1 Objects of Study

Further context for much of the following can be found in the extensive survey[12] and the book [8].

1.1 Preliminary Definitions and Examples

Definition 1.1. A central hyperplane arrangementA in Cd, often called anarrangement, is a collection of finitely many hyperplanes H1, . . . ,Hn passingthrough the origin. A defining polynomial for the arrangement A, denotedQ(A), is a polynomial Q(A) =

∏ni=1 hi, where the hi ∈ C[x1, . . . , xd] are homo-

geneous polynomials of degree one such that the zero set of hi is Hi. Such apolynomial Q(A) is defined up to multiplication by a nonzero scalar.

Definition 1.2. The complementM(A) of an arrangementA = {H1, . . . ,Hn}in Cl is the set Cd \

⋃ni=1Hi. The projectified complement U(A) of A is the

image of M(A) in CPd−1 under the Hopf fibration π : Cd \ {0} → CPd−1.

Definition 1.3. The intersection lattice of an arrangementA = {H1, . . . ,Hn}in Cd is the collection L(A) of all intersections of the Hi, ordered by reverseinclusion. Elements of L(A) are called flats.

Example 1.4. Let A be the Boolean arrangement in C3 with defining poly-nomial Q(A) = xyz. The intersection lattice L(A) is given in Figure 1.

Example 1.5. Let A be the braid arrangement in C3 with defining polyno-mial Q(A) = (x − y)(x − z)(y − z). The intersection lattice L(A) is given inFigure 2.

Example 1.6. Let A be the reflection arrangement of type A3 in C3 withdefining polynomial Q(A) = (x + y)(x − y)(x + z)(x − z)(y + z)(y − z). Theintersection lattice L(A) is given in Figure 3.

Example 1.7. Let A be the reflection arrangement of type B3 in C3 withdefining polynomial Q(A) = xyz(x+ y)(x− y)(x+ z)(x− z)(y+ z)(y− z). Theintersection lattice L(A) is given in Figure 4.

Definition 1.8. Given an arrangement A and a flat X ∈ L(A), the sub-arrangement defined by X, denoted AX , is the collection of hyperplanes Hin A such that H ⊇ X.

Definition 1.9. A property P of a lattice A is said to be combinatoriallydetermined if for any arrangement B with L(B) isomorphic to L(A) as partiallyordered sets, B enjoys property P as well.

There are two standard functions defined on the intersection lattice. Beingso defined makes them combinatorially determined.

1

0

x = 0y = 0

x = 0z = 0

y = 0z = 0

x = 0 y = 0 z = 0

C3

Figure 1: The intersection lattice for the Boolean arrangement in C3 with defin-ing polynomial Q(A) = xyz.

x = y = z

x = y x = z y = z

C3

Figure 2: The intersection lattice for the braid arrangement in C3 with definingpolynomial Q(A) = (x − y)(x − z)(y − z). The colorings partition A into a(3, 1)-net with highlighted base locus. (See Definition 2.5.)

2

0

x=

0y

=0

x=y

x=z

x=−y

x=z

x=

0z

=0

x=y

x=−z

x=−y

x=−z

y=

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=0

x=y

x=−y

x=z

x=−z

y=−z

y=z

C3

Fig

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02

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4

Definition 1.10. Given an arrangement A in Cd, the Mobius function µ onL(A) is defined inductively by setting µ(Cd) = 1 and

µ(X) = −∑Y)X

µ(X).

The rank function on L(A) is given by r(X) = codimX.

1.2 The Orlik-Solomon Algebra

Let A = {H1, . . . ,Hn} be a central arrangement in Cd. The Orlik-Solomonalgebra is a combinatorially defined ring which is isomorphic to the cohomologyring H∗(M(A),Z) of the complement of the arrangement A. The story beginswith a description of this cohomology ring in terms of differential forms due toBrieskorn, and then turns to the construction of Orlik and Solomon.

In Brieskorn’s paper [1] on braid groups, he proves the following decompo-sition of the pth cohomology group of M(A):

Lemma 1.11 (Brieskorn [1]). For each 0 ≤ p ≤ d, let Ap,1, . . . ,Ap,kp be themaximal subsets of A such that the intersection of all hyperplanes in Ap,j hascodimension p.Then the inclusions M(A) ↪→M(Ap,j) induce an isomorphism

Hp(M(A),Z) ∼=kp⊕j=1

Hp(M(Ap,j),Z).

Sketch of Proof. The injectivity is shown by finding open subsets Up,l of M(A)such that the map on cohomology induced by the composition Up,l ↪→M(A) ↪→M(Ap,j) is an isomorphism when l = j and is zero when l 6= j. Surjectivity isfirst proved in that case when p = n by decomposing Cd using a finite numberof real, parallel hyperplanes so that no more than one point of the intersectionof all hyperplanes in A lies between any two of these parallel hyperplanes.Theresult then follows for the case p = n from repeated applications of the Mayer-Vietoris sequence. For general p, one slices with a generic subspace of dimensionp and appeals to a Lefschetz type theorem to reduce to the previous case.

With this result, Brieskorn proves the following:

Proposition 1.12 (Brieskorn [1]). Let A = {H1, . . . ,Hn} be a central arrange-ment in Cd, let hj be a linear form that vanishes on Hj, and let ωj be theholomorphic differential form

ωj =1

2πi

dhjhj

.

Then the images of the ωj generate the cohomology ring H∗(M(A),Z).

5

Sketch of Proof. The proof proceeds by induction on the dimension d of theambient space. If d = 1, the result follows from the residue theorem. Supposingthat the result holds for arrangements in dimension Cd−1, we show that it holdsfor arrangements in Cd. By Lemma 1.11, we have that

Hp(M(A),Z) ∼=kp⊕j=1

Hp(M(Ap,j),Z)

for each 0 ≤ p ≤ d, with notation as in the lemma. For a fixed p and 1 ≤ j ≤ kp,we may write

M(Ap,j) = Cd−p × C∗ ×M(A′p,j),

where A′p,j is an arrangement in Cp−1.The result now follows by the inductionhypothesis.

We turn now to the construction of Orlik and Solomon in [7].

Definition 1.13. Let A = {H1, . . . ,Hn} be an arrangement in Cd and let hibe a linear form that vanishes on Hi. A subset S ⊆ A is dependent if theintersection of all hyperplanes in S has codimension greater than the number ofhyperplanes in S.Equivalently, S is dependent if the linear forms correspondingto the hyperplanes in S are linearly dependent over C.

Given an arrangement A = {H1, . . . ,Hn} in Cd, let E(A) be the exterioralgebra over Con letters e1, . . . en. Define the differential ∂E : E(A)→ E(A) tobe the C-linear map such that ∂E(1) = 0, ∂E(ej) = 1, and ∂E(fg) = ∂E(f)g +(−1)deg ff∂E(g) for all homogeneous f, g ∈ E. Then ∂2E = 0. Let I(A) bethe ideal in E(A) generated by the collection of all elements ∂E(f) with f =es1 · · · esk where {Hs1 , . . . ,Hsk} is a dependent set.

Definition 1.14. The Orlik-Solomon algebra of an arrangementA is definedas the quotient A (A) = E(A)/I(A). By construction, A (A) is combinatoriallydetermined.

The significance of the Orlik-Solomon algebra is expressed in the following:

Theorem 1.15 (Orlik, Solomon [7]). The Orlik-Solomon algebra A (A) ofan arrangement A = {H1, . . . ,Hn} in Cd is isomorphic as graded algebra toH∗(M(A),Z).In particular, the cohomology ring of M(A) is combinatoriallydetermined.

Sketch of Proof. By Proposition 1.12, we have that H∗(M(A),Z) is isomorphicto the algebra R generated by the holomorphic differential forms ωj as in thestatement of the proposition. It therefore suffices to show that A (A) is isomor-phic to R.

Let ϕ : E(A)→ R be defined by sending ej to ωj . We show that this surjec-tion induces a surjection from A (A) to R by showing that ϕ(I(A)) = 0. SinceI(A) is generated by elements ∂E(es1 · · · esk) with {Hs1 , . . . ,Hsk} dependent, it

6

suffices to show that these elements are in the kernel of ϕ. We establish thisusing induction on k, the number of hyperplanes in a dependent set.

If es1 · · · esk is such that S = {Hs1 , . . . ,Hsk} is dependent and no propersubset of S is dependent, we may choose linear forms hs1 , . . . , hsk defining the

hyperplanes in S so that∑kj=1 hsj = 0. Then since

0 =

k∑j=1

dhsj

(dhs1 · · · dhsj dhsj+1 · · · dhsk),

it follows that

dhs1 · · · dhsj dhsj+1· · · dhsk = −dhs1 · · · dhsj dhsj+1

· · · dhsk .

Define ηsj so that

hsjηsj = (−1)j−1ωs1 · · · ωsj · · ·ωsk .

Thenhs1 · · ·hskηsj = hs1 · · ·hskηsj+1 ,

hence ηsj = ηsj+1for all j. Let the common value be denoted η. Then

ϕ(∂E(es1 · · · esk)) =

k∑j=1

hsj

η = 0.

In particular, the result holds if k = 2 providing basis for the induction. On theother hand, if S is a dependent set with a dependent proper subset, the productformula for ∂E and the inductive hypothesis establish the result. We thereforehave an induced surjection ϕ : A (A)→ R.

To see that the map is an isomorphism, it suffices to show that A (A) hasthe same dimension as R. It is shown in [7] that the dimension of of A (A) is∑

X∈L(A)

µ(X)(−1)r(X).

It therefore suffices to show that R has this dimension as well. Appealing toLemma 1.11, we have that

R ∼=⊕p≥0

Hp(M(A),Z) ∼=⊕p≥0

⊕X∈L(A)r(X)=p

Hp(M(AX),Z)

.

Granting that dimHr(X)(M(AX),Z) = µ(X)(−1)r(X) for all X ∈ L(A), weobtain that

dim R =∑

XL(A)

µ(X)(−1)r(X) = dim A (A),

7

establishing the result.It remains to prove that dimHr(X)(M(AX),Z) = µ(X)(−1)r(X) for all X ∈

L(A). This is proved by induction on q = r(X). If q = 0, equality holds witha common value of 1. Suppose now that the result holds for all Y ∈ L(A)with r(Y ) < q and let XL(A) with r(X) = q. It can be shown that the Eulercharacteristic of M(AY ) is 0 when r(Y ) > 0.Using this, we have that

0 =

(q−1∑p=0

(−1)p dimHp(M(AX),C)

)+ (−1)q dimHq(M(AX),C).

Applying 1.11 allows us to continue:

=

q−1∑p=0

(−1)p∑

Y ∈L(AX)r(Y )=p

dimHp(M((AX)Y ),C)

+ (−1)q dimHq(M(AX),C)

=

∑Y ∈L(AX)r(Y )<q

(−1)r(Y ) dimHp(M((AX)Y ),C)

+ (−1)q dimHq(M(AX),C).

Using the inductive hypothesis allows us to re-write the first sum and thenappeal to the inductive definition of the Mobius function:

=

∑Y ∈L(AX)r(Y )<q

µ(Y )

+ (−1)q dimHq(M(AX),C)

= −µ(X) + (−1)q dimHq(M(AX),C).

Our conclusion is that

dimHr(X)(M(AX),C) = µ(X)(−1)r(X),

completing the induction and the proof.

Example 1.16. Let A be the Boolean arrangement in C3 from Example 1.4.Then A has no dependent collections of hyperplanes so that H∗(M(A),Z) ∼=A (A) is the full exterior algebra on letters e1, e2, e3.

Example 1.17. Let A be the braid arrangement in C3 from Example 1.5. Theonly dependent collection of hyperplanes is A itself. Let E(A) be the exterioralgebra on letters e1, e2, e3 and let I(A) be the ideal in E(A) generated by theelement ∂E(e1e2e3) = e1e2 − e1e2 + e2e3. Then

H∗(M(A),Z) ∼= A (A) ∼= Z⊕ Z3 ⊕ Z3

Z(1,−1, 1)⊕ Z

Z.

8

The Theorem 1.15 implies, in particular, that for an arrangmenet A of nhyperplanes, we have that H1(M(A),Z) ∼= Zn with canonical basis e1, . . . , en.Similar arguments can be applied to establish the following

Fact 1.18. The first homology H1(U(A),Z) of the projectivized complement ofA is given by

H1(U(A),Z) = Zn/

(n∑i=1

ei

)∼= Zn−1.

1.3 Further Combinatorial Questions

Given that the cohomology ring of the complement of an arrangement is combi-natorially determined, one might ask if all topological information of the com-plement is so determined. Rybnikov showed in [11] that this is not the case byconstructing two arrangements with isomorphic lattices so that the correspond-ing complements have non-isomorphic fundamental groups.

Theorem 1.19 (Rybnikov [11]). The fundamental group of the complement ofan arrangement is not combinatorially determined.

We now introduce the main object of study.

Definition 1.20. The Milnor fiber F (A) of an arrangement A in Cd withdefining polynomial Q(A) is the subset Q(A)−1(1) of Cd. This is defined up todiffeomorphism provided by multiplication by a nonzero scalar.

Problem 1.21. Determine if the Betti numbers bi = dimC(Hi(F (A),C)) arecombinatorially determined. This problem is even open for the first Betti num-ber of F (A), where A is an arrangement in C3.

2 Studying the Milnor Fiber with CohomologyJump Loci

The group of nth roots of unity acts on the Milnor fiber F (A) of an arrangementA = {H1, . . . ,Hn} via multiplication. Indeed, if Q(A)(z) = 1 and if ζ is an nthroot of unity, then Q(A)(ζz) = ζnQ(A)(z) = 1 since Q(A) is a homogeneouspolynomial of degree n. By functoriality of Hi, the group of nth roots of unityacts on Hi(F (A),C).

Definition 2.1. The (algebraic) monodromy on F (A) is the map

h : Hi(F (A),C)→ Hi(F (A),C)

defined by acting with a primitive nth root of unity.

Studying the monodromy action on F (A) yields a (possibly non-combinatorial)expression for the first cohomology.

9

2.1 Resonance Varieties

Let X be a topological space, let k be a field, and let A be the cohomology ringof X with coefficients in k. For each element ω ∈ A1 of degree 1, multiplicationby ω defines a differential ∂ω on A. Let

Hq(X,ω) = ker(∂ω : Aq → Aq+1)/ im(∂ω : Aq−1 → Aq)

be the corresponding cohomology.

Definition 2.2. Let k be a field, X a topological space, and A the cohomologyring of X. The dimension q depth s resonance variety over k of X is

Rqs(X, k) = {ω ∈ A1 : dimkHq(X,ω) ≥ s}.

Note 2.3. Since the Orlik-Solomon algebra gives a combinatorial description ofthe cohomology of M(A), it follows that the resonance varieties Rqs(M(A), k)are combinatorially determined. Moreover, the degree one part of the Orlik-Solomon algebra is free of dimension |A|, hence we may identify the resonancevarieties with subsets of k|A|.

With this noted, a natural problem arises:

Problem 2.4. Find expressions for the resonance varieties making the combi-natorial dependence explicit.

This problem has an answer for dimension one resonance varieties over Cinvolving the concept of a multinet.

Definition 2.5. A multinet on an arrangement A is a partition of A intosome number k ≥ 3 of disjoint subsets A1, . . . ,Ak together with an assignmentof positive multiplicities m : A → N to each hyperplane such that

(i) there is an integer l with∑H∈Ai

m(H) = l for all 1 ≤ i ≤ k;

(ii) for every flat X in the base locus of the multinet, the set

χ = {H ∩H ′ : H ∈ Ai, H ′ ∈ Aj , i 6= j},

we have thatnX =

∑H∈AiX⊂H

m(H)

is independent of the choice of 1 ≤ i ≤ k;

(iii) for each 1 ≤ i ≤ k, the space(⋃

H∈AiH)\ χ is connected.

We say such a multinet has k classes and weight l, and call it a (k, l)-multinet.If all multiplicities are 1 the multinet is reduced. If in addition to being reducedevery flat in the base locus is contained in exactly one hyperplane in each of theAi, we call the multinet a net.

10

Example 2.6. The arrangements given in Examples 1.5, 1.6, and 1.7 supportmultinets as depicted in the corresponding Figures displaying the intersectionlattices. The Boolean arrangement given in Example 1.4 does not support amultinet.

The following results are due to Pereira and Yuzvinsky

Theorem 2.7. Let A be an arrangement with k classes and base locus χ. Then

(i) If χ has more than one element, then k = 3 or k = 4.

(ii) If there is a hyperplane with multiplicity bigger than 1, then k = 3.

Example 2.8. The Hessian arrangement in C3 has defining polynomialQ(A) = xyz(x3 − y3)(x3 − z3)(y3 − z3). This is the only known example of anarrangement supporting a (4, l)-multinet.

Conjecture 2.9 (Yuzvinsky). The only arrangement supporting a multinet with4 classes is the Hessian arrangement.

The following lemma is stated (with references to further sources) in [6] ascommon knowledge:

Lemma 2.10. Let L(A)2 be the rank 2 elements of L(A), let L(A)′2 be the flatsX in L(A)2 with AX dependent, and let ω =

∑i ωiei ∈ A (A)1 be nonzero.

Then τ =∑i τiei is in ker(∂ω) ∩A (A)1 if and only if both

(i) for every X ∈ L(A)′2 such that∑{i:Hi∈AX}

ωiei 6= 0 and∑

{i:Hi∈AX}

ωi 6= 0,

then∑{i:Hi∈AX} τi = 0.

(ii) for every other X ∈ L(A)2 and i < j with Hi, Hj ∈ AX ,

ωiτj − ωjτi = 0.

Let B be an arrangement in Cd and let M be a multinet on B, so that Mis a partition B1, . . . ,Bk of B into k classes and m : B → N is an assignment ofmultiplicities. Let

ui =∑

{j:Hj∈Bi}

m(Hj)ej .

Then the span over C of the collection {u2 − u1, . . . , uk − u1} is called PM andis contained in R1

1(M(B),C) by the previous lemma.The paper [4] establishes a converse (in a small case) which leads to the

following

Theorem 2.11. The positive dimensional, irreducible components of R11(M(A),C)

are in one-to-one correspondence with the multinets on subarrangements of A,and

R11(M(A),C) = {0} ∪

⋃B⊆A

( ⋃M multinet on B

PM

).

11

2.2 Local Coefficient Systems and Characteristic Varieties

Further background on local coefficient systems can be found in [3].

Definition 2.12. Fix a ring R and an R-module L. A local coefficientsystem L with fiber L on a connected topological space X is a sheaf whichis locally isomorphic to the constant sheaf defined by L.The local system is saidto be a rank one local system if L is a free R-module of rank 1.

Lemma 2.13. A local system L on [0, 1] is constant.

Example 2.14. Let X = C \ {0}, let f : X → X be defined by f(z) = zn forsome n ∈ N, and let L be the nth root sheaf given by

L (U) = {aφ : a ∈ C, φ : U → U such that fφ = idU} ∪ {0}

for all open U in X. Then L is a rank one local coefficient system on X which isnot a constant sheaf. Given γ : [0, 1]→ X, a continuous map with γ(0) = γ(1),the natural isomorphism (γ−1L )0 ∼= (γ−1L )1 is given by multiplication withe2πi/n.

Proposition 2.15. Let k be a field, V a k-vector space, and X a connectedtopological space. There is an equivalence between the categories of

(i) k-local systems on X with fiber V

(ii) Representations ρ : π1(X)→ Aut(V ).

Sketch of Proof. Let L be a k-local system with fiber V on X and let γ : [0, 1]→X be a loop in X. The pullback of L to [0, 1] is a constant sheaf by Lemma2.13. We then get isomorphisms

V = Lx∼= (γ−1L )0 ∼= (γ−1L )1 ∼= Lx = V.

Therefore the composite is an isomorphism which is represented by an elementα of Aut(V ). We then define ρ(γ) = α. It can be shown that this defines arepresentation of the fundamental group.

Conversely, let ρ : π1(X)→ Aut(V ) be a representation of the fundamental

group. Let p : X → X be the universal cover of X and let L be the constantsheaf associated to V on X. Define a sheaf L on X by declaring for each openset U ⊆ X,

L (U) = {s ∈ L (p−1(U)) : ∀u ∈ p−1(U),∀γ ∈ π1(X), s(γu) = ρ(γ)s(u)}.

It can be shown that this is a local system on X.

Definition 2.16. The dimension q depth s characteristic variety of a con-nected space X over an algebraically closed field k is

Vqs (X, k) = {ρ ∈ Hom(π1(X), k∗) : dimkHq(X,Lρ) ≥ s},

where Lρ denotes the rank one local system corresponding to the representationρ.

12

Let A = {H1, . . . ,Hn} be an arrangement in Cd, γi a meridian loop aboutHi in M(A), and k is an algebraically closed field.

Fact 2.17. Given an arrangement A = {H1, . . . ,Hn} in Cd, the fundamentalgroup π1(M(A)) is generated by the images of meridian loops γi about eachhyperplane Hi.

It follows from this fact that any representation ρ : π1(M(A)) → k∗ is de-termined by its action on the classes of the γi. Conversely, let (ai) ∈ (k∗)n begiven. The Orlik-Solomon algebra shows that H1(M(A),Z) ∼= Zn. Composingthe Hurewicz homomorphism π1(M(A)) → H1(M(A),Z) ∼= Zn with the mapsending the image of γi to ai produces a representation of π1(M(A)). In lightof Proposition 2.15, we have the following

Fact 2.18. There is a one-to-one correspondence between rank one local systemson M(A) with fiber k, representations ρ : π1(M(A)) → k∗, and elements of(k∗)n.

Using the Fact 1.18, similar analysis provides the following

Fact 2.19. Let A be an arrangement consisting of n hyperplanes. There is aone-to-one correspondence between rank one local systems on U(A) with fiber k,representations ρ : π1(U(A))→ k∗, and elements of (k∗)n−1.

2.3 First Betti Number of the Milnor Fiber

The connection between the Milnor fiber and characteristic varieties lies in cov-ering space theory. The following basic result for a sufficiently connected topo-logical space X is proved, for example, in [5].

Theorem 2.20. There is a bijection between the set of basepoint preservingisomorphism classes of path-connected covering spaces (X, x0) of (X,x0) and

the set of subgroups of π1(X,x0) which associates a cover p : (X, x0)→ (X,x0)

to the subgroup p∗(π1(X, x0)).

If X is a sufficiently connected topological space and χ : π1(X) � A is asurjection onto a finite cyclic group with cyclic generator α, let Xχ denote thecovering space corresponding to kerχ guaranteed by Theorem 2.20. Let k be analgebraically closed field with characteristic not dividing the order of A. Let hbe the action of α on Xχ, and let h∗ : Hq(X

χ, k) → Hq(Xχ, k) be the induced

map on homology. Let χ : Hom(A, k∗)→ Hom(π1(X), k∗) be the map inducedby χ. This is an injective map and so given ρ ∈ im(χ) there exists a uniquemap ιρ ∈ Hom(A, k∗) such that ρ = ιρ ◦ χ. The following result attributed toLibgober, Sakuma, Hironaka, and Matei-Suciu can be found in [12]:

Theorem 2.21. In the situation described above, the characteristic polynomial∆kχ,q = det(t · id− h∗) of h∗ is given by

∆kχ,q =

∏s≥1

∏ρ∈im(χ)∩Vq

s

(t− ιρ(α)).

13

In order to apply this general covering space theorem to say something aboutthe Milnor fiber, we show that the Milnor fiber acts as a covering of the projec-tivized complement. The following result can be found in [12].

Proposition 2.22. Let A = {H1, . . . ,Hn} be an arrangement in Cd. The re-striction π : F (A)→ U(A) of the Hopf fibration is is a regular Zn-cover classifiedby the homomorphism

δ : π1(U(A)) � Zndefined by sending the image of a meridian loop γi about Hi to 1.

Sketch of Proof. Let Q(A) be a defining polynomial for A. We may identify therestriction of Q(A) to a fiber of U(A) under the Hopf fibration π as the coveringmap q : C∗ → C∗ defined by sending z to zn.The claim that it is a Zn-regularcover follows since the restriction of of the Hopf fibration to F (A) → U(A) isthe orbit map of multiplication by a primitive nth root of unity.

The defining homomorphism is obtained by using the fact that pullbacks ofcovering maps along continuous maps provide covering maps. One pulls backthe covering C∗ → C∗ sending z to zn and the covering F (A) → U(A) to acommon covering which permits the computation via comparison.

The survey [12] states a result of Dimca, Artal Bartolo, Cogolludo, and Mateiwhich builds on work by Arapura. This result describes the dimension one depthone characteristic varieties over C of complements of hyperplane arrangementsusing orbifold fibrations. This result and the Theorem 2.21 yields

Theorem 2.23. Let A = {H1, . . . ,Hn} be an arrangement, F (A) its Milnorfiber, and h the monodromy action. Let Φr be the rth cyclotomic polynomialand let φ be the Euler φ function. Then the characteristic polynomial of h isgiven by

(t− 1)n−1∏

16=r|n

Φr(t)depth(δn/r),

where δ : π1(U(A))→ C∗ is such that δ(ej) = e2πi/n for each 1 ≤ j ≤ n, whereej is the image of a meridian loop γj about Hj, and where depth(ρ) = max{s :ρ ∈ V1

s (U(A)),C}. In particular,

dimCH1(F (A),C) = n− 1 +

∑16=r|n

φ(r) depth(δn/r).

With this expression for the first Betti number of the Milnor fiber, one isleft with the question

Problem 2.24. Determine if the characteristic varieties of the projectivizedcomplement of an arrangement combinatorially determined.

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2.4 Recent Results

In [2] Cohen and Orlik use stratified Morse theory and in [10] Papadima andSuciu use topological considerations to show that

depth(δn/ps

) ≤ βp(σ),

where σ =∑i ei in A (A) ⊗ k and βp(σ) = max{s : σ ∈ R1

r(M(A), k)}. Usinginformation such as this about modular resonance, Papadima and Suciu showin [9] that the characteristic polynomial of the monodromy is

(t− 1)n−1(t2 + t+ 1)β3(σ)

for arrangements with flats of multiplicity no bigger than 3.

References

[1] Egbert Brieskorn. Sur les groupes de tresses [d’apres V. I. Arnol′d]. InSeminaire Bourbaki, 24eme annee (1971/1972), Exp. No. 401, pages 21–44. Lecture Notes in Math., Vol. 317. Springer, Berlin, 1973.

[2] Daniel C. Cohen and Peter Orlik. Arrangements and local systems. Math.Res. Lett., 7(2-3):299–316, 2000.

[3] Fouad El Zein and Jawad Snoussi. Local systems and constructible sheaves.In Arrangements, local systems and singularities, volume 283 of Progr.Math., pages 111–153. Birkhauser Verlag, Basel, 2010.

[4] Michael Falk and Sergey Yuzvinsky. Multinets, resonance varieties, andpencils of plane curves. Compos. Math., 143(4):1069–1088, 2007.

[5] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge,2002.

[6] Anatoly Libgober and Sergey Yuzvinsky. Cohomology of the Orlik-Solomonalgebras and local systems. Compositio Math., 121(3):337–361, 2000.

[7] Peter Orlik and Louis Solomon. Combinatorics and topology of comple-ments of hyperplanes. Invent. Math., 56(2):167–189, 1980.

[8] Peter Orlik and Hiroaki Terao. Arrangements of hyperplanes, volume 300 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principlesof Mathematical Sciences]. Springer-Verlag, Berlin, 1992.

[9] S. Papadima and A. I. Suciu. The Milnor fibration of a hyperplane arrange-ment: from modular resonance to algebraic monodromy. ArXiv e-prints,January 2014.

[10] Stefan Papadima and Alexander I. Suciu. The spectral sequence of anequivariant chain complex and homology with local coefficients. Trans.Amer. Math. Soc., 362(5):2685–2721, 2010.

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[11] G. Rybnikov. On the fundamental group of the complement of a complexhyperplane arrangement. ArXiv Mathematics e-prints, May 1998.

[12] Alexander I. Suciu. Hyperplane arrangements and Milnor fibrations. Ann.Fac. Sci. Toulouse Math. (6), 23(2):417–481, 2014.

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