,l ∈ A and H l / ∈ A . As above, H i , H j , H ak,l ∈ Π must lie in the same block of a neighborly partition Π of A . Usingrank two flats of A of multiplicity two, as in the previous case, any hyperplane H br,s ∈ A must also lie in Π . Since H l / ∈ A , the flat H k ∩ H l ∩ H − k,l ∩ H + k,l ∈ L ( B n )yields a flat of multiplicity two or three in A , which must be monochrome. Hence, H k ∈ Π , and Π is trivial.If |A ′ | = 2, assume that H i , H j ∈ A and H k , H l / ∈ A . If H ak,l ∈ A , then H i , H j , H ak,l ∈ Π must lie in the same block of a neighborly partition Π of A , andone can show that Π must be trivial by considering multiplicity two rank two flatsof A as above. So assume that { H ± k,l } ∩ A = ∅ . Since the hyperplanes of A involve HEN RANKS AND RESONANCE 21 all four indices i, j, k, l , there are hyperplanes H ap,k , H bq,l ∈ A with { p, q } = { i, j } .There is a corresponding multiplicity two rank two flat H ap,k ⋔ H bq,l ∈ L ( A ) (as H ck,l / ∈ A ). So H ap,k , H bq,l ∈ Π must lie in the same block of a neighborly partitionΠ of A . Since H q ⋔ H ap,k , H p ⋔ H bq,l ∈ L ( A ), we have H i , H j ∈ Π , and it followsthat Π must be trivial.If |A ′ | = 1, assume that H i ∈ A and H j , H k , H l / ∈ A . If all hyperplanes of A involve index i , then there are a, b, c ∈ { + , −} so that H ai,j , H bi,k , H ci,l ∈ A . Thesehyperplanes are in general position in A , so lie in the same block Π of a neighborlypartition Π of A . Write, for instance, { a, ¯ a } = { + , −} . If H ¯ ai,j ∈ A , then H ¯ ai,j ∈ Π since H ¯ ai,j ⋔ H bi,k ∈ L ( A ). Similarly, H ¯ bi,k , H ¯ ci,l ∈ Π if either of these hyperplanesis in A . Since H j / ∈ A , the flat H i ∩ H j ∩ H ai,j ∩ H bi,j ∈ L ( B n ) yields a multiplicitytwo or three flat of A , which must be monochrome. So H i ∈ Π , and Π is trivial.Finally, if |A ′ | = 1 and A has a hyperplane which does not involve the index i ,we can assume that H aj,k , H bp,l ∈ A , where p ∈ { i, j, k } . Then H i ⋔ H aj,k ∈ L ( A ),so H i , H aj,k ∈ Π lie in the same block of a neighborly partition Π of A . If p = i ,then H i ⋔ H bp,l ∈ L ( A ), while if p = i , then H aj,k ⋔ H bp,l ∈ L ( A ). So H bp,l ∈ Π aswell. If H cr,s ∈ A and i = r , then H i ⋔ H cr,s ∈ L ( A ), and H cr,s ∈ Π . If H ci,l ∈ A ,then H aj,k ⋔ H ci,l ∈ L ( A ), and H ci,l ∈ Π . Suppose H ci,s ∈ A for s ∈ { j, k } . If { i, s } ∪ { p, l } = { i, j, k, l } , then H ci,s ⋔ H bp,l ∈ L ( A ), and H ci,s ∈ Π . Otherwise,we have either p = i or p = s . If p = i , there is a flat H ci,s ∩ H bp,l ∩ H ds,l ∈ L ( B n ).If H ds,l ∈ A , then H ds,l ∈ Π since i = s . Consequently, this flat yields a flat ofmultiplicity two or three in A , which must be monochrome, so H ci,s ∈ Π . If p = s ,there is a flat H ci,s ∩ H bp,l ∩ H di,l ∈ L ( B n ). If H di,l ∈ A , then H di,l ∈ Π since H aj,k ⋔ H di,l ∈ L ( A ). Consequently, this flat yields a flat of multiplicity two orthree in A , which must be monochrome, so H ci,s ∈ Π . Thus, Π is trivial. (cid:3) Proof of Theorem C. See Example 6.1 for the type A pure braid group. Let P B n = π ( M ( B n )) and P D n = π ( M ( D n )) be the type B and D pure braid groups. Theresolution of the Chen ranks conjecture and the determination of the resonancevarieties of the arrangements B n and D n yield the Chen ranks of these groups.The remaining portions of Theorem C are immediate consequences of Theorem A,Theorem 5.1, Theorem 6.2, and Theorem 6.3. (cid:3) Acknowledgements. Calculations with Macaulay 2 [21] were essential to ourwork. 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Yuzvinsky, Orlik-Solomon algebras in algebra and topology , Russ. Math. Surveys (2001),293–364. Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803 E-mail address : [email protected] URL : Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL61801 E-mail address : [email protected] URL ::