Modular Equalities

for Complex Reflection Arrangements

Daniela Anca Măcinic, Ştefan Papadima ^{1}, and Clement Radu Popescu ^{2}

Received: November 8, 2015 Revised: October 25, 2016

Communicated by Gavril Farkas

Abstract. We compute the combinatorial Aomoto–Betti numbers β_{p} A of a complex reflection arrangement. When A has rank at least 3, we find that β_{p} A 2, for all primes p. Moreover, β_{p} A 0 if p 3, and β_{2} A 0 if and only if A is the Hesse arrangement. We deduce that the multiplicity e_{d} A of an order d eigenvalue of the monodromy action on the first rational homology of the Milnor fiber is equal to the corresponding Aomoto–Betti number, when d is prime. We give a uniform combinatorial characterization of the property e_{d} A 0, for 2 d 4. We completely describe the monodromy action for full monomial arrangements of rank 3 and 4. We relate e_{d} A and β_{p} A to multinets, on an arbitrary arrangement.

2010 Mathematics Subject Classification: Primary 14F35, 32S55; Secondary 20F55, 52C35, 55N25. Keywords and Phrases: Milnor fibration, algebraic monodromy, hyperplane arrangement, complex reflection group, resonance variety, characteristic variety, modular bounds.

Contents

1. Introduction and statement of results  136
2. Non-exceptional reflection arrangements  138
3. Exceptional reflection arrangements  141
4. Multinets and jump loci  144
References  148

1Partially supported by the Romanian Ministry of National Education, CNCS-UEFISCDI, grant PNII-ID-PCE-2012-4-0156

2Supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2012-3-0492

1. Introduction and statement of results

1.1. Milnor fibration and monodromy. The complement M of a degree n complex hypersurface in ^{l}, f 0 , and the associated Milnor fibration, f : M , first analysed by Milnor in his seminal book [15], attracted a lot of attention over the years. Multiplication by exp \frac{2π \overline{1}}{n} induces the geometric monodromy action on the associated Milnor fiber F f^{1} 1 , h : F F, and the algebraic monodromy action, h_{i} : H_{i} F, H_{i} F, . Computing h_{i} is a major problem in the field, when f has a non-isolated singularity at 0. Even for the defining polynomial of a (central) complex hyperplane arrangement A in ^{l} and i 1, the answer is far from being clear. This case was tackled in the recent literature by many authors, who used a variety of tools; see for instance [20] for a brief survey. In this paper, we focus on reflection arrangements, associated to finite complex reflection groups. It is well-known that every arrangement complement M_{A} has the homotopy type of a connected, finite CW-complex with torsion-free homology, whose first integral homology group, H_{1} M_{A}, ^{n}, comes endowed with a natural basis, given by meridian loops around the hyperplanes. It is also well-known that, for an arbitrary arrangement A, h_{1} induces a -module decomposition,

(1)  H_{1} F_{A},  _{d} t Φ_{d} t ^{e_{d} A},

where Φ_{d} is the d-th cyclotomic polynomial, e_{d} A 0 if d n, and e_{1} A n 1. See for instance [14, (1.1)]. A pleasant feature of hyperplane arrangements is the rich combinatorial structure encoded by the associated intersection lattice, L_{}A , whose elements are the intersections of hyperplanes from A, ranked by codimension and ordered by inclusion. In this context, the open monodromy action problem takes the following more precise form: are the multiplicities e_{d} A combinatorial? If so, give a formula involving only L_{}A .

1.2. Characteristic and resonance varieties. Our approach to decomposition (1) is topological, based on two types of jump loci, associated to CW-complexes having the properties recalled for M_{A}, and the interplay between them. The complex characteristic variety 𝒱_{q}^{i} M , sitting inside the character torus M Hom H_{1} M, , ^{}^{}^{n} is the locus of those ρ M for which dim_{}H_{i} M, _{ρ} q, where _{ρ} denotes the associated rank 1 local system on M. Note that Hom H_{1} M, , H^{1} M, ^{n}, and denote by exp : H^{1} M, M the natural exponential map. For an integer d 1, let ρ_{d} M be the exponential of the diagonal cohomology class equal to \frac{2π \overline{1}}{d} , with respect to the distinguished -basis. When M M_{A} is an arbitrary arrangement complement and d 1, it is well-known that

(2)  e_{d} A dim_{}H_{1} M_{A}, _{ρ_{d}} .

See for instance [14, §2.3], and also [3, Theorem 2.5 and Corollary 6.4] for more general results of this type. The resonance variety ℛ_{q}^{i} M, over a field , sitting inside H^{1} M, , is the locus of those σ H^{1} M, for which dim_{}H^{i} H^{}M, , σ q, where σ denotes the left multiplication by σ in the cohomology ring. When is the prime field _{p}, denote by σ_{p} H^{1} M, _{p} the diagonal cohomology class equal to 1, with respect to the distinguished -basis, and define the modulo p Aomoto-Betti number by

(3)  β_{p} M dim_{_{p}} H^{1} H^{}M, _{p} , σ_{p}

When M M_{A} is an arrangement complement, we will replace M by A in the notation. By a celebrated theorem of Orlik and Solomon [16], the cohomology ring of M_{A} is combinatorial. More precisely, β_{p} A may be computed from L_{2} A , as well as ℛ_{q}^{1} A, , for all q and .

1.3. Modular bounds. It follows from Theorem 11.3 in [19] that

(4)  e_{p^{s}} A β_{p} A , for all s 1 ,

when A is an arbitrary arrangement. Actually, the above modular bound holds for all CW-complexes considered in §1.2, with the multiplicity replaced by the value from equality (2), and is in general strict, in the broader context. Our first main result says that the modular bound (4) becomes an equality, for reflection arrangements and s 1.

Theorem 1.1. Let A be a complex reflection arrangement. Then e_{p} A β_{p} A , for all primes p. In particular, e_{p} A is determined by L_{2} A .

1.4. Aomoto-Betti numbers for reflection arrangements. Reflection arrangements have a distinguished history, going back as far as Jordan’s work from 1878 on the symmetry group of the famous Hessian configuration. A related open problem is whether the Hessian arrangement is the only arrangement supporting a 4-net. In Theorem 1.2(2) below, we solve this problem for reflection arrangements. Finite complex reflection groups have been classified by Shephard and Todd [22] (see also [2], [17]). Each such group G gives rise to the complex reflection arrangement A G , consisting of the fixed points of all reflections in G. Among them, we have the monomial arrangements A m, m, l in ^{l} l 2 , with defining polynomials Π_{1 i j l} z_{i}^{m} z_{j}^{m} m 1 , and full monomial arrangements A m, 1, l in ^{l} l 2 , defined by z_{1} …z_{l} Π_{1 i j l} z_{i}^{m} z_{j}^{m} m 2 . We may now state our second main result.

Theorem 1.2. For a complex reflection arrangement A of rank at least 3, the following hold.

(1) If p 3, then β_{p} A 0.

(2) β_{2} A 0 β_{2} A 2 A supports a 4-net A is the Hessian

arrangement.

(3) The only cases when β_{3} A 0 are: A m, 1, 3 with m 1 mod 3 , where β_{3} 1; A m, m, 3 with m 2, where β_{3} 1 if m 0 mod 3 and otherwise β_{3} 2; A m, m, 4 , where β_{3} 1.

(4) In particular, β_{p} A 2, for all primes p.

We imposed the rank condition since the Aomoto-Betti numbers for arrangements of rank at most 2 are known (see for instance [14]). Moreover, when A A m, m, 2 with m 0 mod p it is easy to see that β_{p} A m 2. Thus, the conclusion of Theorem 1.2(4) no longer holds, for m 4. The resonance varieties ℛ_{q}^{1} A, are quite well-understood, due to work by Falk, Libgober, Marco-Buzunáriz and Yuzvinsky; see [10], [13]. There are results in positive characteristic which show a different qualitative behaviour of resonance in this case; see for instance Falk [9]. The complete picture over _{p} largely remains a mystery. Our Theorems 1.2 and 1.1, together with recent vanishing results due to Dimca and Sticlaru ([6], [7]), verify the strong modular conjecture from [20], for the important class of complex reflection arrangements.

1.5. A combinatorial non-triviality test. Dimca, Ibadula and Măcinic asked in [4] the following natural question: if d 1 and e_{d} A 0, does this imply that ρ_{d} exp ℛ_{1}^{1} A, ? A positive answer (for all d) would imply that the non-triviality of h_{1} is combinatorial, since the converse implication is known, for all values of d.

Theorem 1.3. If A is a complex reflection arrangement, then the above question has a positive answer, for 2 d 4.

We derive Theorem 1.3 and Theorem 1.1 from Theorem 1.2 with the aid of a general result (proved in Theorem 4.3) that relates combinatorial structures on arrangements satisfying the key multinet axiom introduced by Falk-Yuzvinsky in [10] to the algebraic monodromy action and the Aomoto-Betti numbers of an arrangement. The tools from our paper also enable us to give a complete, combinatorial description in Proposition 4.4for the monodromy action on H_{1} F_{A}, , in the case of full monomial arrangements of rank 3 and 4. Related results may be found in [12] and [20]. By Theorems 1.1, 1.2 and [6, 7], this complete, combinatorial description holds for arbitrary complex reflection arrangements.

2. Non-exceptional reflection arrangements

We first compute the Aomoto-Betti numbers of monomial and full monomial arrangements.

2.1. The classification. (cf. [22], [2], [17]) A finite reflection group G decomposes as a product of irreducible factors of the same kind. At the level of arrangements, A G G^{}is the product A G A G^{}, and the corresponding complements satisfy M G G^{}M G M G^{}. (Whenever convenient, we will abbreviate notation and replace A G by G, when speaking about associated objects.)

The irreducible reflection arrangements of rank at least 3 comprise the monomial and full monomial arrangements, A m, m, l m 2, l 3 or m 1, l 4 and A m, 1, l l 3 , plus the exceptional arrangements A G_{23} A G_{37} . The Hessian arrangement is A G_{25} . See for instance [17] for the notations.

2.2. Vanishing criteria. Given an arbitrary arrangement A and an r–flat X L_{r} A , set A_{X} H A H X , and define the multiplicity of X to be equal to A_{X} . Using the distinguished -basis, we identify an element η H^{1} M_{A}, _{p} with the family η_{H} _{p} _{H A}. We denote by Z_{p} A H^{1} M_{A}, _{p} the kernel of σ_{p} . Definition (3) implies that β_{p} A dim_{_{p}} Z_{p} A 1. Our computations are based on the following well-known result (see e.g. [20]).

Lemma 2.1. An element η belongs to Z_{p} A if and only if, for any X L_{2} A ,

H A_{X} η_{H} 0  if A_{X} 0 (mod p)
η_{H} η_{H^{}}, H, H^{}A_{X} if A_{X} 0 (mod p)

Clearly, β_{p} A 0 if and only if η Z_{p} A implies that η is constant. A first useful vanishing criterion is due to Yuzvinsky.

Lemma 2.2. ([25]) If A 0 (mod p), then β_{p} A 0.

A second convenient situation is the following.

Lemma 2.3. Assume that A A_{1} A_{2}.

(1) H_{1} M_{A}, _{ρ_{d}} 0, for all d 1.

(2) β_{p} A 0, for all primes p.

Proof. Assuming the contrary, we infer from [18, Proposition 13.1] that ρ_{d} 𝒱_{1}^{1} M_{A_{1}} M_{A_{2}} 1 𝒱_{1}^{1} M_{A_{2}} 𝒱_{1}^{1} M_{A_{1}} 1 , respectively σ_{p} ℛ_{1}^{1} M_{A_{1}} M_{A_{2}}, _{p} 0 ℛ_{1}^{1} M_{A_{2}}, _{p} ℛ_{1}^{1} M_{A_{1}}, _{p} 0 , in contradiction with the fact that all coordinates of ρ_{d} (respectively σ_{p}) are different from 1 (respectively 0).

By Lemma 2.3(2), we only need to compute β_{p} G for an irreducible complex reflection group G. To state the third vanishing criterion, we need to introduce certain simple graphs, associated to an arrangement A and an integer k 2, with vertex set A. The edges of Γ_{k} A are defined by the condition A_{H H^{}}0 mod k , for k 2, and by A_{H H^{}}is either odd or equal to 2, for k 2. The defining property for Γ_{k} A is A_{H H^{}}k. Note that Γ_{2} A is a subgraph of Γ_{p} A , for all primes p. The equivalence relation on A associated to the edge paths of Γ_{k} A , respectively Γ_{k} A , will be denoted by _{k}, respectively _{k}.

Example 2.4. Let A A 1, 1, l be the braid arrangement in ^{l}, with hyperplanes labeled by the two-element subsets of 1, …, l . For ij st, the multiplicity of the 2–flat determined by the hyperplanes z_{i} z_{j} and z_{s} z_{t} is 2 if i, j, s, t 4 and 3 if i, j, s, t 3. Let p be a prime. It follows from the above definition that Γ_{p} A is a complete graph (with full edge set) if p 3. For p 3, ij and st are connected by an edge of Γ_{3} A if and only if i, j, s, t 4. We infer that the graph Γ_{3} A is discrete (with no edges) when l 3, has 3 connected components when l 4 (see the picture of the corresponding graph below), and is connected when l 5.

[figure: Γ_{3} A 1, 1, 4 :; 12; 34; 13; 24; 14; 23]

We obtain the following immediate consequences of Lemma 2.1.

Lemma 2.5. Each of the properties below implies that β_{p} A 0.

(1) The graph Γ_{p} A is connected.

(2) The graph Γ_{2} A is connected.

(3) For all X L_{2} A , p A_{X} .

2.3. Intersection lattices. The Aomoto-Betti numbers for A 1, 1, l were computed in [14]. They verify all statements from Theorems 1.2 and 1.1. Hence, we may suppose from now on that m 2. We need to describe L_{2} A . It will be convenient to label the various hyperplanes as follows. Set ω exp \frac{2π \overline{1}}{m} , H_{i} z_{i} 0 for all 1 i l, and H_{ij^{α}} z_{i} ω^{α}z_{j} 0, for 1 i j l and α m . We go on by listing the 2-flats (identified with the corresponding subarrangements A_{X}).

Case I: A m, 1, l , l 4 :

I_{a}: {H_{i}, H_{j}, H_{ij^{α}} α m }, with multiplicity m 2;

I_{b}: {H_{ij^{α}} , H_{jk^{β}}, H_{ik^{α β}} }, with multiplicity 3;

I_{c}: {H_{ij^{α}} , H_{kh^{β}}}, with multiplicity 2; I_{d}: {H_{i}, H_{jk^{α}}}, with multiplicity 2. Case II: A m, m, l , l 4 : types I_{b} and I_{c}, plus {H_{ij^{α}} α m }, with multi-

plicity m. Case III: A m, 1, 3 : types I_{a}, I_{b} and I_{d}. Case IV: A m, m, 3 : types I_{b} and II.

2.4. β_{p}-vanishing. We will use Lemma 2.5 to treat the cases when β_{p} A 0 in the non-exceptional families. To simplify things, we suppress H from notation and identify the hyperplanes with their labels, i and ij^{α}. We claim that, for A A m, 1, l with l 4, Γ_{2} A is connected. Indeed, given i j we may find h k with i, j, h, k distinct. Hence, i _{2} hk^{0} _{2} j and ij^{α} _{2} k, which proves connectivity. Similar arguments lead to the following conclusions. If A A m, 1, l with l 3 and p 3, then Γ_{p} A is connected; for p 3 and m 1 mod 3 , Γ_{3} A is connected. The remaining full monomial Aomoto-Betti numbers, β_{3} m, 1, 3 with m 1 mod 3 , will be computed later on. If A A m, m, l with l 5, then Γ_{2} A is connected. For l 3, 4 and p 3, Γ_{p} A is connected. So, for monomial arrangements, only β_{3} in ranks 3 and 4 remains to be calculated.

2.5. The remaining non-exceptional cases. A mod 3 cocycle η Z_{3} A is a family of elements of _{3}, η_{k} and η_{ij^{α}}, satisfying the equations from Lemma 2.1, for any X L_{2} A . Case A A m, 1, 3 with m 1 mod 3 . The equations coming from 2-flats of type I_{d} say that η_{jk^{α}} η_{i}, where i is the third element of 1, 2, 3 . The equations of type I_{b} become equivalent to η_{1} η_{2} η_{3} 0, while type I_{a} equations say that η_{i} η_{j} mη_{k} 0, for all i j k. We infer that β_{3} m, 1, 3 1, as asserted in Theorem 1.2. Case A A m, m, 4 . The equations of type I_{c} say that η_{ij^{α}} η_{ij}, and η_{34} η_{12}, η_{24} η_{13}, η_{23} η_{14}. Type I_{b} equations reduce then to η_{12} η_{13} η_{14} 0, while type II conditions follow from η_{ij^{α}} η_{ij}. Again, β_{3} m, m, 4 1, as claimed. Case A A m, m, 3 with m 0 mod 3 . The equations of type II say that η_{ij^{α}} η_{ij}, and the type I_{b} conditions then reduce to η_{12} η_{23} η_{13} 0. This shows that β_{3} m, m, 3 1, as asserted. Case A A m, m, 3 with m 3n. Set η_{12^{α}} a_{α}, η_{23^{α}} b_{α}, η_{13^{α}} c_{α}. With this notation, the equations of type I_{b} are equivalent to the system

(5)  a_{α} b_{β} c_{α β} 0, α, β m ,

while the conditions of type II read

(6)  a_{α}  b_{β}  c_{γ} 0.

We first solve the system (5), as follows. It implies that a_{α} b_{β} a_{α^{}}b_{β^{}}, if α β α^{}β^{}, in particular a_{α} a_{0} b_{α} b_{0} d_{α}, for all α, and d_{α} d_{β} d_{α^{}}d_{β^{}}, if α β α^{}β^{}. We infer that d_{α} αd_{1}, for all α. Hence a_{α} a_{0} αd_{1}, b_{α} b_{0} αd_{1} and c_{α} a_{0} b_{0} αd_{1}, which solves the system (5). In particular, its solution space is 3-dimensional. Finally, it is an easy matter to check that (5) (6), since m 3n. Therefore, β_{3} 3n, 3n, 3 2, as asserted. This proves Theorem 1.2 for non-exceptional reflection arrangements.

3. Exceptional reflection arrangements

We finish the proof of Theorem 1.2, by computing the Aomoto-Betti numbers of the exceptional complex reflection arrangements of rank at least 3, G_{23} G_{37}.

3.1. The groups G_{31}, G_{32}, G_{33}. Case A A G_{31} . The hyperplanes of A live in ^{4}. Their defining equations are as follows (see [11]). Set ω exp \frac{2π \overline{1}}{4} . The hyperplanes of A are:

H_{i} z_{i} 0 1 i 4 ;

H_{ij^{β}} z_{i} ω^{β}z_{j} 0 1 i j 4, β 4 ;

H\overline{α} z_{1} _{2 i 4} ω^{α_{i}}z_{i} 0 \overline{α} α_{2}, α_{3}, α_{4} 4 ^{3}, α_{2} α_{3} α_{4} 0 mod 2 .

By Lemma 2.5(2), it is enough to show that Γ_{2} G_{31} is connected. This can be seen as follows. Clearly, the 2-flat H_{k} H_{ij^{β}} has multiplicity 2, when i j k. This implies that i _{2} j _{2} kh^{β}, for all 1 i j 4, 1 k h 4 and β 4 . Given any H\overline{α}, it is not hard to see that the multiplicity of H\overline{α} H_{12^{α_{2}}} is 2. This proves connectivity, as claimed. Case A A G_{32} . Set ω exp \frac{2π \overline{1}}{3} . The arrangement A consists of the following hyperplanes in ^{4} (see [24]):

H_{i} z_{i} 0 1 i 4 ;
H_{1^{αβ}} z_{2} ω^{α}z_{3} ω^{β}z_{4} 0;
H_{2^{αβ}} z_{1} ω^{α}z_{3} ω^{β}z_{4} 0;
H_{3^{αβ}} z_{1} ω^{α}z_{2} ω^{β}z_{4} 0;
H_{4^{αβ}} z_{1} ω^{α}z_{2} ω^{β}z_{3} 0 (α, β 3 ).

Clearly, the 2-flats H_{i} H_{j} i j and H_{i} H_{i^{αβ}} have multiplicity 2. This shows that Γ_{2} G_{32} is connected and we are done. Case A A G_{33} . Here, ω exp \frac{2π \overline{1}}{3} and the hyperplanes (in ^{6}) are as follows (see [2, 21]):

H_{ij^{β}} z_{i} ω^{β}z_{j} 0 1 i j 4, β 3 ;
H\overline{α}  _{1 i 4} ω^{α_{i}}z_{i} z_{5} z_{6} 0 \overline{α} α_{1}, α_{2}, α_{3}, α_{4}  3 ^{4}, α_{i} 0 .

Plainly, ij^{β} _{2} kh^{0} _{2} ij^{β^{}}, for all β, β^{}(where i j k h). Like in the Case G G_{31}, it can be checked that H\overline{α} H_{ij^{β}} has multiplicity 2, if β α_{j} α_{i}. We infer that Γ_{2} G_{33} is connected, and we are done.

3.2. More vanishing criteria. We will no longer need defining equations to settle the remaining cases. We will use instead a couple of new vanishing arguments. For the beginning, let us recall from [17, pp. 224-225] the following very useful properties of reflection groups and arrangements, derived from a key result of Steinberg [23]. For any complex reflection group G and any X L_{r} A G , the fixer subgroup G_{X} g G gx x, x X is a reflection group, and A G _{X} A G_{X} is again a reflection arrangement, of rank r. By the construction of A G , the group G acts on the arrangement A G , hence on the intersection lattice L_{}G . Let us denote by O_{X} the G-orbit of X L G . Let Type X be the isomorphism type of the reflection group G_{X}. It follows from [17, Lemma 6.88] that the type is constant on each orbit O_{X}. Moreover, Table C from [17] gives A G and the orbit partition of L_{}G for all exceptional groups, in terms of types of orbits. This leads to the quick computation of the sets

𝒫 G : p prime X L_{2} G such that A G_{X} 0 mod p

In particular, 𝒫 G 2, 3, 5 , for every exceptional arrangement of rank at least 3. We infer from Lemma 2.5(3) that β_{p} G 0, if p 5. Hence, we may suppose from now on that p 5. For an arbitrary arrangement A, H A and a prime p, we define

(7)  m_{p} H 1  A_{X} 1 ,

where the sum is taken over those X L_{2} A such that X H and A_{X} 0 mod p . The numbers from (7)may be extracted from Table C in [17], for exceptional reflection arrangements of rank at least 3. This is based on the following fact, valid for an arbitrary reflection group G. For X, Y L G , let u X, Y be the number of Z L G such that Z O_{Y} and Z X. Clearly, this number depends only on O_{X} and O_{Y}. The values u H, X may be found in Table C, for all orbit types corresponding to H L_{1} G and X L_{2} G .

Lemma 3.1. For an arrangement A and a prime p, the following hold.

(1) If m_{p} H \frac{A}{2} for all H A, then β_{p} A 0.

(2) If m_{p} H \frac{A}{3} for all H A and A has no rank 2 flats of multiplicity p r

with r 1, then β_{p} A 0.

Proof. If β_{p} A 0, there is a non-constant function η _{p}^{A} satisfying all equations from Lemma 2.1. Fix H A and set η_{H} α. We claim that η α m_{p} H . Indeed, if X L_{2} A is contained in H and A_{X} 0 mod p , then η must have the constant value α on A_{X}. An easy count of all these hyperplanes gives the claimed inequality. In Part (1), this implies that η must be constant, a contradiction. In Part (2), we infer that η has only two distinct values. By adding constant functions and multiplying by non-zero elements in _{p}, we may assume that these values are η_{H} 0 and η_{H^{}}1. By Lemma 2.1, the flat X H H^{}has multiplicity p r, imposing the condition K A_{X} η_{K} 0. Since necessarily r 1, we arrive again at a contradiction.

3.3. Induction on rank. We start with a couple of general considerations. A subarrangement B A is called line-closed in A (see the first definition from [8, Definition 1.1]) if B_{X} A_{X}, for all X L_{2} B . This property implies that the restriction map, _{p}^{A} _{p}^{B}, sends Z_{p} A into Z_{p} B . Clearly, A_{Y} is line-closed in A, for any Y L A .

Lemma 3.2. Let G be a complex reflection group. Assume that 2 r rank A G and β_{p} A G^{}0, for all irreducible groups G^{}Type L_{r} A G . Then β_{p} A G 0.

Proof. Assuming the contrary, there exist η Z_{p} G and H_{1}, H_{2} A G such that η_{H_{1}} η_{H_{2}}. From our assumption on r, we may find H_{3}, …, H_{r} A G such that X H_{1} H_{2} H_{r} L_{r} G . Set B A G _{X} A G_{X} . We deduce that β_{p} G_{X} 0. If G_{X} is reducible, this contradicts Lemma 2.3. Otherwise, our second assumption is violated.

3.4. The rank 3 case. The rank 3 exceptional groups are H_{3}, G_{24}, G_{25}, G_{26} and G_{27}. Table C from [17] provides the following information on each group:

𝒫 G 2, 3, 5 ; 2, 3 ; 2 ; 2, 5 ; 2, 3, 5 ;
A G = 15; 21; 12; 21; 45.

By Lemma 2.2 and Lemma 2.5(3), the only primes p which might give β_{p} G 0 are as follows:

3, 5 H_{3} ; 3 G_{24} ; 2 G_{25} ; G_{26} ; 3, 5 G_{27} .

Using Lemma 3.1(1), we obtain β_{3} H_{3} β_{3} G_{24} β_{3} G_{27} β_{5} G_{27} 0, and Lemma 3.1(2) gives β_{5} H_{3} 0. Finally, β_{2} G_{25} 2, cf. [20]. Thus, Theorem 1.2 is proved in this case. Indeed, the Hessian arrangement A G_{25} supports a 4-net (see e.g. [26]). This implies that β_{2} 0, by [20]; the other implications from Theorem 1.2(2) are obvious. Applying Lemma 3.2 for r 3 and p 5, we also infer that Theorem 1.2(1) holds for all complex reflection arrangements of rank at least 3. Thus, we only need to show that β_{2} G β_{3} G 0, when G is exceptional of rank at least 4, in order to complete the proof of Theorem 1.2.

3.5. The remaining cases. The only rank 5 exceptional arrangement is G_{33}, for which we know from §3.1 that all β_{p} vanish. By the computations from Section 2, the same thing happens for non-exceptional irreducible arrangements of rank 5. Lemma 3.2, applied for r 5, guarantees then that we may reduce our proof to the rank 4 case. Here, the list is G F_{4}, G_{29}, H_{4}, G_{31}, G_{32}, and the last two groups were treated in §3.1. Case G F_{4}. The irreducible rank 3 types are listed in Table C from [17]: B_{3} and C_{3}; in both cases, the arrangement A G^{}is A 2, 1, 3 , for which all β_{p} vanish, cf. Section 2. We may conclude by resorting to Lemma 3.2 for r 3. Case G G_{29}. The list of irreducible rank 3 types is: G^{}A_{3}, B_{3}, G 4, 4, 3 . Taking r 3 and p 2 in Lemma 3.2, we deduce from Section 2 that β_{2} G_{29} 0. Since A G_{29} 40, β_{3} G_{29} 0, by Lemma 2.2. Case G H_{4}. The irreducible types of L_{3} G are: G^{}A_{3}, H_{3}. Again by Lemma 3.2 and previous computations, β_{2} H_{4} 0. Finally, β_{3} H_{4} 0, as follows from Lemma 3.1(1). The proof of Theorem 1.2 is complete.

4. Multinets and jump loci

In this section we prove Theorems 1.1 and 1.3. Along the way, we establish a useful general result that relates combinatorial structures on arrangements satisfying the main multinet axiom to the algebraic monodromy of its Milnor fibration and its Aomoto-Betti numbers.

4.1. Multinets and weighted partitions. The work of Falk and Yuzvinsky from [10] gives, among other things, a description of the resonance variety of an arrangement A, ℛ_{1}^{1} A, , in terms of multinets on the associated matroid. A k multinet on L_{2} A is a partition Π with k 3 non-empty blocks, A _{α k} A_{α}, together with a function, : A _{0} , satisfying certain axioms. The most important is the following: For any H A_{α} and H^{}A_{β} with α β, and every γ k ,

(8)  n_{X} _{K A_{γ} A_{X}}  _{K}

is independent of γ, where X H H^{}L_{2} A . Let Π be a partition of A with k 3 non-empty blocks, as above. Let : A be an assignment of arbitrary integer weights to the hyperplanes of A.

Definition 4.1. The pair 𝒩 Π, is a weighted k-partition if axiom (8) is satisfied. We will say that 𝒩 is h-reduced h 1 if _{K} 1 mod h , for all K A.

The underlying partition of a k-multinet, together with its positive weight function, is a weighted k-partition. Moreover, the usual notion of reduced multinet corresponds to h 1. We will need a result from [20] related to axiom (8). To recollect it, we start with a few notations. Set H_{A} H_{1} M_{A}, and denote by a_{H} _{H A} the distinguished -basis. Let S be ^{1} k points and set H_{S} H_{1} S, span c_{α} α k _{α k} c_{α}, where c_{α} is the class of a small loop in S around the point α. Let _{A} : ^{2} H^{1} M_{A}, H^{2} M_{A}, be the cup product. Recalling from [16] that H^{}M_{A}, has no torsion, we denote by ∇_{A} : H_{2} M_{A}, ^{2} H_{A} the -dual comultiplication map. The next result improves Theorem 2.4 from [10], in several ways. The hypothesis of Proposition 4.2is reduced to the key axiom iii from [10, Definition 2.1]. The weight function may take arbitrary integer values, while in [10] positivity plays a crucial role. The conclusion in [10] is that im ϕ ^{}H^{1} M_{A}, is an isotropic subspace, for a characteristic 0 field (in positive characteristic, an additional condition on is needed in [10]), while Proposition 4.2 gives the conclusion over . For the reader’s convenience, we include the proof.

Proposition 4.2 ([20]). Let 𝒩 Π, be a weighted k-partition. Then ^{2} ϕ ∇_{A} 0, where ϕ : H_{A} H_{S} sends a_{H} to _{H}c_{α}, for H A_{α}. Therefore, _{A} 2 ϕ^{}0, by taking -duals.

Proof. Let A be the -form of the holonomy Lie algebra of A, appearing in Proposition 5.2 from [20]. By definition, this is the graded -Lie algebra quotient of the free -Lie algebra generated by H_{A}, ^{}H_{A} , graded by bracket length, by the graded Lie ideal generated by im ∇_{A} ^{2} H_{A}, where ^{2} H_{A} is identified with ^{2} H_{A} via the Lie bracket. Let ^{}ϕ : ^{}H_{A} ^{}H_{S} be the graded -Lie algebra map extending ϕ. Our claim says that ^{}ϕ factors through A . To check this, we recall from [20, (30)] that the defining Lie relations of A are:

_{K A_{X}} a_{K}, a_{L} , for X L_{2} A and L A_{X}.

Thus, we have to show that _{K A_{X}} ϕ a_{K} , ϕ a_{L} 0 ^{2} H_{S} . There are two cases to consider. When X is mono-coloured, i.e., A_{X} A_{α} for some α k , by construction ϕ a_{K} c_{α}, for all K A_{X}, and we are done. Otherwise, X H H^{}, with H A_{α}, H^{}A_{β} and α β. Again by construction, K A_{X} ϕ a_{K} _{γ k} _{K A_{γ} A_{X}} _{K} c_{γ}, which equals n_{X} _{γ k} c_{γ} , by axiom (8). This implies that _{K A_{X}} ϕ a_{K} 0 ^{1} H_{S} , by the definition of H_{S}, which completes the proof.

4.2. Relating weighted partitions to jump loci. We are now ready to state our result, keeping the previous notation.

Theorem 4.3. Assume that 𝒩 Π, is a k-reduced weighted k-partition. Then the following hold, for all divisors p, d of k with p prime and d 1.

(1) ρ_{d} A exp ℛ_{1}^{1} A, , in particular e_{d} A 0.

(2) β_{p} A 0.

Proof. Part (2). Since the weighted partition is k-reduced and p k, ϕ is surjective, by construction, for _{p} and . It follows from Proposition 4.2 that im ϕ H^{1} M_{A}, is a k 1 -dimensional subspace, isotropic with respect to the cup product. The linear map sending each c_{α} to 1 _{p} defines an element of H^{1} S, _{p} , denoted σ_{p} S . Clearly, ϕ^{}σ_{p} S σ_{p} A , since 𝒩 is in particular p-reduced. By definition (3), β_{p} A 0 as claimed, since k 3. Part (1). Note that ϕ : H_{A} H_{S} induces homomorphisms ϕ^{}: S M_{A} and ϕ^{}: H^{1} S, H^{1} M_{A}, , compatible with the surjective exponential maps of S and M_{A}. The map sending each c_{α} to exp \frac{2π \overline{1}}{k} defines an element of the character torus, ρ_{k} S S . Plainly, ϕ^{}ρ_{k} S ρ_{k} A , since 𝒩 is k-reduced. We also have ρ_{k} ^{k d} ρ_{d}, for both S and A, since we assumed that d divides k. Hence, ϕ^{}ρ_{d} S ρ_{d} A exp ϕ^{}H^{1} S, exp ℛ_{1}^{1} A, , where the last inclusion follows from the argument in Part (2). Indeed, we know that ϕ^{}H^{1} S, is an isotropic subspace in H^{1} M_{A}, , of dimension at least 2, and we may simply use the definition of ℛ_{1}^{1}. The conclusion e_{d} A 0 is a direct consequence of equality (2), since it is well-known that exp ℛ_{1}^{1} A, 𝒱_{1}^{1} M_{A} ; see e.g. [5, Theorem D] for a more general result.

4.3. Reduced weighted partitions on complex reflection arrangements. We begin the proof of Theorem 1.3. Let A be a complex reflection arrangement and assume e_{d} A 0, with 2 d 4. We will show that ρ_{d} A exp ℛ_{1}^{1} A, with the aid of Theorem 4.3(1), which requires the existence of a certain weighted partition on A. An easy preliminary remark is that the question from [4] always has a positive answer, for any arrangement A of rank at most 2. To see this, note first that the assumption e_{d} A 0 is equivalent to ρ_{d} 𝒱_{1}^{1} M_{A} , by equality (2). When rank A 2, it is known that 𝒱_{1}^{1} M_{A} exp ℛ_{1}^{1} A, , so the conclusion follows trivially. Consequently, we may also suppose that the rank is at least 3. On the other hand, e_{d} A 0 and d p^{s} together imply, via the modular bound (4), that β_{p} A 0. Therefore, A must be either the Hessian arrangement, or one of the arrangements from Theorem 1.2(3). To apply Theorem 4.3(1), we need to describe suitable weighted partitions on these arrangements, in each case. The Hessian arrangement supports a reduced 4-multinet (actually, a 4-net). The monomial arrangement A m, m, 3 has a reduced 3-multinet (in fact, a 3-net), as noted in [10]. A (non-reduced) 3-multinet on the full monomial arrangement A m, 1, 3 was constructed in [10]. It is immediate to check that the weighted partition associated to this multinet is 3-reduced, when m 1 mod 3 .

The last case is A A m, m, 4 , with hyperplanes H_{ij^{μ}} z_{i} ω^{μ}z_{j} 0, where 1 i j 4, μ m and ω exp \frac{2π \overline{1}}{m} . We define a partition Π with three blocks, H_{ij^{μ}} , H_{kh^{ν}} μ, ν m , and set 1 on A. It is straightforward to verify axiom (8) by using the description of 2-flats given in §2.3. (Actually, this is a 3-net on L_{2} A .)

4.4. Proof of Theorem 1.3 completed. In case A A G_{25} , d must be 2 or 4. We may take k 4 in Theorem 4.3 to obtain the desired conclusion. The remaining cases, described in Theorem 1.2(3), lead to d 3. Taking k 3, we conclude as before.

4.5. Proof of Theorem 1.1. We may suppose that rank A 3, since otherwise the conclusion is known (see [14, p. 773]). By the modular bound (4) and Theorem 1.2, e_{p} A β_{p} A , when A is not A G_{25} or one of the arrangements listed in Theorem 1.2(3). Moreover, we have to verify the conclusion only for p 2 (in case A G_{25} ) or for p 3 (in the remaining cases). The equality e_{2} G_{25} β_{2} G_{25} 2 is well-known (see e.g. [20]). When A is not A m, m, 3 with m 0 mod 3 , we know that β_{3} A 1. In these cases, we may use the 3-reduced weighted 3-partitions from §4.3, for d k 3, exactly as in §4.4, to obtain that e_{3} A 0. Now we are done, since the modular bound (4) implies that e_{3} A 1. The last case, A A m, m, 3 with m 3n and p 3, when β_{3} A 2, requires a more careful treatment. The (relabeled) hyperplanes of A are H_{12^{α}}, H_{23^{β}}, H_{13^{γ}} α, β, γ m . We recall from §2.3 the two types of 2-flats: H_{ij^{α}} α m and H_{12^{α}}, H_{23^{β}}, H_{13^{γ}} with α β γ 0. We have to show that e_{3} A 2, in order to finish the proof. To this end, we need two reduced weighted 3-partitions on A. The first one, 𝒩 , is constructed in [10]. The blocks of the partition Π are given by H_{ij^{α}} α m _{1 i j 3} . The blocks of the second one, 𝒩 ^{}, are defined by H_{ij^{α}} 1 i j 3, α τ mod 3 _{τ _{3}} , with α replaced by α when i, j 1, 3 . Let us check axiom (8) for 𝒩 ^{}. The 2-flats X appearing in axiom (8) clearly coincide with those A_{X} that contain two hyperplanes with different colours with respect to Π^{}. For X H_{ij^{α}} α m we find that n_{X} n. For X H_{12^{α}}, H_{23^{β}}, H_{13^{γ}} with α β γ 0, the condition on colours translates to α β γ mod 3 , and implies that n_{X} 1. Hence Π^{}defines a reduced weighted 3-partition 𝒩 ^{}. Consider the two (surjective) homomorphisms from Proposition 4.2, ϕ, ϕ^{}: H_{A} H_{S}. Clearly, ϕ^{}S exp ϕ^{}H^{1} S, is a positive dimensional subtorus of M_{A} , and similarly for ϕ^{}. Moreover, ϕ^{}S 𝒱_{1}^{1} M_{A} , since ϕ^{}H^{1} S, H^{1} M_{A}, is isotropic of dimension 2, hence contained in ℛ_{1}^{1} A, , and likewise for ϕ^{}. Therefore, we may find two irreducible components of 𝒱_{1}^{1} M_{A} , W and W^{}, such that ϕ^{}S W and ϕ^{}S W^{}. On the other hand, ρ_{3} A W W^{}, by the argument from the proof of Theorem 4.3(1).

In this situation, it follows from a result of Artal Bartolo, Cogolludo and Matei [1, Proposition 6.9] that ρ_{3} A 𝒱_{2}^{1} M_{A} , if W W^{}. Hence, e_{3} A 2, by equality (2), and we are done. Suppose then that W W^{}. We know from [10] that actually W ϕ^{}S , since ϕ comes from a 3-net. Taking tangent spaces at the origin 1 M_{A} , we infer that ϕ^{}H^{1} S, ϕ^{}H^{1} S, . We identify H^{1} M_{A}, with ^{A}, using the distinguished -basis. In this way, the subspace ϕ^{}H^{1} S, (respectively ϕ^{}H^{1} S, ) is identified with the subset of those elements η ^{A} (respectively η^{}^{A}) taking the constant values a, b, c (respectively a^{}, b^{}, c^{}) on the blocks of Π (respectively Π^{}), where a b c a^{}b^{}c^{}0. Now, it is an easy matter to check that ϕ^{}H^{1} S, ϕ^{}H^{1} S, 0. This contradiction finishes the proof of Theorem 1.1.

4.6. Full monodromy action. It follows from decomposition (1) that the characteristic polynomial Δ_{A} t t 1 ^{A 1} Π_{1 d n}Φ_{d} t ^{e_{d} A} encodes the full monodromy action on H_{1} F_{A}, . The approach via modular bounds works only for prime power monodromy multiplicities, e_{p^{s}} A . One way to avoid this inconvenience is to impose restrictions on multiplicities of 2-flats, like in [20] for instance, to arrive at full monodromy computations. Unfortunately, as we saw in §2.3, arbitrarily high flat multiplicities may appear for non-exceptional complex reflection arrangements. Even in this kind of situation, there is hope related to the following well-known vanishing criterion (see e.g. [14]): if d A_{X} for any X L_{2} A , then e_{d} A 0. It turns out that this works for full monomial arrangements of small rank. The result below verifies in particular the strong form of the conjecture from [20].

Proposition 4.4. For A A m, 1, l , with l 3 or 4, Δ_{A} t t 1 ^{A 1} t^{2} t 1 , if l 3 and m 1 mod 3 , and Δ_{A} t t 1 ^{A 1} , otherwise.

Proof. We have to compute e_{d} A for all divisors d 1 of A . If d is prime, this was done in Theorem 1.1 and Theorem 1.2. It follows from §2.3 that the 2-flats of A have multiplicities 2, 3 or m 2, A 3 m 1 for l 3 and A 2 3m 2 for l 4. If d is not prime and e_{d} A 0, the vanishing criterion forces m 2 mod d . Writing that A 0 mod d , we obtain for l 3 that 3 0 mod d , a contradiction, and 8 0 mod d , for l 4. In the second case, the modular bound implies that β_{2} A 0, contradicting Theorem 1.2(2).

Acknowledgments. We are grateful to Alex Suciu for helpful discussions. The second author thanks Université de Nice Sophia Antipolis, where this work was completed, for support and hospitality.

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Daniela Anca Măcinic Simion Stoilow Institute of Mathematics P.O. Box 1-764 RO-014700 Bucharest Romania Anca.Macinic@imar.ro Ştefan Papadima Simion Stoilow Institute of Mathematics P.O. Box 1-764 RO-014700 Bucharest Romania Stefan.Papadima@imar.ro

Clement Radu Popescu

Simion Stoilow Institute of Mathematics unit no. 4 P.O. Box 1-764 RO-014700 Bucharest Romania Radu.Popescu@imar.ro