Hyperplane Arrangements an Introduction Universitext Universitext

Universitext

Alexandru Dimca Hyperplane Arrangements An Introduction Universitext Universitext

Series editors Sheldon Axler San Francisco State University, San Francisco, CA, USA

Carles Casacuberta Universitat de Barcelona, Barcelona, Spain

Angus MacIntyre Queen Mary University of London, London, UK

Kenneth Ribet University of California, Berkeley, CA, USA

Claude Sabbah École polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France

Endre Süli University of Oxford, Oxford, UK

Wojbor A. Woyczyński Case Western Reserve University, Cleveland, OH, USA

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Hyperplane Arrangements An Introduction

123 Alexandru Dimca Université Côte d’Azur Nice France

ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-3-319-56220-9 ISBN 978-3-319-56221-6 (eBook) DOI 10.1007/978-3-319-56221-6

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This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface

Hyperplane arrangement theory is a very active area of research, combining ideas from combinatorics, algebraic topology and algebraic geometry in a blend that is both tasty and useful. The classical textbook was written by Orlik and Terao [180] about 25 years ago, when the subject was already vast and mature. A continuation by the same authors was produced 10 years later, to discuss important progress involving local systems and twisted cohomology, see [183]. On the combinatorial side, the book by Stanley [213] is an excellent introduction, see also the monograph [29] by Björner, Las Vergnas, Sturmfels, White, Ziegler, as well as Cartier’s report [40]. The recent and detailed monograph by De Concini and Procesi [58] adds new important directions to the subject, by looking at polytopes and polyhedra, matroids and root systems, splines and numbers of integral points in polytopes. The results continued to accumulate over recent years, and a number of leading experts in the area joined efforts to produce a rather complete survey of the current situation, but their book is still a project, see [50] for a preliminary version. Our aim in writing these notes was more modest: We intended to write an introduction to hyperplane arrangement theory which is both accessible and motivating. On the accessibility side, we have recalled in the ﬁrst few chapters many of the basic results from the book by Orlik and Terao [180], choosing to avoid the proofs that are long and technical. As for the motivating side, our choice of topics in the latter chapters was inspired by the current frontiers of research and includes both new results and open problems. As a result, some important topics are not treated at all, e.g., the Lie algebras attached to arrangements, which have a whole chapter devoted to them in [50]. Other subjects, such as free arrangements, are treated from a very personal viewpoint. One of our main concerns was to keep the book at a reasonable size, so that the resulting text is more an invitation to explore a beautiful area of mathematics and/or to embark on a related research project, rather than a reference monograph. The exercises at the end of each chapter are a good test of the understanding of the material and should make the book (or parts of it) readily usable in a graduate course.

v vi Preface

We describe the contents of this book in more detail. Chapter 1 is a brief introduction, in which we point out the interplay between combinatorics, topology, geometry and arithmetic in the realm of line arrangements in the plane, where the reader’s intuition can be strongly supported by drawings. Each of the themes introduced here is developed fully in a later chapter. We include a discussion of the Sylvester–Gallai property for real line arrangements, both the classical projective version in Theorem 1.5 and an affine version in Theorem 1.4. The proof of both results is inspired by Hirzebruch’s approach in [135]. In Chap. 2, we collect the basic definitions and results involving the intersection lattice LðAÞ of a hyperplane arrangement A; we explain the key induction technique based on triples of hyperplane arrangements (A,A0,A00) and apply it to deduce the main properties of the characteristic polynomial v(A; t) and of the Poincaré polynomial p(A; t). The characteristic polynomial enters into Zaslavsky’s Theorem 2.8, expressing the number of regions (resp. bounded regions) of the complement of a real arrangement A in terms of v(A, Æ1). These numbers are also related to the number of singularities of the polynomial Q on the complement MðAÞ, for any essential affine arrangement A : QðxÞ¼0, see Theorem 2.9.We also introduce the supersolvable arrangements and state the factorization property of their Poincaré polynomials in Theorem 2.4. In the last section, we define the graphic arrangements and state the fact that the chromatic polynomial of a simple graph C coincides with the characteristic polynomial of the associated hyperplane arrangement AC, see Theorem 2.10. Finally, we discuss the reflection arrangements and introduce the main arrangements in this class, namely the monomial arrangements Aðr; r; nÞ in Example 2.24 and the full monomial arrangements Aðr; 1; nÞ in Example 2.23. The purely combinatorial definition of the Orlik–Solomon algebra AÃðAÞ of a hyperplane arrangement A is given in Chap. 3, and the fundamental result stating that this algebra is isomorphic to the cohomology algebra of the complex hyperplane arrangement complement MðAÞ is proved in Theorem 3.5. To do this, we assume a technical result on the behavior of the Orlik–Solomon algebras with respect to triples, see Theorem 3.1. In this chapter, we also mention a tensor product decomposition of the Orlik–Solomon algebra of a supersolvable arrangement, see Theorem 3.3, as well as an alternative definition of the Orlik–Solomon algebra of a projective hyperplane arrangement, see Theorem 3.4. In Chap. 4, we discuss the minimality of the complement MðAÞ and its relation to the degree of the gradient mapping of the defining equation for A, see Theorem 4.4. In Remark 4.2, we collect some basic results on the topology of the n union of the hyperplanes in A, namely on the hypersurface NðAÞ¼fx 2 C : QðxÞ¼0g. Then, we mention two beautiful results of June Huh, the first on the log-concavity of the coefficients of the Poincaré polynomial pðA; tÞ, see Theorem 4.6, and the second on the relation between the degree of the gradient mapping of a projective hypersurface V and the multiplicities of its singularities, see Theorem 4.7. When V is a line arrangement, we give an elementary new proof of a more precise version of the latter result in Theorem 4.8. In this chapter, we also discuss the fundamental group of the complement MðAÞ and the arrangements Preface vii whose complements are Kðp; 1Þ-spaces. We state Deligne’s result which says that real simplicial hyperplane arrangements give rise to such Kðp; 1Þ-spaces, see Theorem 4.12, as well as Bessis’ result which says that the same holds for the complex reflection arrangements, see Theorem 4.15. We introduce the fiber type arrangements, show their relation to the Kðp; 1Þ-spaces in Theorem 4.15, and then state the surprising fact that a central arrangement is supersolvable if and only if it is fiber type, see Theorem 5.3. This is another deep connection between combinatorics and topology in this subject. Some very interesting groups occur as fundamental group of complements MðAÞ, for instance the Stallings group and the Bestvina– Brady groups are discussed in Remark 4.11. In Chap. 5, we start our discussion of the Milnor fiber F associated to a central hyperplane arrangement A, the monodromy action on the cohomology HÃðFÞ and the relation between monodromy eigenspaces and the twisted cohomology of the complement MðA0Þ, where A0 is the projective arrangement associated to A, see Proposition 5.4. Then, we state a very general vanishing result on the twisted cohomology of the complement MðA0Þ with coefficients in a rank one local system, a result due to D. Cohen, P. Orlik and the author, see Theorem 5.3, whose consequences are used several times in the sequel. A largely unexplored subject, namely a natural candidate for the Milnor fiber of an affine hyperplane arrangement, is discussed in Remark 5.3, where the Bestvina–Brady groups occur again naturally. Characteristic varieties (resp. resonance varieties) are jumping loci for some cohomology groups which are topologically (resp. algebraically) defined, and are at the center of many research papers published in the last decade in relation to both hyperplane arrangements and, more generally, the fundamental groups of smooth quasi-projective algebraic varieties. We discuss the relation between the characteristic varieties and the homology of finite abelian covers, in particular cyclic covers of prime order in Proposition 6.3 and congruence covers in Theorem 6.2. This brings us to the polynomial periodicity properties of the first Betti numbers of such covers, see Theorem 6.3, and to the smooth surfaces obtained as coverings of P2 ramified over a line arrangement, in particular to the Hirzebruch covering surfaces, a.k.a. Hirzebruch–Kummer surfaces, see Theorem 6.4. The main results in this chapter are the Tangent Cone Theorem 6.1 and its relation with the multinet structures introduced by M. Falk and S. Yuzvinsky, see Theorem 6.6. This brings in unexpected and beautiful relations with the theory of pencils of projective plane curves studied in classical algebraic geometry, see Theorem 6.5. After a brief discussion of the translated components of the characteristic varieties, we treat in great detail the deleted B3-line arrangement. This was the first example of a hyperplane arrangement having a translated component of a characteristic variety, and it was discovered by Suciu in [217]. In Chap. 7,wefirst consider more general smooth quasi-projective varieties and prove a version of the Tangent Cone Theorem in this setting, using the logarithmic connections, and following the main ideas from [112], [204], see Theorem 7.3. Then, we discuss the mixed Hodge structure on the cohomology of the hyperplane viii Preface complement MðAÞ and of the Milnor fiber F.Wedefine the corresponding spectrum and state the key results of N. Budur and M. Saito which say that this spectrum is determined by the intersection lattice and which give an explicit formula in the case of a line arrangement in P2, see Theorems 7.6 and 7.8. In Remark 7.4, we relate the topology of the Milnor fiber F associated to a projective line arrangement to the topology of two natural compactifications of F, one with isolated singularities and the other a smooth surface. Next, we discuss an arithmetic property of algebraic varieties Y defined over the rationals Q, namely the polynomial count property, see Definition 7.5. Following Katz [131], this property is related to Hodge theoretic properties of Y, i.e., to the property of Y being cohomologically Tate. This property holds when Y ¼ MðAÞ, the complement of a central or projective arrangement A. When Y is the Milnor fiber F of such an arrangement, then this property is related to the triviality of the monodromy action on all the cohomology groups of F, see Propositions 7.6 and 7.8, Theorem 7.10 and Example 7.10. A discussion of Hodge–Deligne polynomials of line arrangements with only nodes and triple points completes this chapter. Chapter 8 starts with a discussion of free projective hypersurfaces, stressing the relation with the Jacobian syzygies of the defining equation f ¼ 0. After giving the general definitions, we give a proof of Kyoji Saito’s Criterion in this setting, see Theorem 8.1, valid for any projective hypersurface and not only for hyperplane arrangements. The factorization property for pðA; tÞ, when A is a free arrangement, is given in Theorem 8.3, while the fact that any supersolvable arrangement is free occurs in Theorem 8.4. The freeness of reflection arrangements is stated in Theorem 8.2. For the monomial line arrangement, the full monomial line arrangement and the Hessian arrangement, bases of the Jacobian syzygy modules are given in Example 8.6. The locally free arrangements A occur in Theorem 8.5, and the Chern classes of their logarithmic 1-form vector bundles X1ðlog AÞ are determined in Theorem 8.6. Tame hyperplane arrangements are shown to play a key role in the Logarithmic Comparison Theorem 8.7 due to J. Wiens and S. Yuzvinsky. Then, we move to the case of curves (and in particular, line arrangements) in P2 and state several characterizations of such free curves in Proposition 8.2 and Theorem 8.8. We state a recent result of ours, see Theorem 8.9, relating the freeness of a line arrangement to the multiplicities of its intersection points. This result (or the freeness of supersolvable arrangements) is used to construct free line arrangements with arbitrary exponents in Theorem 8.10. In the third section, we discuss a spectral sequence approach to the computation of the Alexander polynomial of a plane curve, see Theorems 8.12 and 8.15. The initial term of this spectral sequence is given by the cohomology of the Koszul complex associated to the partial derivatives fx; fy; fz, and this is why the Jacobian syzygies play a major role in this approach. First, we consider the case when the curve C : f ¼ 0 and describe an algorithm to compute the second page of this spectral sequence. This algorithm is very fast, as we can use the software Singular to determine a basis of the Jacobian syzygy module. The general case is considered Preface ix next, and here, we reduce everything to some huge systems of linear equations, see Eq. (8.42). In this case, the necessary computer time increases. In the final section, we show that this approach is effective, by looking at the monomial line arrangement and the Hessian arrangement: In each case, we construct explicit bases for some eigenspaces of the monodromy action on H1ðFÞ, see Theorems 8.17 and 8.19. Some examples are included to show that our new approach can be successfully applied even beyond the class of free line arrangements. To state the results in a convenient way, we introduce the pole order spectra, in analogy to the spectrum, see Eq. (8.44). For some of these results, the computations based on effective algorithms developed jointly with Sticlaru in [99–101] and using the computer algebra systems Singular [55] and CoCoA [45] play an important role. These notes have grown out of several sources. First, there were my lectures on hyperplane arrangements in Nice, followed in particular by my former Ph.D. stu- dents N. Abdallah, P. Bailet, T.A.T. Dinh and H. Zuber. Then, my lectures in ASSMS GCU Lahore, followed by S. Ahmad, I. Ahmed, S. Nazir, K. Shabbir, H. Shaker and R. Zahid. And finally, the series of lectures which I gave in the USTC summer school on “Hyperplanes arrangements” in Hefei, attended by Y. Liu, Z. Wang, K.T. Wong and S. Yun. I thank all of them for their interest in the subject, their questions and their comments, which shaped the presentation into its present form. Hyperplane arrangements were not my “first love.” I became interested in this beautiful subject when I was already a (relatively) mature mathematician. In spite of that, the specialists in this field have treated me extremely well: They took the time to explain to me many things, and above all they allowed me to participate in their very friendly meetings all over the world. For all this, I thank them all. Discussions with G. Lehrer, S. Papadima, M. Saito, A. Suciu and M. Yoshinaga had a direct impact on these notes, and I am very grateful to them. Contents

1 Invitation to the Trip ...... 1 1.1 Linear Partitions and the Topology of Complements ...... 1 1.2 Finite Fields and Cohomology ...... 9 1.3 Exercises ...... 13 2 Hyperplane Arrangements and Their Combinatorics ...... 15 2.1 Central, Afﬁne and Projective Arrangements ...... 15 2.2 The Intersection Lattice and the Möbius Function ...... 21 2.3 Deletion-Restriction Theorems and the Number of Regions...... 31 2.4 Graphic Arrangements and Reﬂection Arrangements ...... 37 2.5 Exercises ...... 40 3 Orlik–Solomon Algebras and de Rham Cohomology...... 45 3.1 Orlik–Solomon Algebras for Hyperplane Arrangements...... 45 3.2 The Arnold–Brieskorn and Orlik–Solomon Theorems ...... 53 3.3 Exercises ...... 59 4 On the Topology of the Complement MðAÞ...... 61 4.1 Complements of Projective Hypersurfaces ...... 61 4.2 Minimality of MðAÞ and the Degree of the Gradient Map ...... 66 4.3 The Fundamental Group of the Complement MðAÞ ...... 71 4.4 Exercises ...... 82 5 Milnor Fibers and Local Systems ...... 85 5.1 Milnor Fibers and Monodromy ...... 85 5.2 Monodromy Eigenspaces and Twisted Cohomology ...... 90 5.3 Exercises ...... 97 6 Characteristic Varieties and Resonance Varieties ...... 99 6.1 Topological and Algebraic Jumping Loci...... 99 6.2 Jumping Loci and Pencils of Plane Curves ...... 108 6.3 Translated Components in the Characteristic Varieties...... 114

xi xii Contents

6.4 The Deleted B3-Line Arrangement...... 118 6.5 Exercises ...... 122 7 Logarithmic Connections and Mixed Hodge Structures ...... 125 7.1 Two Theorems of Pierre Deligne ...... 125 7.2 Mixed Hodge Structure and Spectrum ...... 135 7.3 Polynomial Count Varieties ...... 140 7.4 Exercises ...... 147 8 Free Arrangements and de Rham Cohomology of Milnor Fibers ...... 151 8.1 Free Divisors and Logarithmic Differential Forms ...... 151 8.2 Free Curves, Free Line Arrangements, and Terao’s Conjecture ... 159 8.3 Spectral Sequences and Alexander Polynomials...... 168 8.4 The de Rham Cohomology of Milnor Fibers ...... 178 8.5 Exercises ...... 184 References ...... 187 Index ...... 197 Chapter 1 Invitation to the Trip

Abstract This chapter is a quick introduction, in which we point out the interplay between combinatorics, topology, geometry and arithmetic in the realm of real and complex hyperplane arrangements. After giving some basic definitions, most of the results are stated in the plane, where the reader’s intuition can be strongly supported by drawings. Each of the themes introduced in this chapter is fully developed in a later chapter. We include a discussion of the Sylvester–Gallai property for real line arrangements, both the classical projective version and a new affine version. The proof of both results is inspired by Hirzebruch’s approach. The main topic of this book, the study of the monodromy of the Milnor fiber of a hyperplane arrangement, is also introduced in a very simple setting.

1.1 Linear Partitions and the Topology of Complements

= 2 We begin with a classical basic problem: What is the number Rd Rd of pieces obtained by cutting a cake d times? More precisely, consider the cake to be the plane R2, a piece to correspond to a region, i.e. a connected component of the plane with d lines removed, and let us start with d = 3. The following cases may occur, see Fig. 1.1. • three cutting lines forming a triangle (a so-called generic arrangement), and then clearly R3 = 7; • three cutting lines passing through the same point (a so-called central arrangement), and then R3 = 6; • two cutting parallel lines and a third transversal one, and then R3 = 6; • three cutting parallel lines, and then one has R3 = 4. There is a simple explicit formula which gives the complete answer for the case of an arbitrary number of lines in a plane, see [180, Introduction].

Theorem 1.1 (Roberts’ formula, 1889) Let A be a collection of d lines L1, ..., Ld in the plane R2. Then the corresponding number of regions is given by

Fig. 1.1 Three lines in the plane

p d k k − 1 l R = 1 + d + − n − j , d 2 k 2 2 i=1 j=1 where nk is the number of k-fold intersection points in A ,fork≥ 3, and there are p families of parallel lines, containing respectively l1, ..., l p lines, with l j ≥ 2.

It is easy to verify this formula in the above examples. (It is always important to check whether formulas, either given in a paper or a book, or discovered by oneself, are correct in computable situations.) For more on this formula, the interested reader n can refer to Exercise 2.9. In higher dimensions, we have a bound for the number Rd of regions in Rn obtained by cutting d times, i.e. the number of connected components of a complement of d afﬁne hyperplanes in Rn,see[180, Introduction].

Theorem 1.2 (L. Schläfli, 1901) Let A be a collection of d hyperplanes in the Rn n affine space . Then the corresponding number of regions Rd satisfies the following inequality d d d Rn ≤ + + ... + , d 0 1 n and equality holds when the hyperplanes are in generic position.

For the last claim below, see also Example 2.19. The case n = d = 3 and when the hyperplanes are in generic position is represented in Fig. 1.2. As one may have already guessed, a hyperplane arrangement is a finite collection of hyperplanes in an affine space (or, equivalently, of affine hyperplanes in a vector space) over some field K . Toavoid treating the case of parallel hyperplanes separately, as in Theorem 1.1 for instance, one often considers hyperplane arrangements in an n-dimensional projective space Pn over the field K . In the modern approach, where Algebraic Geometry plays a key role, one often considers hyperplane arrangements in Cn, instead of arrangements in Rn. Indeed, C is an algebraically closed field, while R is not. Since Cn can be regarded as a real 2n-dimensional vector space, which we cannot visualize for n > 1, we have to study the arrangements and their complements by using the tools of Algebraic Topology, for instance, fundamental groups, homology and cohomology groups, Betti numbers and Euler characteristics. For these notions we refer to [130, 211]. We now fix some notations. For an arrangement A ={H1, ..., Hd } of d hyper- n planes in K , where each hyperplane H j is given by an equation 1.1 Linear Partitions and the Topology of Complements 3

x = 0

z = 0

y = 0 3 = 3 = 3 + 3 + 3 + 3 Fig. 1.2 R3 2 0 1 2 3

j (x) = a1x1 + a2x2 +···+an xn = 0, we set d Q(A ) = j (x) ∈ K [x], j=1 and call Q(A ) the defining equation of the arrangement A . When A is an arrangement in Rn, and hence Q(A ) ∈ R[x], then the arrangement in Cn defined by the same equation Q(A ) = 0 is called the complexified arrangement associated to A , and is usually denoted by the same symbol A ,orbyAC. The connected components of the complement

n M(A ) = M(A )K ={x ∈ K | Q(A )(x) = 0} in the case K = R are convex sets, and hence contractible, uninteresting spaces for the topologists. Recall the following, and see [130, 211] if necessary. Deﬁnition 1.1 The j-th Betti number of a topological space X is deﬁned by

j b j (X) := dimQ H (X, Q) = rankH j (X, Z), 4 1 Invitation to the Trip as soon as the dimension is finite. For spaces X having only finitely many nonzero Betti numbers, the Euler characteristic χ(X) is defined to be the alternating sum j χ(X) = (−1) b j (X). j

It follows that for a real arrangement A , the only non-zero Betti number of the complement is b0(M(A )), which is exactly the number of connected components of M(A ). In this case, one clearly has b0(M(A )) = χ(M(A )), the Euler characteristic of the real complement. On the other hand, when K = C, the complement M(A ) is connected. Indeed, if p, q ∈ M(A ) are two distinct points and L is the complex n line in C determined by p and q, then it is clear that L0 = L ∩ M(A ) is just a copy of C with finitely many points deleted. It follows that the points p, g ∈ L0 can be joined by a path contained in L0. Moreover, note that M(A ) can have a very rich topology. ∗ n Example 1.1 If Q(A ) = x1x2 ···xn, then M(A ) is equal to the affine torus (C ) , which is homotopy equivalent to (S1)n, the real n-dimensional torus. By using the Künneth formula, see [130, 211], one clearly gets n b (M(A )) = b ((S1)n) = j j j for 0 ≤ j ≤ n and b j (M(A )) = 0for j > n. We will prove later the following result, showing that the topology (expressed by the Betti numbers) is determined by the combinatorics of the hyperplane arrangement, see Corollary 3.6. In particular, we have the next result, a reformulation of Corollary 3.4 below. Proposition 1.1 For any line arrangement A in the plane C2, one has the following: b0(M(A )) = 1,b1(M(A )) = d, the number of lines in A , and the second Betti number of the complement is given by b2(M(A )) = nk (k − 1), k≥2 where nk is the number of k-fold intersection points in A . Example 1.2 Consider the line arrangement defined by

Q(A ) = xy(x + y − 1)(x − y) = 0, having one triple point at the origin O = (0, 0) and three double points, namely ( , ) ( , ) ( 1 , 1 ) : = : = : + − 1 0 , 0 1 and 2 2 , see Fig. 1.3.Let’ssetL1 x 0, L2 y 0, L3 x y 1 = 0 and L4 : x − y = 0. Then by the above formulas, one has b0(M(A )) = 1, b1(M(A )) = 4, and 1.1 Linear Partitions and the Topology of Complements 5

Fig. 1.3 Four lines in the plane L1

L3 L2 O

b2(M(A )) = 3 · 1 + 1 · 2 = 5.

Note that the real complement M(A ) has two bounded components, and one can check that the real function Q : M(A ) → R has one critical point inside each bounded region, and these critical points are non-degenerate Morse singularities. In fact, this property holds in general, see [181], as well as Theorem 2.9 below. The following deﬁnition introduces a convenient way to put together information about the Betti numbers of a topological space and also to relate them to the number of n A Rn n-dimensional regions Rd in the case of a hyperplane arrangement in with d =|A |.

Definition 1.2 The Betti polynomial of a topological space X with only finitely many non-zero Betti numbers is defined as k B(X, t) = bk (X)t . k≥0

The union of the hyperplanes in an afﬁne complex hyperplane arrangement A ,usually denoted by N(A ), also has an interesting topology, usually described in terms of bouquets of spheres, see Remark 4.2. We state now, in the special case n = 2, a fundamental result to be discussed later, just to give a glimpse of the surprising inter- actions present in this beautiful subject. For the n-dimensional version of Zaslavsky’s theorem, see Theorem 2.8. The last claim follows from Remark 4.2.

Theorem 1.3 (Zaslavsky, 1975) For a line arrangement A in the plane R2, one has

2 = ( (A )) + ( (A )) + ( (A )) = ( (A ), ) = + + ( − ), Rd b0 M C b1 M C b2 M C B M C 1 1 d nk k 1 k≥2

2 where M(AC) is the complement in C of the complexiﬁed arrangement AC associated to the real line arrangement A . Moreover, the number b(A ) of bounded regions of the real complement M(A ) of A in R2 coincides with the number of circles in 6 1 Invitation to the Trip the bouquet giving the homotopy type of the complex hypersurface N(AC), and is given by b(A ) = χ(M(A )) = B(M(A ), −1) = 1 − d + nk (k − 1). k≥2

Sometimes we like to have more information on the partition of the plane R2 given by a real line arrangement A . Let us denote as above by nk ,fork ≥ 2, the number of k-fold intersection points in A and by pk ,fork ≥ 2, the number of polygonal regions in the complement M(A ) having k-sides. We assume that d =|A | > 1 and not all the lines in A are parallel. With this notation, the total number of intersection points in A is f0 = nk , (1.1) k≥2 and the total number of regions is f2 = pk . (1.2) k≥2

If we wish to compute the number of (bounded or unbounded) intervals determined by the f0 points on all the lines in A we can proceed in two ways. First, if we look at A L a single line L in , we see that the number of points f0 on this line and the number L L = L + of intervals f1 on this line are related by the formula f1 f0 1. By summing such equalities for all the d lines in A , we get f1 = knk + d. (1.3) k≥2

Indeed, each k-multiple intersection point is counted k times in this sum. On the other hand, each interval occurs as a side for exactly two regions, and hence one has 2 f1 = kpk . (1.4) k≥2

To ﬁnd one relation involving all these numbers, consider a closed disc DR centered at the origin and of a large radius R, such that all the intersection points in A are inside A 1 = ∂ DR and all the lines in are secants, i.e. meet the corresponding circle SR DR in two distinct points. Then the line arrangement A induces a cellular decomposition of the closed disc DR, as follows. The 0-cells correspond to the f0 intersection points in A ,plusthe2d intersection A 1 = + points between the lines in and the circle SR. Hence there are f0 f0 2d cells of dimension 0. The 1-cells correspond to the f1 intervals on the lines in A ,plusthe 1 2d arcs on the circle SR determined by the 2d intersection points between the circle = + and the lines. Hence there are f1 f1 2d cells of dimension 1. Finally, the 2-cells 1.1 Linear Partitions and the Topology of Complements 7

(A ) = correspond to the regions in M , and hence their number is f2 f2. Since the Euler characteristic of a disc is well known, i.e.

χ(DR) = b0(DR) − b1(DR) + b2(DR) = 1, and since it can be computed using any cellular decomposition, we get

− + = . f0 f1 f2 1

This implies − + − = − ( + ) + ( + + + ) − = . 3 3 f0 3 f1 3 f2 3 3 nk 6d knk 3d kpk 4d 3 pk 0 k≥2 k≥2 k≥2 k≥2 In other words, we get d + 3 − n2 − p2 + (k − 3)nk + (k − 3)pk = 0, k≥3 k≥3 which implies the following result.

Theorem 1.4 (afﬁne Sylvester–Gallai property) Let A be a real line arrangement in the afﬁne plane R2 such that d =|A | > 1 and not all the lines in A are parallel. Then n2 + p2 ≥ d + 3.

The classical Sylvester–Gallai property, see [135, 140, 169], as well as [180], pp. 8–9, refers to a line arrangement A in the real projective plane P2. We now describe this result, following [135], which was also our inspiration for the proof of Theorem 1.4 above. Now we assume that the projective real line arrangement A satisfies d =|A | > 2, and nd = 0, i.e. not all the lines in A meet at one point. We recall that the Euler characteristic of the real projective plane P2 is equal to 1: indeed, one can construct P2 using one 0-cell, one 1-cell and one 2-cell. The line arrangement A induces a cellular decomposition of P2 such that one has the following, exactly as above in the affine case. = (i) The number of 0-cells is f0 k≥2 nk , where nk denotes the number of k- multiple points in A ,fork ≥2. = (ii) The number of 2-cells is f2 k≥2 pk , where pk denotes the number of regions in the projective complement M(A ) having k sides. Note that by our assumption pk = 0fork < 3. (iii) The number f1 of 1-cells satisfies the equalities f1 = knk and 2 f1 = kpk . k≥2 k≥2 8 1 Invitation to the Trip

Using the same approach as in the afﬁne case, one gets 3 − n2 + (k − 3)nk + (k − 3)pk = 0, k≥3 k≥3 which implies the following result. Theorem 1.5 (Sylvester–Gallai property) Let A be a line arrangement in the real projective plane P2 such that d =|A | > 2 and not all the lines in A meet in one point. Then the number of double points n2 in A satisﬁes the inequality

n2 ≥ 3.

A very important invariant of the complement of a complex hyperplane arrangement M(A ) is its fundamental group. This group can be quite complicated, and it is very difficult to determine in general. A general approach to such computations is the braid monodromy method, see [47]. We consider the very simple line arrangement from Example 1.2 as an illustration. Example 1.3 Consider again the affine line arrangement A in C2 defined by

Q(A ) = xy(x + y − 1)(x − y) = 0, and recall the notation from Example 1.2. The fundamental group of the complement M(A ) for this arrangement is determined in [216, Example 2.5] using the braid monodromy approach. Here we present a different viewpoint. First we replace the line arrangement A by the line arrangement B in P2, obtained by adding the line at inﬁnity L0 : z = 0. Hence the deﬁning equation for B is

Q(B) = xyz(x + y − z)(x − y) = 0, and the complement of A in C2 coincides with the complement of B in P2.Next we pass again to the afﬁne situation, by discarding this time the line L2 : y = 0. Hence we obtain a new afﬁne arrangement C in C2 with coordinates x, z, given by the equation Q(C ) = xz(x + 1 − z)(x − 1) = 0, obtained from Q(B) by setting y = 1. This arrangement is drawn in Fig. 1.4. This new arrangement has only double points and moreover M(C ) = M(A ) by construction. Now we can use Theorem 4.11 further on in this book and conclude that 2 π1(M(A )) = π1(M(C )) = Z × Z × F2 = Z × F2, where F2 denotes the free group on two generators. This is the same result for π1(M(A )) as that given in [216, Example 2.5]. 1.2 Finite Fields and Cohomology 9

Fig. 1.4 The line arrangement C

1.2 Finite Fields and Cohomology

Now assume that the hyperplanes H j in A have equations j in Z[x], in which case we say that the arrangement A is deﬁned over Z.Letp be a prime number, F = Z/ Z ¯ Fn set p p and consider the hyperplanes H j in p obtained by taking all the coefﬁcients in j modulo p.LetN p(A ) be the number of points in the complement, that is, (A ) =|Fn \∪d ¯ |. N p p j=1 H j

Theorem 1.6 For any hyperplane arrangement A in Rn deﬁned over Z, there is a polynomial PA ∈ Z[t] of degree n such that

(i) PA(p) = N p(A ) for all but ﬁnitely many primes p. ( (A ), ) = n ( −1 ) (A ) (ii) B M t t PA t , where M denotes the corresponding complex n = (− ) hyperplane arrangement complement. In particular, Rd PA 1 . For a proof refer to Proposition 7.5 below. The above result leads to a very interesting computational problem: estimating Betti numbers of arrangement complements by counting lots of N p’s. It also bring into the picture the speciﬁc mixed Hodge structure properties of the complex hyperplane arrangement complements, namely the fact that they are cohomologically Tate, see Theorem 7.7 (i). For more on this topic, see Proposition 7.5, Corollary 3.6 and Exercise 2.6 below, as well as [80] and especially [131]. Now we introduce some basic notations and results about differential forms related to our subject. If U ⊂ Rn is an open set, a k-form ω = ( ) ∧ ... ∧ , aI x dxi1 dxik I

where I ={1 ≤ i1 < i2 < ···< ik ≤ n}, is smooth if all the coefﬁcients aI are smooth functions on U and it is rational if aI = PI /Q I , with PI , Q I polynomials and Q I (x) = 0 for any x ∈ U. A similar deﬁnition applies for holomorphic (resp. regular) differential forms on an open subset U ⊂ Cn (resp. on an algebraic 10 1 Invitation to the Trip variety). Differential forms play a key role in computing topological invariants, as the following classical results show. Theorem 1.7 (de Rham) The cohomology groups H ∗(X, R) of a smooth manifold X can be computed using smooth forms. More precisely, one has

m( , R) = m( 0( ,Ω∗ ), ), H X H H X X d the cohomology groups of the complex of global sections of the de Rham complex of smooth forms on X. Theorem 1.8 (Grothendieck) If X is a smooth afﬁne algebraic variety over C, then H ∗(X, C) can be computed using only regular forms. More precisely, one has

m( , C) = m( 0( ,Ω∗ ), ), H X H H X X d the cohomology groups of the complex of global sections of the de Rham complex of regular forms on X. For more on this, have a look at [74, 77, 127, 128]. The above general results are to be compared to the following very explicit result, for more details see Corollary 3.8. Theorem 1.9 (Arnold–Brieskorn, 1971) Let A be a hyperplane arrangement in Cn such that a hyperplane H ∈ A is deﬁned by a linear equation H = 0. Then the cohomology algebra H ∗(M(A ), C) is isomorphic to the subalgebra of the algebra of all rational differential forms on M(A ) spanned by 1 and the family of 1-forms

dH ωH = H for H ∈ A . Example 1.4 Consider the simplest case when n = 1 and the arrangement A consists of a single hyperplane given by z = 0. Then M(A ) = C∗, and clearly 1( (A ), C) = C · dz ( (A ), C) = C · γ γ :[ , ]→ H M z and H1 M , where the path 0 1 M(A ), γ(t) = exp(2πit), is the standard loop around the origin. By the Cauchy formula, or just using z = exp(2πit) and dz = 2πi exp(2πit),wehave

dz dz 1 γ, = = 2πidt = 2πi. z γ z 0

It follows that the pairing

1 −, − : H1(M(A ), C) × H (M(A ), C) → C is non-degenerate, which ﬁts well with the fact that the cohomology and the homology are dual to each other. A similar computation shows that the complex conjugation 1.2 Finite Fields and Cohomology 11 c : C∗ → C∗, z → z, induces the multiplication by −1onH 1(C∗, C). Indeed, one has dz 1 =− 2πidt =−2πi. γ z 0

This simple example shows a general pattern: describing the cohomology classes in such an explicit way can be useful in many situations, e.g. if a finite group G acts on M(A ) and we want to study the cohomology groups H ∗(M(A ); C) not only as vector spaces, but as G-modules, see Exercise 1.7 below and [148, 150]. We conclude this Introduction by pointing out one of the main open questions in the study of hyperplane arrangements. Let F(A ) be the hypersurface in Cn defined by Q(A ) = 1, and assume that A is a central arrangement, i.e. Q(A ) is a homogeneous polynomial. Then F(A ) is a smooth affine hypersurface, called the Milnor fiber of the arrangement. We would like to know whether the cohomology algebra H ∗(F(A )) is combinatorial, as is the case for H ∗(M(A )), see Corollary 3.7.Even the corresponding question for the Betti numbers is open, and this is so even for central plane arrangements in C3. If d is the number of hyperplanes in A , then there is a monodromy transformation

h : F(A ) → F(A ), given by h(x) = λ · x, with λ = exp(2πi/d). This induces the monodromy operators

h j : H j (F(A ), C) → H j (F(A ), C) in cohomology, and one may ask whether they are determined by the combinatorics of A . The answer is known only in very special cases, see for related results [17–19, 82, 159, 219] and especially [184].

Example 1.5 Let n = 2 and A : Q(x, y) = xy(x − y) = 0. Then Q has an isolated singularity at the origin and the monodromy morphisms h j are easy to describe. More precisely, one has H 0(F(A ), C) = C and h0 is the identity Id, while 1 4 H (F(A ), C) = C and a basis is given by the following regular 1-forms ω1 = α, ω2 = xα, ω3 = yα and ω4 = xyα, with α = ydx − xdy,see[74, Remark (6.2.11) and Example (6.2.13)]. It follows that h1 is semisimple of order 3 and has the following characteristic polynomial, also known as the Alexander polynomial of the arrangement A Δ(t) = det(t · Id − h1) = (t − 1)(t3 − 1).

Moreover, F(A ) is homotopically equivalent to a bouquet of four circles by Milnor’s results [170], and hence H j (F(A ), C) = 0 for any j > 1. Similar results hold for any central line arrangement in C2, see Example 5.1.

By deﬁnition, the monodromy transformation h has order d, and hence any monodromy operator h j is semi-simple and has eigenvalues which are d-th roots of unity. 12 1 Invitation to the Trip

When n ≥ 3 and when we look only at the monodromy operator h1, the only eigenvalues that are observed are roots of unity of order 1, 2, 3 or 4, see [184]. It is an open question whether this is a general property of the monodromy operator h1 for n ≥ 3 and any d, or if it is just a reflection of our partial understanding of the Milnor fibers of hyperplane arrangements. The computation of the Alexander polynomial Δ(t) in general, which is one of our main themes in this book, is closely related to the study of the cohomology groups H ∗(M(A ), L ) of the complement M(A ) with coefficients in a rank one local system L , see for instance Proposition 5.4. The dimension of these twisted cohomology groups can be estimated using the so-called resonance varieties, see for instance Theorem 6.1. The resonance varieties in turn can be computed using the cohomology algebra H ∗(M(A ), C), and they are related to some classical objects in Algebraic Geometry, the pencils of plane curves, see for instance Theorem 6.6. Here is one example of such a pencil.

Example 1.6 Recall first the duality between the projective plane P2, with coordinates (x : y : z), and the dual projective plane Pˇ 2, parametrizing the lines in P2, and having coordinates (u : v : w). More precisely, a point p = (a : b : c) ∈ P2 corresponds to the set in Pˇ 2 parametrizing the lines passing through the point p, i.e. to the line au + bv + cw = 0 in Pˇ 2. Consider now the smooth cubic curve C : f = x3 + y3 + z3 = 0. As for any smooth curve, the inflection points of C are the common solutions of the equations f = 0 and H( f ) = 0, where H( f ) is the Hessian of f , i.e. the determinant of the matrix of second-order partial derivatives of f . Up to a constant factor, in our case H( f ) = xyz. Therefore, there are 3 inflection points for z = 0, namely (1 : α : 0) with α3 =−1. The dual lines corresponding to these 3 inflection points under the above duality are the lines u + αv = 0. Their union is defined by the product of these 3 linear equations, hence by the equation u3 − v3 = 0. By the same argument, we see that the union of the 9 lines in Pˇ 2, dual to the nine inflection points of the cubic curve C above, give rise to a line arrangement A in Pˇ 2 with equation

Q = (u3 − v3)(u3 − w3)(v3 − w3) = 0.

This is a special case of the Ceva arrangement, or monomial arrangement, see 3 3 3 3 Example 5.4. A special feature is that, if we set Q1 = u − v , Q2 = u − w and 3 3 Q3 = v − w , then we have Q = Q1 Q2 Q3 and Q3 = Q2 − Q1. This can be restated by saying that A is obtained by taking the union of all the singular members in the 1 pencil P = sQ1 + tQ2 = 0 of plane cubic curves, where (s : t) ∈ P . Note that this pencil induces a regular mapping

f : M(A ) → P1 \{(1 : 0), (0, 1), (1 : 1)}, given by (u : v : w) → (Q1(u, v, w) : Q2(u, v, w)). 1.2 Finite Fields and Cohomology 13

A large part of our book, namely most of Chaps. 5–8, is devoted to the study of Milnor ﬁbers of central hyperplane arrangements and of their monodromy operators, and this takes the reader into a very active research area with many open questions to be addressed.

1.3 Exercises

Exercise 1.1 In the notation from Theorem 1.1, show that one has k = p = 0 if and 2 2 only if the line arrangement in the real projective plane P = R ∪ L∞ obtained by adding the line at inﬁnity L∞ to the d lines L1, ...Ld has only intersection points of multiplicity 2.

Exercise 1.2 Using Theorem 1.1, give a proof of Theorem 1.2 for n = 2 and describe exactly what “generic” means in this case, i.e. describe the line arrangements for which one has equality in Theorem 1.2.

Exercise 1.3 Compare Zaslavsky’s Theorem 1.3 and Roberts’ formula 1.1. Can you 2 explain directly why the two expressions for the number of regions Rd are equal? Exercise 1.4 With the notation from Theorem 1.3, find a direct elementary proof of 2 − (A ) = the equality Rd b 2d, i.e. without using Theorem 1.3. Exercise 1.5 Consider the line arrangement A in the affine plane K 2 defined by

Q(A ) = xy(x + y − 1)(x − y) = 0.

Compute the number of points N p(A ) for all the prime numbers p. Can you see geometrically why the prime p = 2 is special?

Exercise 1.6 Show that the complement M(A ) of any hyperplane arrangement is an affine hypersurface, i.e. can be described as the zero set of a single polynomial in an affine space K n+1. This shows that the complex complement M(A ) is an affine smooth variety, in particular one may apply Grothendieck’s Theorem 1.8 to it.

Exercise 1.7 Using Theorem 1.9, describe the cohomology of X = M(A ), where A is the line arrangement in C2 given by Q(x, y) = xy = 0. Let the multiplicative group G ={±1} act on X via the formula g · (x, y) = (gx, gy) for g =±1. Show that H ∗(X/G, C) = H ∗(X, C). What happens when we change the G-action by setting (−1) · (x, y) = (y, x)? Hint: Recall that if a ﬁnite group G acts on a topological space X, then the cohomology of the quotient space X/G is given by H j (X/G, C) = H j (X, C)G , the invariant part of H j (X, C) under the induced G-action.

Exercise 1.8 Using Theorem 1.9, describe the cohomology of the complex complement X = M(A ), where A is the line arrangement in C2 given by 14 1 Invitation to the Trip

(i) (ii)

Fig. 1.5 n = 3andm = 2

Q = (x − a1) ···(x − an)(y − b1) ···(y − bm ) = 0, for some integers n > 0, m > 0, and with ai = a j and bi = b j for i = j. See Fig. 1.5 (i) for the case n = 3 and m = 2. When A is a real arrangement, i.e. when ai ∈ R for any i and b j ∈ R for any j, then describe the relation between the bounded regions of the real complement M(A )R and the critical points of the function Q : M(A )R → R. Explain why all these singularities are non-degenerate, i.e. of Morse type.

Exercise 1.9 Using Theorem 4.11 as in Example 1.3, describe the fundamental group of the complex complement X = M(A ), where A is the line arrangement in C2 given by

Q = (x − a1) ···(x − an)(y − b1) ···(y − bm )(x + y − c) = 0, for some integers n > 0, m > 0, and with ai = a j , bi = b j for i = j, and c = ai + b j for any pair (i, j). See Fig. 1.5 (ii) for the case n = 3 and m = 2.

Exercise 1.10 Show that the Ceva line arrangement of 9 lines in P2 described in Example 1.6 cannot be deﬁned, up to a linear change of coordinates, by a product of real linear forms. Chapter 2 Hyperplane Arrangements and Their Combinatorics

Abstract In this chapter we collect the basic deﬁnitions and results involving the intersection lattice of a hyperplane arrangement. Then we explain the key induction technique called deletion-restriction and apply it to deduce the main properties of the characteristic polynomial and of the Poincaré polynomial of an arrangement. These polynomials enter into Zaslavsky’s Theorem expressing the number of regions (resp. bounded regions) of the complement of a real arrangement. In this chapter we also introduce several important classes of hyperplane arrangements: the supersolvable arrangements, the graphic arrangements and the reﬂection arrangements.

2.1 Central, Afﬁne and Projective Arrangements

Definition 2.1 Let K be a field and V be an n-dimensional vector space over K . A hyperplane in V is an affine linear subspace of V of codimension 1. An (affine) arrangement A is a finite collection of hyperplanes in V . The notation (A , V ) is also used in this situation to emphasize the ambient space V . A projective arrangement is a finite set of projective hyperplanes in a given projective space Pn over K .

Deﬁnition 2.2 An arrangement A is called centerless if its center

C(A ) =∩H∈A H is empty. If 0 ∈ C(A ), then A is called central.

In other words, A is central if all the hyperplanes in A pass through the origin of V . An arrangement which is not central (and which cannot be made central by a simple translation in V ) is sometimes called an afﬁne arrangement to emphasize this fact.

Definition 2.3 The defining equation of an arrangement A is the polynomial Q = (A ) ∈ [ ] ( ) = ( ) ∈ A Q K x given by Q x H∈A H x , where each H is given by a linear equation H (x) = 0, after fixing a basis of V , yielding the coordinates x1,...,xn on V . Note that Q(x) is defined up to a unit in K and a linear change of coordinates in Gn(K ). For an arrangement A in V , we define the variety of A

© Springer International Publishing AG 2017 15 A. Dimca, Hyperplane Arrangements, Universitext, DOI 10.1007/978-3-319-56221-6_2 16 2 Hyperplane Arrangements and Their Combinatorics by N(A ) =∪H∈A H and the complement of A by M(A ) = V \ N(A ). Similar deﬁnitions apply to a projective arrangement.

Note that the arrangement A is central if and only if the deﬁning polynomial Q(A ) is homogeneous.

Remark 2.1 Let L be a field extension of K .Let(AK , VK ) be an arrangement with defining polynomial Q(AK ).TheL-extended arrangement is the arrangement in the vector space VL = VK ⊗K L consisting of the hyperplanes AL obtained as the ∼ n zero sets of H = 0inVL = L for H ∈ AK . In particular, the two arrangements have the same defining equations Q(AL ) = Q(AK ). The most common situation of this extension of scalars is when K = R and L = C, and then AL is called the complexification of the real arrangement AK .

Remark 2.2 There is an increasing interest in, and a rapidly growing literature on the so-called toric (or toral) arrangements, i.e. ﬁnite sets of translates of codimension 1 subgroups in an afﬁne torus (C∗)n, see for instance [10, 54, 57, 58, 85, 151, 171, 172]. Note that the obvious exponential map exp : Cn → (C∗)n sends a hyperplane arrangement into a toric arrangement.

Similarly, one can consider an abelian arrangement, namely a ﬁnite set of translates of codimension 1 subgroups in an abelian variety A,see[26]. A special case of this, when A is a product of elliptic curves, is also called an elliptic arrangement, see [153]. Many of the results for hyperplane arrangements discussed in this book have their analogs for toric and abelian arrangements.

Let P(V ) =∼ Pn−1 be the associated projective space of V and recall the usual n n identiﬁcation K = P \ H∞, where H∞ is the (projective) hyperplane at inﬁnity.

Definition 2.4 (i) If A is a central arrangement in the vector space V , then there is an associated (projective) hyperplane arrangement A in the associated projective space P(V ), because each linear equation H = 0 for a hyperplane H ∈ A defines a projective hyperplane H there. Conversely, if B is an arrangement in Pn−1, there is an associated central arrangement Bc in K n, obtained by using the same equations for the hyperplanes in B and in Bc. In Algebraic Geometry, the hypersurface N(A ) (resp. N(Bc)) is called the cone over the hypersurface N(A ) (resp. N(B)). (ii) If A is an affine arrangement in V = K n, then there is an associated projective arrangement A p in Pn, obtained by adding the hyperplane at infinity to the closures of hyperplanes in A . Conversely, if B is a non-empty arrangement in Pn, and H ∈ B, then we can define an affine arrangement (B, H)a by taking the n n traces of the hyperplanes H1 ∈ B, H1 = H on the affine space K = P \ H. Sometimes the simpler notation Ba is used in this context, if the choice of H is clear or not important.

Note that the second construction has already been used in Exercise 1.1. 2.1 Central, Afﬁne and Projective Arrangements 17

Remark 2.3 If A is an afﬁne arrangement in K n, then the composition

A → A p → (A p)c yields a non-empty central arrangement in K n+1, called the cone over A and denoted by cA .IfQ(x1,...,xn) = 0 is a deﬁning equation for A , with d = deg Q, then x x d+1 1 , ··· , n = x0 Q 0 x0 x0 is an equation for cA . When A is central, the above equation takes the simpler form

x0 Q(x1,...,xn) = 0, but one should note that N(cA ) is not the cone over N(A ) in the sense of our discussion in Deﬁnition 2.4 (i) above. Conversely, if B is a non-empty central arrangement in K n+1, and H ∈ B, then the composition

B → B → (B , H )a yields an afﬁne arrangement in K n, called the decone of B (with respect to H) and denoted by dB,see[180], pp. 14–15. It is useful to note that

|dB|=|B|−1.

Example 2.1 The 3-line central arrangement A : Q = xy(x + y) = 0inK 2 corresponds to a projective arrangement A consisting of 3 points in P1. Similarly, starting from the central plane arrangement A : Q = xyz(x + y − z) = 0inK 3,we get by deconing with respect to the plane z = 0 the following afﬁne line arrangement dA = (A )a : xy(x + y − 1) = 0inK 2.

Proposition 2.1 (i) If A is an affine arrangement and A p is the corresponding projective arrangement as in Definition2.4 (ii), then A and A p have the same complement. (ii) Let A be the arrangement in Pn−1 corresponding to the central arrangement A in K n as in Definition2.4 (i). If |A | > 0, then there is an isomorphism of K -algebraic varieties M(A ) → M(A ) × K ∗.

Proof The claim (i) is obvious. To prove the claim (ii), deﬁne the mapping u : ∗ M(A ) → M(A ) × K by u(x) = ([x],H (x)) for some hyperplane H ∈ A and note that the inverse mapping is given by a v([x], a) = x. H (x) 18 2 Hyperplane Arrangements and Their Combinatorics

The upshot of Proposition 2.1 is that, in order to study the complement of an arrangement, we can consider only central (or only projective) arrangements. The following notions are introduced in [180], p. 27 and [183], p. 25.

Deﬁnition 2.5 (i) Let (A1, V1) and (A2, V2) be two arrangements and let V = V1 × V2. Deﬁne the product arrangement (A1 × A2, V ) by

A1 × A2 ={H1 × V2 | H1 ∈ A1}∪{V1 × H2 | H2 ∈ A2}.

(ii) An arrangement A ⊂ V is called reducible if there exist V1, V2, A1 and A2 such that dim Vi > 0fori = 1, 2, V = V1 × V2 and A = A1 ×A2. Otherwise, we say that A is irreducible. (iii) If A is a central arrangement, then A is called decomposable if there exist V1, V2, A1 and A2 such that A = A1 ×A2, where dim Vi > 0fori = 1, 2 and both (A1, V1) and (A2, V2) are non-empty central arrangements. An arrangement which is not decomposable is called indecomposable.

Note that any irreducible arrangement is indecomposable, and any decomposable arrangement is reducible. Now we introduce some examples of central arrangements.

Example 2.2 The Boolean arrangement Bn : Q(x) = x1 ...xn = 0 is just the union n ∗ n of the coordinate hyperplanes in K . Clearly M(Bn) = (K ) .LetVi = K with a coordinate xi and Ai be deﬁned by Q(Ai ) = xi = 0 for all i = 1, ··· , n. It n =×n A =×n A . is clear that K i=1 K and i=1 i Thus, the Boolean arrangement is decomposable.

Remark 2.4 Any hyperplane arrangement in K n can be thought of as a linear section d of a Boolean arrangement in K , where d =|A |. Indeed, if A ={H1,...,Hd } and H j : j (x) = 0for j = 1, ··· , d, consider the map

n d eA : K → K , x → (1(x),...,d (x)).

The map eA is linear when A is central, and is affine when A is affine. Hence the d image EA = im(eA ) is a linear (resp. affine) subspace in K and it is clear that

−1 A = eA (Bd ).

Example 2.3 The braid arrangement is given by Br n : Q(x) = (xi − x j ) = 0. 1≤i< j≤n

n Here M(Br n) ={x ∈ K |xi = x j , ∀i = j} is the configuration space of ordered n-tuples of distinct elements of K . This is also the simplest example of a reflection group arrangement. For more in this direction, see [148, 150, 180] and our discussion in Sect. 2.4 below. Indeed, the hyperplanes in Br n are precisely the reflecting 2.1 Central, Affine and Projective Arrangements 19 hyperplanes of the symmetric group Sn, which is a Coxeter group of type An−1, acting on K n by permutation of the coordinates. This explain why this arrangement is sometimes denoted by An−1. Assume now that char(K ) = 0 and define

n V ={x ∈ K |x1 + x2 +···+xn = 0}

and A to be the trace on V of the braid arrangement Br n.Let

n V ={x ∈ K |x1 = x2 =···= xn},

and A be the empty arrangement in V . It is clear that A × A = Br n, and hence the braid arrangement Br n is indecomposable but reducible. The subscript n − 1 in the notation An−1 is explained by the fact that the linear representation of the symmetric group Sn in the (n − 1)-dimensional space V is irreducible, while that in V = K n is not.

Example 2.4 Let V = R3 and deﬁne a central arrangement A in V by

Q(A ) = xyz(x + y)(x − y)(x + z)(x − z)(y + z)(y − z) = 0.

The nine planes are the symmetry planes of the cube in V with vertices (±1, ±1, ±1). They intersect in lines which are axes of rotational symmetry for this cube. This arrangement is known as the B3-arrangement, since the group of symmetries of the cube is precisely the Coxeter group of type B3. The corresponding decone arrange- 2 ment dB3 lies in R and is deﬁned by the equation obtained from Q by setting z = 1, which gives

Q(dB3) = xy(x − 1)(x + 1)(y − 1)(y + 1)(x − y)(x + y).

The arrangement dB3, unlike the B3-arrangement itself, is easy to picture, see Fig. 2.1, and this is one of the uses of the deconing construction in general. Note, moreover, that both the dB3- and B3-arrangements are irreducible.

Fig. 2.1 The dB3-arrangement 20 2 Hyperplane Arrangements and Their Combinatorics

Example 2.5 Some interesting hyperplane arrangements cannot be deﬁned using only equations with real coefﬁcients. Here is such an example. Let α be a root of the equation t2 + t + 1 = 0. The line arrangement in the complex projective plane P2 given by

A (α) : Q = xyz(y − x)(z − x)(z + αy)(z + α2x + αy)(z − x − α2 y) = 0 is called the MacLane arrangement associated to α. There are two possibilities for α, and the corresponding line arrangements are closely related. For more details, refer to [216, Example10.7] and [13, Example1.7]. Definition 2.6 An affine arrangement A is essential if there is a subarrangement B ⊂ A such that the center C(B) is a point. In particular, |A |≥n for an essential affine arrangement in K n. A projective arrangement A in Pn is essential if the associated central arrangement A c in Kn+1 is essential. This is equivalent to asking n that ∩H∈A H =∅in P .

The Boolean arrangements Bn are essential, while the braid arrangements Br n n are not. Indeed, C(Br n) ={(t,...,t) ∈ K | t ∈ K }=V in the notation from Example 2.3. Remark 2.5 A hyperplane arrangement A in K n is essential if and only if the map n d eA : K → K constructed in Remark2.4 is injective. If this is the case, then there is an obvious identiﬁcation A = Bd |EA , where the arrangement Bd |EA consists of all the intersections H ∩ EA for H ∈ Bd .

Let now A be central but not essential, let c = dim C(A ) and say y1,...yn−c are linear coordinates such that the equations y1 = ...= yn−c = 0 deﬁne C(A ) in V . So, C(A ) ⊂ H if and only if the equation of H is of the type a1 y1 +...+an−c yn−c = 0. n ∼ n−c c n−c Therefore, K = K × K and there is a hyperplane arrangement A1 in K such that A = A1 × Ae, (2.1)

c where Ae is the empty arrangement in K . Moreover, A1 is essential because C(A ) = C(A ) × K c, which implies C(A ) = 0. Deﬁnition 2.7 The integer codim(C(A )) = n − c is called the rank of A and is denoted by rank(A ).

As an example, in the case of a braid arrangement Br n, the rank is n − 1. If n−c n char(K ) = 0, we can take the hyperplane K to be {x ∈ K |x1 + ...+ xn = 0}, which is transversal to the line C(An). For an afﬁne arrangement A which is not essential, we also have a splitting as in (2.1). To see this, consider the central arrangement A0 obtained by taking for each hyperplane H ∈ A the hyperplane H0 which is parallel to H and passes through theoriginofV .IfA is not essential, then A0 is not essential either. If we apply the above discussion to the central arrangement A0, we get a spitting (2.1) which works for the afﬁne arrangement A as well. 2.2 The Intersection Lattice and the Möbius Function 21

2.2 The Intersection Lattice and the Möbius Function

We now introduce the notions used to describe the combinatorics of an arrangement A . We recall that a lattice is a poset (=partially ordered set) such that the joins and meets exist for all pairs of elements, while a semi-lattice is a poset such that only the meets are required to exist for all pairs of elements.

Definition 2.8 Let A be an arrangement in V = K n. (i) A non-empty intersection X of a family of hyperplanes in A is called an edge or a flat of A . Note that V itself is always an edge, the intersection of the empty family of hyperplanes. (ii) The intersection poset of A is the set L(A ) of all the flats X of A . The order is defined by X ≤ Y if and only if Y ⊂ X. (iii) The rank function r : L(A ) → Z is defined by r(X) = n − dim X = codimX. (iv) Two arrangements A and B in K n are said to have the same combinatorics, or to be combinatorially equivalent, if they have isomorphic intersection posets L(A ) and L(B).

Remark 2.6 Let A be an arrangement in V = K n. (i) V is the minimal element in (L(A ), ≤), and it is sometimes denoted by 0.ˆ (ii) The hyperplanes H ∈ A are the atoms, that is 0ˆ < H and there is no ﬂat X in L(A ) such that 0ˆ < X < H. (iii) The meet X ∧ Y = inf(X, Y ) is given by ∩X∪Y ⊂Z∈L(A ) Z (the meet always exists because V ∈ L(A )). (iv) The join X ∨ Y = sup(X, Y ) is given by X ∩ Y (the join of two elements in an afﬁne arrangement may not exist, as this intersection may be empty).

It follows that the intersection lattice of a central arrangement is indeed a lattice. The intersection poset of an afﬁne arrangement is a semi-lattice and may not be a lattice. However, in the sequel, we often use the term lattice for the poset L(A ) in order to simplify the terminology, see for instance Example2.8 below.

Theorem 2.1 For a central arrangement A , the intersection lattice L(A ) is a geometric lattice, i.e. the following properties hold. (i) For any ﬂat X ∈ L(A ), all maximal chains

V = X0 < X1 <...

have the same length p = r(X) = codim(X) (the Jordan–Dedekind condition). (ii) Any ﬂat X ∈ L(A ) is a join of atoms, that is X = H1 ∨ H2 ∨ ...∨ Hq ,for some hyperplanes H j ∈ A . Moreover, one can always choose q = r(X). (iii) For any pair of ﬂats X, Y ∈ L(A ), one has r(X ∧Y )+r(X ∨Y ) ≤ r(X)+r(Y ). 22 2 Hyperplane Arrangements and Their Combinatorics

Proof We give the proof only for the claim (iii). We know that X ∨ Y = X ∩ Y and it is clear that X + Y ⊂ X ∧ Y . The classical relation

dim(X) + dim(Y ) = dim(X + Y ) + dim(X ∩ Y ) from Linear Algebra yields the claimed result.

Example 2.6 Let A be the Boolean arrangement Bn. Then L(A ) is isomorphic to the power set of [n]={1, 2,...,n}, i.e. the set of all subsets of [n], denoted by P([n]), with the order relation being inclusion. The lattice isomorphism : (A ) → P([ ]) =∩k f L n is given as follows: if X j=1 Hi j , for distinct i j ’s, then take f (X) ={i1,...,ik } and note that r(X) = k. A B (A ) Example 2.7 Let be the braid arrangement r n. Then L is isomorphic to the lattice of partitions of [n], denoted by ([n]).IfA, B are two such partitions, then A ≤ B if A is ﬁner than B, that is, any block in B is a union of blocks in A. Deﬁne f : L(A ) → P([n]) by f (X) ={A1,...,Al }∈ ([n]) by

i, j ∈ Ak ⇔∀x ∈ X, xi = x j .

Note that dim X is equal to number of blocks in f (X). The following result is easy to prove. Lemma 2.1 All maximal elements in L(A ) have the same rank, namely rank(A ). This motivates in part the following.

Deﬁnition 2.9 Let L p(A ) ={X ∈ L(A ) | r(X) = p}. The Hasse diagram of L(A ) has vertices labeled by the elements of L(A ) arranged on rows L p(A ) for p ≥ 0, the p-th row having height p. Suppose X ∈ L p(A ) and Y ∈ L p+1(A ).An edge in the Hasse diagram connects X with Y if X < Y. If L is a poset and x < y,wesayy covers x if there does not exist a z such that x < z < y. Hence the Hasse diagram of L shows all the covering relations among the elements in L,see[180] for more on this notion. Example 2.8 Consider the complex line arrangement A in P2 given by

Q(A ) = xyz(x − y)(x − z)(y − z)(x − y − z)(x − y + z).

This arrangement is called the deleted B3-arrangement because it is obtained from the projectivized B3-arrangement by deleting the line x + y − z = 0. This arrangement was introduced by A. Suciu in [217] and will frequently appear in this book. The decone dA , obtained by setting z = 1, is depicted in Fig. 2.2. Let L1 : x = 0, L2 : y = 0, L3 : x = y, L4 : x = 1, L5 : y = 1, L6 : 2 x − y − 1 = 0, L7 : x − y + 1 = 0 be the lines of dA in C . Then the lattice L(dA ) is the set of all non-empty intersections of lines in dA , i.e. 2.2 The Intersection Lattice and the Möbius Function 23

L6 L1 L4

Fig. 2.2 The decone dA of the deleted B3-arrangement

2 L(dA ) ={C , L1,...,L7,(0, 0), (1, 1), (0, ±1), (±1, 0), (1, 2), (2, 1)}.

The Hasse diagram of L(dA ) appears in Fig.2.3.

We introduce now a class of central hyperplane arrangements which enjoy many special properties.

Definition 2.10 Let A be a central hyperplane arrangement in V . (i) A flat X ∈ L(A ) is modular if X + Y ∈ L(A ) for any other flat Y ∈ L(A ). (ii) The arrangement A is supersolvable if the intersection lattice L(A ) has a maximal chain V = X0 < X1 < ···< Xr = C(A )

of modular ﬂats with r = rank(A ) and C(A ) the center of A .

Example 2.9 (i) If Bn is the Boolean arrangement, then one can easily see that any ﬂat in L(Bn) is modular and hence the Boolean arrangement is supersolvable. n (ii) If Br n is the braid arrangement in V = C as in Example2.7, then the following

V = X0 < {x1 = x2} < {x1 = x2 = x3}···< {x1 = x2 =···= xn}=C(A ) 24 2 Hyperplane Arrangements and Their Combinatorics

(0,0)(0,1)(0,−1)(1,0)(−1,0)(1,2)(2,1)(1,1) ••••••••

•••••••L1 L2 L3 L6 L4 L7 L5

• C2

Fig. 2.3 The Hasse diagram of L(dA )

is a maximal chain of modular flats. Hence the braid arrangements are supersolvable. (iii) The plane arrangement in C3 given by xy(x+z)(y+z) = 0 is not supersolvable. Indeed, any flat of rank 2 in L(A ) is not modular. (iv) The deleted B3-arrangement A introduced in Example2.8, but regarded now as a central arrangement in C3, is supersolvable. Indeed, one can get a maximal modular chain by using the line X determined by the point (1 : 1 : 0) ∈ P2, situated on the line at infinity z = 0. More precisely, a maximal chain

X0 < X1 < X2 < X3

3 is given by X0 = C , X1 ={x = y}, X2 = C · (1, 1, 0) and X3 ={0}. Deﬁnition 2.11 The Möbius function μ of a poset L is deﬁned as the (unique) function μ : L × L → Z such that (i) μ(x, x) = 1 for any x ∈ L; 2.2 The Intersection Lattice and the Möbius Function 25 μ( , ) = , ∈ < (ii) x≤z≤y x z 0, for all x y L with x y; (iii) μ(x, y) = 0, for x y. If L has a smallest element 0,ˆ then we set μ(x) = μ(0ˆ, x). Note that for any hyperplane arrangement, i.e. when L = L(A ), one has 0ˆ = V , and hence μ(V ) = 1 and μ(H) =−1. The unicity of such a function μ is obvious. To prove the existence, one can use the following formula: p μ(x, y) = (−1) cp(x, y), p≥0 where cp(x, y) is the number of p-chains x = x0 < x1 <...

Example 2.10 For the decone dA of the deleted B3-arrangement in Example2.8,by using the Hasse diagram given in Fig.2.3, we get μ(Li ) =−1, for all i = 1,...,7 and μ((0, 0)) = μ((0, 1)) = μ((1, 0)) = μ((1, 1)) = 2 and μ((0, −1)) = μ((2, 1)) = μ((−1, 0)) = μ((1, 2)) = 1. Example 2.11 Let A be the Boolean arrangement in K n. We already know the lattice isomorphism L(A ) =∼ P([n]). We show, by induction on |B \ A|, that for any A ≤ B ≤[n] in P([n]),

μ(A, B) = (−1)|B\A|.

It is clear that μ(A, A) = 1 = (−1)0 and if B covers A, then μ(A, A)+μ(A, B) = 0, which implies that μ(A, B) =−1 = (−1)1. For the inductive step, suppose the above formula is true for all pairs (A, B) such that |B \ A| < p, and assume for now | \ |= μ( , ) = B A p. By the property A≤X≤B A X 0, we have p p 1 + (−1)1 + ...+ (−1)p−1 + μ(A, B) = 0 = (1 − 1)p. 1 p − 1

Hence, μ(A, B) = (−1)p. Back to A , μ(X) = μ(V, X) = (−1)r(X) = (−1)codimX . An important property of the Möbius function of arrangements is the following, see [180, Theorem 2.47] for a proof. 26 2 Hyperplane Arrangements and Their Combinatorics

Theorem 2.2 For an arrangement A ,ifX ≤ Y ∈ L(A ), then μ(X, Y ) = 0 and the sign of μ(X, Y ) is equal to (−1)r(Y )−r(X). Now we introduce the main numerical combinatorial invariants associated to an arrangement A . Deﬁnition 2.12 The characteristic polynomial of a hyperplane arrangement A is deﬁned by χ(A , t) = μ(X)tdim X . X∈L(A )

The Poincaré polynomial of a hyperplane arrangement A is deﬁned by π(A , t) = μ(X)(−t)r(X). X∈L(A )

By deﬁnition, if A and B are combinatorially equivalent, it follows that χ(A , t) = χ(B, t) and π(A , t) = π(B, t).

Example 2.12 Using Example2.11, it follows that for the Boolean arrangement A = Bn, one has χ(A , t) = (−1)r(X)tn−r(X) = (t − 1)n X∈L(A) and π(A , t) = (−1)r(X)(−t)r(X) = (t + 1)n. X∈L(A)

Example 2.13 For the decone dA of the deleted B3-arrangement in Example2.8, by using the computations done in Example2.10, we get the following Poincaré polynomial π(dA , t) = 1 + 7t + 12t2 = (1 + 3t)(1 + 4t).

Similarly, for the deleted B3-arrangement A introduced in Example2.8, but regarded now as a central arrangement in C3, one has

π(A , t) = (1 + t)(1 + 3t)(1 + 4t).

It is not easy to determine the Möbius function μ in general, so the polynomials χ(A , t) and π(A , t) may not be computable directly from their deﬁnitions. Here is one example where we can calculate the characteristic polynomial χ(A , t) indirectly, using the Inversion Formula described below.

Theorem 2.3 (Inversion Formula) If L is a poset, μ : L×L → Z the corresponding Möbius function and f : L → Z is any function, then we have two pairs of equivalent statements: 2.2 The Intersection Lattice and the Möbius Function 27

(i) g(x) = f (y) ⇐⇒ f (x) = μ(y, x)g(y). y≤x y≤x

(ii) g(x) = f (y) ⇐⇒ f (x) = μ(y, x)g(y). y≥x y≥x

Proof We prove only one implication of (i) and refer the reader to [180]or[213]for a complete proof. μ(y, x)g(y) = μ(y, x)( f (z)) y≤x y≤x z≤y = f (z)( μ(y, x)) = f (x) + f (z)(0) = f (x). z≤x z≤y≤x z n. Any function u :[n]→H induces a −1 partition of [n] by taking (u (y))y∈H (and ignoring the empty sets in this family). Call this partition ker u. Deﬁne a map f : ([n]) → Z by setting f (π) to be the number of maps u b(π) as above such that ker u = π. Obviously one has g(π) := π ≥π f (π ) = h , where b(π) denotes the number of blocks in the partition π. Hence by the inversion formula,

f (π) = μ(π, π )hb(π ). π ≥π

Take π = 0ˆ ={{1},...,{n}}, and note that f (0ˆ) = h(h − 1)...(h − n + 1) equals the number of injective maps from [n] to H. Therefore,

h(h − 1)...(h − n + 1) = μ(π )hb(π ) = μ(X)hdim X = χ(A , h), π X

where X ∈ L(A ) corresponds to the partition π under the lattice isomorphism described in Example 2.7. Since the above equation is true for all h > n, and χ(A , h) is a polynomial, it follows that

χ(Br n, t) = t(t − 1)...(t − n + 1).

Those readers who ﬁnd the trick in the above example too mysterious should refer to Theorem 2.10, which reveals that this unexpected equality between two apparently unrelated polynomials happens for a large class of hyperplane arrangements, the so-called graphical arrangements. 28 2 Hyperplane Arrangements and Their Combinatorics

Wehave seen that the braid arrangement is supersolvable in Example2.9.Itfollows that the factorization result for χ(A , t) can also be regarded as a consequence of the following general property, see [180, Theorem 2.63].

Theorem 2.4 Let A be a supersolvable arrangement with a maximal chain of mod- = < < ··· < = (A ). =|A \ A | ular ﬂats V X0 X1 Xr C Let bi Xi Xi−1 for i = 1,...,r. Then the following factorization holds

π(A , t) = (1 + b1t)(1 + b2t) ···(1 + br t).

Note that the formula for π(A , t) given in Example2.13 can be seen as a special case of this result. Indeed, with the maximal chain of modular ﬂats given in Example 2.9 (iv), and using the same notation for a line in P2 and the correspond- C3 A \ A ={ } A \ A ={ , , } ing plane in , one has X1 X0 L3 , X2 X1 L0 L6 L7 , where : = A \ A ={ , , , } L0 z 0 is the line at inﬁnity, and X3 X2 L1 L2 L4 L5 . Hence π(A , t) = (1 + t)(1 + 3t)(1 + 4t), exactly as in Example2.13. One can use Exercise2.6 to relate the formula for χ(A , t) given by Example2.14 and the formula for π(A , t) given by Theorem 2.4 when A is the braid arrangement with the maximal modular chain given in Example2.9.

Example 2.15 Consider the following two distinct realizations of the conﬁguration (93),see[216, Examples 10.9 and 10.10]. The ﬁrst one is the Pappus line arrangement 2 (93)1 in P given by

A : = ( − )( − )( − − )( + + )( + − )( − + ) = . 1 Q1 xyz x y y z x y z 2x y z 2x y z 2x 5y z 0

The line at inﬁnity L0 : z = 0 is not drawn, and the other lines are numbered according to the position of the corresponding factor in Q1, see Fig. 2.4. The second 2 one is the non-Pappus line arrangement (93)2 in P given by

A : = ( + )( + )( + )( + + )( + + )( + + ) = . 2 Q2 xyz x y x 3z y z x 2y z x 2y 3z 2x 3y 3z 0

See Fig. 2.5 A A Note that 1 and 2 both consist of 9 lines, and have the same number of double and triple points, namely 9 double points and 9 triple points. It follows that there is a bijection φ : L(A1) → L(A2) between the intersection lattices of the corresponding two central arrangements in C3, such that codimφ(X) = codim(X) and μ(φ(X)) = μ(X), for any X ∈ L(A1). In particular, one has

χ(A1, t) = χ(A2, t).

Note also that each line in both arrangements contains exactly 3 triple points and 2 double points. However, the two lattices L(A1) and L(A2) are not isomorphic. A This can be seen as follows. In the line arrangement 1 we can ﬁnd three lines, for instance L1 : x = 0, L0 : z = 0 and L8 : 2x − 5y + z = 0, such that all the triple points are situated on these three lines. Other choices of such three lines: L2 : y = 0, 2.2 The Intersection Lattice and the Möbius Function 29

Fig. 2.4 The Pappus (93)1-arrangement

Fig. 2.5 The non-Pappus (93)2-arrangement

L5 : x − y − z = 0, L7 : 2x + y − z = 0, or L3 : x − y = 0, L4 : y − z = 0, : + + = A L5 2x y z 0. The line arrangement 2 does not have this property. Hence the arrangements A1 and A2 are not combinatorially equivalent. 30 2 Hyperplane Arrangements and Their Combinatorics

Remark 2.7 One can construct various moduli spaces for hyperplane arrangements. We briefly discuss this issue in this remark in the case of line arrangements in P2 and refer to [7–9, 13, 177, 231] for more information and extended classification results. Let S = C[x, y, z] be the graded polynomial ring in the variables x, y, z with complex coefficients, and Sm be the vector space of degree m homogeneous polynomials d in S. For a fixed degree d,letX(d) be the set of linear forms (1,...,d ) ∈ P(S1) such that i = j for i = j and define A(d) ⊂ P(Sd ) to be the image of the map

d ψ : P(S1) → P(Sd ) (2.2) given by (1,...,d ) → 1 ·2 ·...·d . Then X(d) and A(d) are irreducible algebraic varieties of dimension 2d, and X(d) is in addition smooth. Indeed, X(d) is a Zariski d 2 open subset in the smooth irreducible variety P(S1) .LetG = Aut(P ), and note that G is a connected algebraic group of dimension 8. This group acts naturally on the varieties X(d) and A(d) by coordinate changes. We denote by L (d) the set of all possible intersection lattices L(A ), for line arrangements A in P2 consisting of d distinct lines. For a lattice L ∈ L (d), we denote by X(L) the set of all elements (1,...,d ) ∈ X(d) such that the line arrangement A : 1 = ... = d = 0 has an intersection lattice L(A ) isomorphic to L. One has constructible partitions X(d) =∪L∈L (d) X(L) and A(d) =∪L∈L (d) A(L), where A(L) = ψ(X(L)).A major question is whether a given constructible set X(L) or A(L) is irreducible. The variety X(L) is just the variety of all ordered complex realizations ord(C ) of the ordered combinatorics C ord considered in [13], where C ord is the ordered combinatorial type associated to the lattice L with a ﬁxed numbering of the lines. The quotient X(L)/G is the ordered moduli space M ord(C ) considered in [13]. The variety A(L) is nothing else but the variety of all complex realizations (C ) of the combinatorics C as considered in [13], while A(L)/G is the moduli space M (C ) of the combinatorics C .IfL is the lattice corresponding to the MacLane line arrangement discussed in Example 2.5, it follows from [13, Example1.7] that X(L) is not connected, being the union of two G-orbits, while A(L) is irreducible, being a G-orbit. Let A be a non-empty arrangement in V = K n.

Definition 2.13 For any flat X ∈ L(A ) and any hyperplane H0 ∈ A , we define

(i) AX ={H ∈ A |X ⊂ H}, a subarrangement in A ; (ii) A X ={H ∩ X =∅|H ∈ A , X H}, an arrangement in X;

(iii) A = A \{H0}, the deleted arrangement (with respect to H0); H0 (iv) A = A ={H ∩ H0 =∅|H = H0}, the restricted arrangement in H0, a.k.a. the contraction of A to H0; (v) If A , A and A are hyperplane arrangements as above, then (A , A , A ) is called a triple of arrangements with respect to the distinguished hyperplane H0. Such triples (A , A , A ) are very useful in proofs by induction. Situations where the arrangement A occurs at the same time as the projective arrangement associated 2.2 The Intersection Lattice and the Möbius Function 31 to a central arrangement A , usually also denoted by A in this book, are very rare, and we trust the reader to avoid any confusion. In the rest of this section let A be a real hyperplane arrangement in Rn.Any n connected component of the complement M(A ) = R \∪H∈A H is called a chamber (or a region) of A , and we denote by C (A ) the set of all chambers of the real arrangement A .

Deﬁnition 2.14 The set

X L (A ) =∪X∈L(A )C (A ) partially ordered by P ≤ Q if Q ⊂ P is called the face poset of A . An element P ∈ L (A ) is called a face of A .

Assume that d =|A |, and let j = 0 be the deﬁning equations for the hyperplanes H j ∈ A , where j = 1,...,d.LetJ ={+, −, 0}, with the partial order given by + < 0 and − < 0. Consider J d with the induced partial order and deﬁne a map n d n d σ : R → J by setting, for x ∈ R , σ(x) = (y1,...,yd ) ∈ J , where yk = 0if k (x) = 0, and yk = sign(k (x)) if k (x) = 0.

Deﬁnition 2.15 The set G (A ) = σ(Rn) with the induced order from J d is a poset called the oriented matroid of A .

In fact, σ induces an isomorphism of posets between L (A ) and G (A ), and hence the poset L (A ) can also be regarded as the oriented matroid of A . For more on oriented matroids, see [29] as well as Theorem4.14 further on.

2.3 Deletion-Restriction Theorems and the Number of Regions

Let A be a non-empty arrangement in V = K n.

Deﬁnition 2.16 If K = R,letr(A ) be the number of regions of the arrangement A .IfA is essential, then a region R is bounded if its closure R¯ is compact. If A is not essential, then we have Rn =∼ Rs × Rn−s , where s = rank(A )

Example 2.16 A =∅. Then rank(A ) = n − n = 0 and Rn = 0 × Rn. So, Rn is a relatively bounded region and b(A =∅) = 1.

The following result is essentially linear algebra, and its proof (though a bit tedious) is left as an exercise. See also [213, Lemma2.1] and Exercise3.3 below. 32 2 Hyperplane Arrangements and Their Combinatorics

Proposition 2.2 Let (A , A , A ) be a triple of arrangements. Then the following hold. (i) r(A ) = r(A ) + r(A ). (ii) b(A ) = b(A ) + b(A ),ifrank(A ) = rank(A ). (iii) b(A ) = 0,ifrank(A ) = rank(A ) + 1. The next result gives a useful property of the Möbius function μ of a ﬁnite lattice L. For a proof we refer to [213, Theorem 2.2]. Theorem 2.5 (Cross-Cut Theorem) Let L be a ﬁnite lattice and T a subset of L such that 0ˆ is not in T . Suppose that for any y ∈ L, y = 0ˆ, there is an element x ∈ T such that x ≤ y. Let Nk be the number of k-element subsets of T with join equal to 1ˆ, the maximal element of L. Then one has ˆ ˆ k μ(0, 1) = (−1) Nk . k≥0

Example 2.17 For L = L(A ), take the set T to be just the set of hyperplanes in A . In the case of the Boolean arrangement Bn, we clearly have N j = 0for j = n ˆ ˆ n and Nn = 1. This gives μ(0, 1) = (−1) , which is in accord with our results in Example 2.11.

The main consequence of the above theorem for hyperplane arrangements is the following result. Theorem 2.6 (Whitney’s Theorem) If A is an arrangement in K n, then χ(A , t) = (−1)|B|tn−rank(B), B⊂A ,B central where here B central means C(B) =∅.

Proof Choose z ∈ L(A ) and notice that Lz ={x ∈ L(A )|x ≤ z} is a lattice. Apply the Cross-Cut Theorem to Lz and T = Az ={H ∈ A |z ⊂ H}. Then we have k |B| μ(z) = (−1) Nk = (−1) .

k≥0 B⊂Az ,z=C(B)=∩H∈B H

Since tdim z = tn−rank(B), we ﬁnish the proof by multiplying these two equations side by side and summing over z ∈ L(A ).

Using the above preliminary results, we can now prove the following fundamental result about triples of hyperplane arrangements.

Theorem 2.7 (Deletion-Restriction Theorem) For a triple (A , A , A ) of arrangements with respect to the distinguished hyperplane H0, one has 2.3 Deletion-Restriction Theorems and the Number of Regions 33

χ(A , t) = χ(A , t) − χ(A , t) and π(A , t) = π(A , t) + tχ(A , t).

Proof In view of Exercise2.2 it is enough to establish the ﬁrst equality. By Whitney’s Theorem we have χ(A , t) = (−1)|B|tn−rank(B) + (−1)|B|tn−rank(B).

H0∈/B⊂A ,B central H0∈B⊂A ,B central

Note that the ﬁrst summand is equal to χ(A , t). For the second summand, we ﬁrst show the following.

Lemma 2.2 For a central subarrangement B of A , one has

(−1)|B| = (−1)|B |+1. H H0∈B⊂A ,B central,B =B 0

B ,..., Proof Assume that consists of m hyperplanes H1 Hm and for each such H hyperplane H one has k j =|A |≥1. Then B = B 0 if and only if B is j H j obtained by taking p j ∈[1, k j ] hyperplanes from the set A for j = 1, 2,...,m H j and adding the hyperplane H0. If we ﬁx the multi-index p = (p1,...,pm ), then there are exactly k1 k2 ··· km p1 p2 pm possibilities for such an arrangement B. Moreover, one has |B|=p1 +···+pm +1. It follows that the sum in the Lemma can be rewritten as +···+ + k1 k2 km (−1)p1 pm 1 ··· p p p p 1 2 m ⎛ ⎞ ⎛ ⎞ k1 km + =−⎝ (−1)p1 − 1⎠ ···⎝ (−1)pm − 1⎠ = (−1)m 1. p1 pm p1=0,k1 pm =0,km

Indeed, one clearly has k (−1)p = (1 − 1)k = 0. p p=0,k

To complete the proof of Theorem 2.7, just note that any arrangement B occurring in the above Lemma clearly satisﬁes rank(B) = rank(B ) + 1. 34 2 Hyperplane Arrangements and Their Combinatorics

Remark 2.8 Our proof of Theorem 2.7 is inspired by the proof of the corresponding result, stated as Lemma2.2 in [213]. However, the proof there seems to us to contain an inaccuracy, the fact that there are several arrangements B having the same trace

B on H0 being apparently overlooked. We ﬁrst state and prove the celebrated result due to Zaslavsky, announced in the Introduction in the special case n = 2.

Theorem 2.8 Let A be an arrangement in Rn. Then the following equalitiesr(A ) = (−1)nχ(A , −1) = π(A , 1) and b(A ) = (−1)rank(A )χ(A , 1) hold.

Note that only the intersection lattice L(A ) is used to deﬁne χ(A , t) and π(A , t). So, the numbers r(A ) and b(A ) are determined by the lattice L(A ), a fact not at all obvious.

Proof For any arrangement A , deﬁne a new numerical invariant by the formula s(A ) = (−1)nχ(A , −1). Then it is clear by Theorem2.7 that s(A ) = s(A ) + s(A ). Since we know that r(A ) = r(A )+r(A ) by Proposition2.2, we ﬁnish the proof by induction on |A |. The initial case is as follows: when A =∅, χ(A , t) = tn, hence s(A ) = 1 = r(A ). For the second claim, consider the numerical invariant d(A ) = (−1)rankA χ(A , 1). It is easy to see that b(A ) = 1 = d(A ),forA =∅. To prove the claim by induction using Theorem 2.7, we have to discuss two cases. Firstly, if the rank of A is equal to the rank of A , then the rank of A equals the rank of A minus one and we can repeat the above induction process. Secondly, if rank(A ) = rank(A ) + 1, we know b(A ) = 0. On the other hand, in this case there is a lattice isomorphism L(A ) =∼ L(A ) coming from a splitting A = A × R, implying that χ(A , t) = tχ(A , t). Hence, χ(A , 1) = 0, by Theorem 2.7 and so d(A ) = 0.

If A : Q(x) = 0 is an essential arrangement in Rn, it follows that each bounded region R has a compact closure. Since the real function Q is not constant on R,ithas an extremum on R, so there is at least one singular point of Q inside each bounded region R. Before stating the next result, we recall two basic numerical invariants from Singularity Theory, see [72, 74, 170] for more information.

Deﬁnition 2.17 Let On be the C-algebra of holomorphic function germs at the origin n 0ofC .For f ∈ On, we consider the Jacobian ideal J f spanned by the ﬁrst-order partial derivatives ∂ f/∂x j for j = 1,...,n in On. Then f has an isolated singularity at the origin 0 of Cn if and only if the vector space dimension

On μ( f, 0) = dimC J f is ﬁnite. If this is the case, then μ( f, 0) is also denoted by μ( f ) or μ(X), and it is called the Milnor number of the isolated hypersurface singularity X : f = f (0). Under the same conditions, the vector space dimension 2.3 Deletion-Restriction Theorems and the Number of Regions 35

On τ(f, 0) = dimC ( f − f (0)) + J f is also denoted by τ(f ) or τ(X), and it is called the Tjurina number of the isolated hypersurface singularity X : f = f (0).Here( f − f (0)) is the ideal in On spanned by f − f (0). A real analytic germ g at the origin of Rn can be regarded as the restriction of a unique holomorphic function germ f at the origin 0 of Cn and we deﬁne μ(g) = μ( f ) and τ(g) = τ(f ) when these numbers are ﬁnite.

Example 2.18 We say that a germ f ∈ On deﬁnes a node singularity, or an A1- singularity, or again a Morse non-degenerate singularity, if f can be written in the form = ( ) + 2 + 2 +···+ 2, f f 0 x1 x2 xn after a suitable local holomorphic change of coordinates at the origin of Cn.Itiswell known that f is an A1-singularity if and only if μ( f ) = 1. If g is a real analytic function germ as above, then μ(g) = 1 if and only if

= ( ) ± 2 ± 2 ±···± 2, g g 0 x1 x2 xn after a suitable local real analytic change of coordinates at the origin of Rn,see [72]. Clearly, if g has a minimum (resp. a maximum) at 0, then all the signs are + (resp. −). With this notation, we have following result, to be compared with [181, 225]. Theorem 2.9 Let A : Q(x) = 0 be an essential hyperplane arrangement in Kn, with K = R or K = C. Then the following hold. (i) The function Q : M(A ) → K has only isolated singularities. (ii) The total Milnor number of the function Q, namely μ(Q) = μ(Q, pi ), i

where the sum is over the finite set of singular points pi of Q in M(A ), satisfies the equality μ(Q) = (−1)nχ(A , 1). (iii) When the arrangement A is real, then all the singular points are of Morse nondegenerate type, i.e. A1-singularities. Moreover, each bounded region of the complement M(A ) contains exactly one such singular point, and this point is either a local minimum or a local maximum for Q, while any unbounded region contains no singular point. Proof The first two claims are special cases of [76, Corollary2.2], in view of Corol- lary3.6 below. To prove the last claim, note that the discussion before the theorem shows that μ(Q) ≥ b(A ), while the second claim (ii) tells us that in fact we have an equality μ(Q) = b(A ). This fact implies the claim (iii), since an A1 singularity is characterized by having the Milnor number equal to 1, as recalled in Example2.18. 36 2 Hyperplane Arrangements and Their Combinatorics

Deﬁnition 2.18 Let A be an arrangement in V = K n. Then A is said to be in general position, or to be a generic arrangement, if for any H1,...,Hp ∈ A , one has dim(H1 ∩ ...∩ Hp) = n − p,forp ≤ n and H1 ∩ ...∩ Hp =∅when p > n. Example 2.19 Let A be an arrangement in general position in K n with |A |=d. Then it is easy to see that the intersection lattice L(A ) is isomorphic to a truncation of the intersection lattice L(Bd ) of the Boolean arrangement Bd . With an obvious notation, we have ∼ L(A ) = {S ⊂[d]:|S|≤n}=L(Bd )≤n, where L(Bd )≤n ={X ∈ L(Bd ) : codimX ≤ n}. Then, by Whitney’s Theorem, it follows that d d d χ(A , t) = (−1)|S|tn−|S| = tn − tn−1 + tn−2 − ...+ (−1)n . 1 2 n S⊂[d],|S|≤n

= R (A ) = d + ...+ d In particular, when K ,wehaver 0 n , i.e. we get the equality claim in Schläﬂi’s Theorem1.2. Using the well-known formula − − d = d 1 + d 1 , k k k − 1

(A ) = d−1 we also get b n . The corresponding notion for projective arrangements is discussed next. Deﬁnition 2.19 Let A be an arrangement in Pn = P(K n+1). Then A is said to be in general position, or to be a generic arrangement, if for any H1,...,Hp ∈ A , one has dim(H1 ∩ ...∩ Hp) = n − p for p ≤ n and H1 ∩ ...∩ Hp =∅when p > n. Note that the associated central arrangement B = A c in K n+1 is not in general position when d =|A | > n +1, since 0 ∈ H1 ∩...∩ Hp for any H j ∈ B. However, exactly as above, we have a nice description of the intersection lattice L(B), namely

L(B) =∼ {S ⊂[d]:|S|≤n}∪{1ˆ}, where 1ˆ is the maximal element of the lattice on the right-hand side, and it corresponds to the trivial subspace 0 ⊂ K n+1. It follows that χ(B, t) = (−1)|S|tn+1−|S| + μ(1ˆ). S⊂[d],|S|≤n

In other words, we have d d χ(B, t) = (−1) j tn+1− j + (−1) j , (2.3) j j j=0,n j=n+1,d 2.3 Deletion-Restriction Theorems and the Number of Regions 37 as follows from Theorem 2.5 by taking T to be the set of all hyperplanes. On the other hand, if we pick a hyperplane H ∈ A and we consider the associated affine arrangement B = (A , H)a as in Definition2.4 (ii), then clearly B is an arrangement in general position as in Definition 2.18, see also Exercise2.18.

2.4 Graphic Arrangements and Reﬂection Arrangements

Now we deﬁne the graphic arrangements and list some of their basic properties. For all of these results we refer to [213], Sect. 2.3. Let Γ be a simple graph on the vertex set [n]={1,...,n} for n ≥ 1. Let E(Γ ) denote the set of edges of Γ , i.e. ij ∈ E(Γ ) if and only if the vertices i and j are connected by a (unique) edge in Γ . Here and in the rest of this section we assume i < j.

n Definition 2.20 For any field K , we define the graphic arrangement AΓ in K to be the set of all hyperplanes Hij : xi − x j = 0forij ∈ E(Γ ). For instance, if Γ is the complete graph on [n], i.e. any two vertices in Γ are joined by an edge, then AΓ = Br n, the braid arrangement defined in Example2.3. Definition 2.21 A coloring of a graph Γ on [n] is a function κ :[n]→N.The coloring κ is proper if κ(i) = κ(j) whenever ij ∈ E(Γ ).Ifq ∈ N, q > 0, let χΓ (q) denote the number of proper colorings κ :[n]→[q]. The function χΓ : q → χΓ (q) turns out to be a polynomial for q large enough, and the resulting polynomial is called the chromatic polynomial of the graph Γ and is denoted by χΓ (t). For instance, if Γ is the complete graph on [n], it is easy to see that

χΓ (q) = q(q − 1) ···(q − n + 1).

If we recall Example2.14, the following beautiful result is not so surprising.

Theorem 2.10 For any simple graph Γ , we have the equality χ(AΓ , t) = χΓ (t) in the polynomial ring Z[t]. There is a second numerical invariant of a graph Γ related to the associated graphic arrangement AΓ when K = R. Deﬁnition 2.22 An orientation O of a graph Γ is an assignment of a direction i → j or j → i for any edge ij ∈ E(Γ ). A directed cycle of O is a sequence of vertices i0,...,ik of Γ , with k ≥ 2, such that i0 → i1 → ...→ ik → i0. An orientation O is acyclic if it contains no directed cycles. One has the following interesting result, related to Theorem2.8. Theorem 2.11 For any simple graph Γ with n vertices, the number of acyclic orientations of Γ coincides with the number of regions r(AΓ ) of the associated graphic n arrangement AΓ and hence it is given by (−1) χΓ (−1). 38 2 Hyperplane Arrangements and Their Combinatorics

If Γ is a simple graph on the vertex set [n], an induced subgraph Γ of Γ is a subgraph obtained by fixing a subset of vertices V0 ⊂[n], and deleting all the edges in Γ with vertices outside of V0. An induced cycle of Γ is a cycle which is at the same time an induced subgraph. One says that the graph Γ is chordal if all the minimal induced cycles are triangles. This notion enters into the following beautiful result, which describes which graphic arrangements are supersolvable, see [212]. Theorem 2.12 For any simple graph Γ with n vertices, the associated graphic arrangement AΓ is supersolvable if and only if the graph Γ is chordal. Example 2.20 Let n ≥ 5 and consider the simple graph Γ on the set [n] obtained as follows. Consider a regular polygon P with n − 1 vertices and its symmetry center O. Then the vertices of Γ are the vertices of the regular polygon P plus its center O. The edges in Γ are the n − 1 edges of the regular polygon P plus the n − 1 rays joining the center O to these vertices. The corresponding graphic arrangement AΓ is not supersolvable. Indeed, if we choose V0 ={O}, then the induced subgraph is a minimal cycle of length n − 1 ≥ 4, and hence it is not a triangle. Finally, we discuss reflection arrangements. Let V be a finite-dimensional vector space over C. An element r ∈ GL(V ) is a reflection if it has finite order and it fixes all the points of a hyperplane Hr . Then Hr is called the reflecting hyperplane of r.A finite subgroup G ⊂ GL(V ) is called a (complex) reflection group if it is generated by reflections. When G is obtained as the complexification of a subgroup of GL(V ), where V is a real vector space, then G is called a Coxeter group. A reflection group G ⊂ GL(V ) is reducible if V = V1 ⊕ V2, where the vector subspaces Vi ⊂ V are stable under G. In such a case, the group G induces two reflection subgroups Gi ⊂ GL(Vi ),fori = 1, 2. The group is called irreducible if it is not reducible. Definition 2.23 Let G ⊂ GL(V ) be a reflection group. Then the set A (G) of reflecting hyperplanes of G in V is called the reflection arrangement of G.Inthe special case when G is a Coxeter group, the reflection arrangement A (G) is called a Coxeter arrangement. For the following result, see [180, Corollary6.28]. Theorem 2.13 Let G ⊂ GL(V ) be a reflection group. If X ∈ L(A (G)), then the following hold.

(i) G X ={g ∈ G : g(x) = x for any x ∈ X} is a reflection group. (ii) A (G)X = A (G X ). For a reflection group G we define its rank to be the rank of the corresponding reflection arrangement A (G).

Example 2.21 (The cyclic groups)LetC(r) = μr be the group of r-roots of unity. It acts on C as a reflection group, and the corresponding reflection arrangement A (C(r)) consists of the origin. 2.4 Graphic Arrangements and Reflection Arrangements 39

Example 2.22 (The symmetric groups) It is known that a symmetric group An−1 = Sn is generated by the set of its transpositions (i, j) for 1 ≤ i < j ≤ n. When n Sn acts on C by permuting the coordinates, as in Example2.3, the corresponding linear transformation r(i, j) is a reflection, since r(i, j)2 = Id and r(i, j) fixes the hyperplane xi = x j .ItfollowsthatAn−1 = Sn is a reflection group (in fact a Coxeter group since the permutation action already exists on Rn)ofrankn − 1, and the corresponding Coxeter arrangement is the braid arrangement Br n. Example 2.23 (The full monomial groups) We define the full monomial group G = n G(r, 1, n) for r, n ≥ 2 as the set of all linear transformations g(σ, θ) of C for σ ∈ Sn n and θ ∈ (μr ) , with μr the group of r-roots of unity. Here g(σ, θ) acts on the standard n basis e1,...,en of C by g(σ, θ)(ei ) = θi eσ(i), where θ = (θ1,...,θn).Itiseasy to check that the transformation g((ij), θ0) is a reflection, for any transposition (ij), ( j) ( j) and θ0 = (1, 1,...,1). The transformation g(Id,θ ) is also a reflection, where θ θ ( j) = = has all the components i 1fori j. These two types of reflections generate the group G. By imposing the condition that the fixed point set of g(σ, θ) is a hyperplane, one can see that the reflection arrangement associated to the group G = G(r, 1, n) is given by the following equation in Cn: ··· ( r − r ) = . x1x2 xn xi x j 0 1≤i< j≤n

This hyperplane arrangement is called the full monomial arrangement and is usually denoted by A (r, 1, n). Note that the Coxeter groups of type Bn = Cn coincide with G(2, 1, n). The corresponding arrangement for B3 = G(2, 1, 3) was discussed in Example 2.4.

Example 2.24 (The monomial groups) Now we introduce the monomial group G = G(r, p, n) for r, n ≥ 2 and p a divisor of r as the set of all linear transformations g(σ, θ) ∈ G(r, 1, n) such that θ1 ···θn is a power of exp(2πip/r). It is easy to check that G = G(r, p, n) is a reﬂection group such that A (G(r, p, n)) = A (G(r, 1, n)) is the full monomial arrangement when p < r. However, for p = r, we get a new arrangement given by ( r − r ) = . xi x j 0 1≤i< j≤n

This hyperplane arrangement is called the monomial arrangement and is usually denoted by A (r, r, n). The Coxeter groups of type Dn coincide with the groups G(2, 2, n), while the Coxeter groups I2(m) are the groups G(m, m, 2). Remark 2.9 The classification of finite, irreducible, complex reflection groups is due to Shephard and Todd and is described in detail in [180, Chapter 6] or [152]. This classification consists of three infinite families and 34 exceptional groups. The three infinite families are the cyclic groups C(r), the symmetric groups An and the monomial groups G(r, p, n) described above. The exceptional groups are usually denoted by G j for 4 ≤ j ≤ 37. More precisely, G j has rank 2 for 4 ≤ j ≤ 22, 40 2 Hyperplane Arrangements and Their Combinatorics rank 3 for 23 ≤ j ≤ 27, rank 4 for 28 ≤ j ≤ 32, rank(G33) = 5, rank(G34) = rank(G35) = 6, rank(G36) = 7 and rank(G37) = 8. The Coxeter groups of type E6, E7, E8 correspond to G35, G36, G37. The reflection arrangement A (G25) is the Hessian arrangement, see Exercise5.5. Note that the intersection lattices for the reflection arrangements are described in [180, Chapter 6 and Appendix C], essentially using Theorem2.13 above.

2.5 Exercises

Exercise 2.1 Let V be an n-dimensional vector space over the ﬁnite ﬁeld Fq with q elements. Let A be the central arrangement in V consisting of all the hyperplanes passing through the origin. Show that

|A |=1 + q + q2 +···+qn−1.

Exercise 2.2 Show that a central essential arrangement is irreducible if and only if it is indecomposable.

Exercise 2.3 Prove the ﬁrst claim in Remark2.5.

Exercise 2.4 Prove the last two claims in Remark2.6.

Exercise 2.5 Show that for any hyperplane arrangement A , the Poincaré polyno- π(A , ) j mial t can be written as a sum j=0,rank(A ) p j t , with the coefﬁcients p j strictly positive integers for any j with 0 ≤ j ≤ rank(A ).

Exercise 2.6 For any hyperplane arrangement A in K n, show that the characteristic polynomial and the Poincaré polynomial determine each other via the formulas

π(A , t) = (−t)nχ(A , −t−1) and χ(A , t) = tnπ(A , −t−1).

Exercise 2.7 Let A be an arrangement in V = K n. Show that rank(A ) = n − c if and only if tc divides χ(A , t) and tc+1 does not divide it.

Exercise 2.8 Let A be a line arrangement in the projective plane P2 and d =|A |.

Assume that there are nk points of multiplicity k in A , i.e. points where exactly k lines in A meet, for k ≥ 2. Prove the following equality k d n = . k 2 2 k≥2 2.5 Exercises 41

Exercise 2.9 Let A be a line arrangement in the afﬁne plane V = K 2. Assume that there are nk points of multiplicity k in A ,fork ≥ 2. (i) Determine the characteristic polynomial χ(A , t) and the Poincaré polynomial π(A , t) in terms of the integers nk . (ii) Explain the relation with Roberts’ formula Theorem 1.1 giving the number of regions for the complement of A when K = R.

Exercise 2.10 Let A be a line arrangement in the plane V = C2. Determine all the possibilities for the poset L(A ) when A consists of d lines, 1 ≤ d ≤ 4. For each possible poset L(A ), produce a deﬁning equation Q = 0 corresponding to this poset.

Exercise 2.11 Show that the two line arrangements in the projective plane V = P2 given by A = A (2, 2, 3) : (x2 − y2)(x2 − z2)(y2 − z2) = 0 and B : xyz(x − y)(x − z)(y − z) = 0 differ by a linear transformation in G3(R). One can represent this arrangement by a complete quadrangle as in Fig.2.6.

4 Exercise 2.12 The braid arrangement Br 4 in C has 6 hyperplanes and it is not B C3 essential. But there is an essential, central arrangement r 3 in , obtained by taking the intersection with a transversal hyperplane as in the formula (2.1). Show B that this arrangement r 3 may be given by the following equation

A : ( 2 − 2)( 2 − 2)( 2 − 2) = . 3 x y y z z x 0

Fig. 2.6 The A (2, 2, 3) arrangement

•

•• 42 2 Hyperplane Arrangements and Their Combinatorics

In particular, the two reﬂection arrangements Br 4 and A (2, 2, 3) essentially coincide.

n Exercise 2.13 Let Br n be the braid arrangement in R . Show that r(A ) = n! in two distinct ways: ﬁrst by using the formula from Example2.14 and secondly, by using the natural action of the symmetric group Sn on the complement M(Br n).

Exercise 2.14 Let A be a central arrangement in Rn such that d =|A | > 0. Show that r(A ) ≥ 2r where r = rank(A ). Moreover, show that r(A ) is even and b(A ) = 0.

n Exercise 2.15 (i) Let B be an arrangement in R , with the coordinates x1,...,xn. Consider the arrangement B˜ in Rn+1 obtained by taking for each hyperplane H ˜ n+1 in B the hyperplane H in R , with coordinates x1,...,xn, xn+1, deﬁned by the same equation as H and by adding the hyperplane Hn+1 : xn+1 = 0. Show that b(B˜) = 0. (ii) Let (A , A , A ) be a triple of real arrangements with respect to the distin- n+1 guished hyperplane H0, and A an arrangement in R . Then show that

rank(A ) = rank(A ) + 1

if and only if up to a linear change of coordinates in Rn+1, one has the identiﬁ- ˜ n cations A = B and H0 = Hn+1 for some arrangement B in R .

Exercise 2.16 Show that a general position arrangement A in K n is essential (resp. indecomposable) if and only if d =|A |≥n (resp. d =|A |≥n + 1).

Exercise 2.17 Let A be a central arrangement in Kn with d =|A |. Show that if A is in general position, then d ≤ n. Moreover, if A is in general position and essential, n then up to a linear change of coordinates A = Bn, the Boolean arrangement in K .

Exercise 2.18 Show that an afﬁne arrangement A in K n is in general position if and only if the associated projective arrangement A p in Pn, obtained by adding the hyperplane at inﬁnity, is in general position.

Exercise 2.19 Show that an arrangement A in Pn such that |A | > n is in general position if and only if any n +1 hyperplanes in A have an empty intersection. Let Sn be the vector space of homogeneous polynomials of degree n in x and y. Any point 1 n p ∈ P determines a hyperplane Hp in P = P(Sn), by taking all the polynomials 1 f ∈ Sn such that f (p) = 0. Show that a family of distinct points pi ∈ P gives rise Pn = P( ) to a general position arrangement in Hpi in Sn .

n Exercise 2.20 Let A be an arrangement in K .LetP(S≤1) be the projective space associated to the vector space S≤1 of polynomials in x1,...,xn of degree at most 1. For each hyperplane H j in A , we denote by δ(H j ) ∈ P(S≤1) the class of the deﬁning equation j for H j . Show that the arrangement A is in general position if and only if the following condition holds. 2.5 Exercises 43

( ) , ··· , A ≥ δ( ) C : for any set H j1 H jk of k hyperplanes in , with k 1, the points H j1 , ··· δ( ) = ( − , ) P( ) , H jk span a linear subspace E of dimension e min k 1 n in S≤1 , such that [1] ∈/ E for e < n.Here[1] is the class in P(S≤1) of the constant polynomial equal to 1.

Exercise 2.21 Let A be an affine essential arrangement in Kn with d =|A |. Show d that A is in general position if and only if the affine subspace EA ⊂ K defined in Remark2.4 is transversal to the Boolean arrangement Bd in the following sense: d for any X ∈ L(Bd ), the codimension of X in K coincides with the codimension of X ∩ EA in EA , where by definition codim(∅) =−1. Chapter 3 Orlik–Solomon Algebras and de Rham Cohomology

Abstract In this chapter we give the purely combinatorial deﬁnition of the Orlik– Solomon algebra of a hyperplane arrangement. A fundamental result says that this algebra is isomorphic to the cohomology algebra of the complex hyperplane arrangement complement. This is the ﬁrst instance of a recurring theme which says that the topology is often determined by the combinatorics. A tensor product decomposition of the Orlik–Solomon algebra of a supersolvable arrangement, as well as an alternative view of the Orlik–Solomon algebra of a projective hyperplane arrangement, can also be found here.

3.1 Orlik–Solomon Algebras for Hyperplane Arrangements

Let A be a central hyperplane arrangement in V = K n and R be a unitary commutative ring. If A =∅,wesetE0 = E0(A ) = R and E p = E p(A ) = 0 for any integer p > 0. Suppose now that A ={H1,...,Hd },ford ≥ 1, and associate to 1 1 each hyperplane H j a symbol e j . Denote by E = E (A ) the free R-module with the base {e j } j=1,...,d . In other words, ⎧ ⎫ ⎨d ⎬ 1 = | ∈ . E ⎩ c j e j c j R⎭ j=1

Let E∗ = E∗(A ) =∧(E1(A )) be the exterior algebra of E1, which is equal to

E0(A ) ⊕ E1(A ) ⊕···⊕Ed (A ),

0 p with E = R and, for p > 0, E ={ cI eI | cI ∈ R},afreeR-module with the = ∧···∧ = ( ,..., ) ≤ basis eI ei1 ei p , where I i1 i p is a multi-index satisfying 1 i1 < ···< i p ≤ d. We will also use the notation

= ... = . eI ei1 ei p ei1...i p

Note that E∗ is a graded commutative R-algebra, namely αβ = (−1)|α||β|βα, where |γ | denotes the degree of a homogeneous element γ ∈ E∗. Deﬁne an R-linear deriva- p p−1 tion ∂ : E → E by ∂(1) = 0, ∂ei = 1, for i = 1,...,d, and

p ∂ = (− )k−1 ...... , ei1...i p 1 ei1 eik ei p (3.1) k=1 ∂ ∂∂ = where eik means that the element eik is deleted. Notice that satisﬁes 0 and ∂(αβ) = ∂(α)β + (−1)|α|α∂(β). Hence one can deﬁne the following complex of free R-modules ∂ ∂ ∂ 0 → Ed → Ed−1 →···→ E0 → 0. (3.2)

Proposition 3.1 The complex (3.2) is acyclic if |A | =∅.

Proof This complex is exactly the Koszul complex K (1,...,1) of the sequence (1,...,1) in R, where 1 occurs d-times. Note that K (1,...,1) = K (1)⊗d , where

Id K (1) : 0 → R → R → 0.

So, H ∗(K (1)) = 0 and we ﬁnish the proof with the aid of the Künneth formula, see [130, 211].

Everything up to here only depends on d =|A |, it has nothing to do with the combinatorics contained in the intersection lattice L(A ). Now we bring this lattice = ( ,..., ) A ≤ < into play. If S Hi1 Hi p is an ordered set of hyperplanes in ,for1 i1 ···< ≤ = ∈ ∗(A ) i p d,leteS ei1...i p E be the corresponding element and deﬁne the ∩ = ∩···∩ = =∅ ∩∅ = associated ﬂat S Hi1 Hi p . When p 0, then we set S , V and e∅ = 1.

Deﬁnition 3.1 The set S of hyperplanes in A is said to be dependent if

codim(∩S)<|S|.

Note that codim(∩S) ≤|S| in general, and equality holds if and only if the defining equations of the hyperplanes in S are linearly independent as linear forms. ∗ Definition 3.2 The ideal I = I (A ) in E (A ) is the ideal generated by ∂eS, for all dependent sets S.IfA =∅,wesetI = I (A ) = 0. =⊕d = ∩ k (A ) The ideal I is homogeneous, i.e., I k=0 Ik , where Ik I E . Definition 3.3 The Orlik–Solomon algebra of the central arrangement A is the graded commutative quotient R-algebra A∗(A ) = E∗(A )/I (A ). 3.1 Orlik–Solomon Algebras for Hyperplane Arrangements 47

When we want to emphasize the dependence on the ring R, we write A∗(A , R). It follows that A∗(A ) is a graded commutative algebra, generated as an algebra by ∗ the classes ai =[ei ], and A (A ) only depends on the intersection lattice L(A ).We ∗ write aS =[eS]∈A (A ). Lemma 3.1 Ak (A ) = 0 if k > dork> n.

Proof The ﬁrst claim is obviously from the deﬁnition. For the second, take eS ∈ k E (A ) with |S| > n. Then, S is dependent and ∂eS ∈ I .NowifH ∈ S, then eH eS = 0. By applying the derivation ∂,wehave

0 = ∂(eH eS) = ∂eH eS − eH ∂eS = eS − eH ∂eS,

k which implies that eS ∈ I . It follows that Ik = E (A ) for k > n, which yields the second claim. Example 3.1 For any central arrangement A we have Ai (A ) =∼ Ei (A ) for i = 0, 1. Indeed, I0 = I1 = 0, since a dependent set S satisﬁes |S|≥3. Example 3.2 Consider the central line arrangement A in K 2 consisting of d lines through the origin. Then the set S = (Hi , H j , Hd ) is dependent for any indices i < j < d, and hence ∂eS = e j ed − ei ed + ei e j ∈ I.

2 This implies aij = aid − a jd, and hence the R-module A (A ) is generated by the (d − 1) elements aid for 1 ≤ i < d. It is left as an exercise to check that these elements aid are indeed linearly independent. As a conclusion, we have in this case,

2 ∼ d−1 A (A ) = R < ai ad | 1 ≤ i ≤ d − 1 >= R .

Note that, as in Proposition 2.1, in the case K = C, we have for the complements

M(A ) = M(A ) × C∗ = C \{d − 1 points}×C∗ =∼ ∨d−1 S1 × S1, where A is the corresponding projective arrangement. Using the Künneth formula, we see that the corresponding Betti numbers are b0(M(A )) = 1, b1(M(A )) = d, k and b2(M(A )) = d − 1, which agrees with the rank of A (A ),fork = 0, 1, 2 respectively. Example 3.3 Let A be the Boolean arrangement in K n. Since there is no dependent family S in this case, A∗(A ) =∼ E∗(A ) is a free exterior R-algebra on n generators. On the topological side, if K = C, then M(A ) = (C∗)n =∼ (S1)n and hence

∗ H (M(A ), R) =∧R(x1,...,xn) is also a free exterior R-algebra on n generators. Hence in this case it is clear that the Orlik–Solomon algebra A∗(A ) agrees with the algebra structure of the cohomology ring of M(A ) with R-coefﬁcients. 48 3 Orlik–Solomon Algebras and de Rham Cohomology

Fig. 3.1 A simple afﬁne line arrangement

We have defined the Orlik–Solomon algebra for central arrangements. Now we define it for any affine arrangement in V = K n.

Deﬁnition 3.4 Let A be an afﬁne arrangement in V = K n. The corresponding Orlik–Solomon algebra A∗(A ) is the quotient E∗/I , where the homogeneous ideal I in E∗ is generated by the following two types of elements:

(i) all the elements ∂eS for sets S with ∩S =∅and codim(∩S)<|S|; (ii) all the elements eS for sets S with ∩S =∅.

Notice that Lemma 3.1 applies to the Orlik–Solomon algebra A∗(A ) of an afﬁne arrangement as well, with exactly the same proof.

Example 3.4 Consider the afﬁne line arrangement

H1 : x = 0, H2 : x = 1, H3 : y = 0, H4 : y = 1,

represented in Fig. 3.1 below. There is no contribution to the ideal I from the elements of type (i), since the arrangement has only double points. It follows that I =< e1e2, e3e4 >, correspond- 0 ∼ ing to the parallel lines H1||H2 and H3||H4. Hence we get the following: A = R, 1 ∼ 4 2 ∼ 4 A = R , and A =< e13, e14, e23, e24 >= R . Notice that here

M(A ) = (C \{0, 1}) × (C \{0, 1}) =∼ (S1 ∨ S1) × (S1 ∨ S1).

k So, the Betti numbers bk (M(A )) match with the rank of A for any integer k and the two algebra structures also match each other. Indeed, if we write

H ∗(C \{0, 1}, R) = R · 1 + R · u + R · v,

with u, v a basis for H 1(C \{0, 1}, R), then the cup product u ∪ v is trivial, since H 2(C \{0, 1}, R) = 0. Recall also Exercise 1.8.

The following is one of the main theorems in the theory of hyperplane arrangements, which is extremely important for proofs by induction. Its proof is long and technical, see [180, Theorem 3.65]. 3.1 Orlik–Solomon Algebras for Hyperplane Arrangements 49

Theorem 3.1 Let (A , A , A ) be a triple of hyperplane arrangements with distinguished hyperplane H0 ∈ A . Then there are exact sequences for all p ≥ 0,

i j 0 → A p(A ) → A p(A ) → A p−1(A ) → 0

( ) = ( ) = ∈/ ( ) = where i a a ,ja 0,ifH S and j a ... − a ∩ ... ∩ ,if S S S 0 0i1 i p 1 H0 Hi1 H0 Hi p−1 H0 ∈ S.

∩ ... ∩ = Here and in the sequel, by deﬁnition, we set aH0 Hi H0 Hi − 0 if two intersections ∩ ∩ = 1 p 1 H0 Hia and H0 Hib coincide for some a b. Corollary 3.1 For any integer p and any arrangement A ,Ap(A ) is a free R-module of ﬁnite type.

Proof We prove this by induction on |A |.IfA =∅, then A∗(A ) = R (in degree zero), and hence the claim holds. Notice that if

0 → M → M → M → 0 is an exact sequence of R-modules, with M and M free, then M = M ⊕ M, and hence M is also free.

Corollary 3.2 Consider the Poincaré polynomial of the graded algebra A∗(A ) given by ∗ k k P(A (A ), t) = rank R(A (A ))t . k≥0

Then, for any afﬁne hyperplane arrangement A , one has P(A∗(A ), t) = π(A , t).

Proof Theorem 3.1 implies that P(A∗(A ), t) = P(A∗(A ), t) + tP(A∗(A ), t). Recall that by Theorem 2.7, the Poincaré polynomial satisﬁes

π(A , t) = π(A , t) + tπ(A , t).

Hence, by induction on |A |, it is enough the check the claim for the case A =∅,in which case both polynomials are equal to 1.

Example 3.5 Consider the decone dA of the deleted B3-arrangement deﬁned in Example 2.8. Then ( A ) =⊕2 , A d i=0 Ai

= =⊕7 , , where A0 R, A1 j=1 Raj where a j corresponds to the line L j for all j, and A2 = Ra12 ⊕ Ra13 ⊕ Ra15 ⊕ Ra16 ⊕ Ra17 ⊕ Ra24 ⊕ Ra26 ⊕ Ra27 ⊕ Ra34 ⊕ Ra35 ⊕ Ra47 ⊕ Ra56. Here ai ∧ a j = ai a j = aij, aii = 0 and aij =−a ji, for all i, j. Moreover, a1a4 = a2a5 = a3a6 = a3a7 = 0, since the corresponding pairs of lines do not intersect. Each of the four triple points gives rise to a relation, 50 3 Orlik–Solomon Algebras and de Rham Cohomology

Table 3.1 The product A1(dA ) × A1(dA ) → A2(dA )

a1 a2 a3 a4 a5 a6 a7 a1 0 a12 a13 0 a15 a16 a17 a2 −a12 0 a13 − a12 a24 0 a26 a27 a3 −a13 a12 − a13 0 a34 a35 0 0 a4 0 −a24 −a34 0 a35 − a34 a26 − a24 a47 a5 −a15 0 −a35 a34 − a35 0 a56 a17 − a15 a6 −a16 −a26 0 a24 − a26 −a56 0 0 a7 −a17 −a27 0 −a47 a15 − a17 0 0

namely we get a2a3 =−a12 + a13, a5a7 =−a15 + a17, a4a5 =−a34 + a35, a4a6 = −a24 + a26. The product

A1(dA ) × A1(dA ) → A2(dA ) is given in Table3.1. This determines completely the R-algebra structure of A∗(dA ). Note that rank(A2) = 12 as computed using Example 2.13 and Corollary 3.2. Corollary 3.3 If A is a central arrangement in K n, then the rank of Ar (A ) is given by (−1)r μ(1ˆ)>0, where r = rank(A ). Moreover, for any k, and any arrangement A , one has k k rank R A (A ) = rank R A (AX ),

X∈Lk (A ) where Lk (A ) is the set of ﬂats of codimension k in L(A ) and AX is the subarrangement of A determined by X, as in Deﬁnition 2.13 (i).

Proof The first claim follows by comparing the coefficients of tn in the polynomials P(A∗(A ), t) and π(A , t). For the second claim, note that the comparison of the coefficients of tk in the two polynomials in Corollary 3.2 yields k k rank R A (A ) = (−1) μ(X). (3.3)

X∈Lk (A )

Then apply the ﬁrst claim to the central arrangements AX . The corollary above has the following more precise version, see [180, Corollary 3.73]. Theorem 3.2 (Brieskorn decomposition) The natural morphism of R-modules

⊕ k (A ) → k (A ) X∈Lk (A ) A X A is an isomorphism. 3.1 Orlik–Solomon Algebras for Hyperplane Arrangements 51

=[ ]∈ k (A ) [ ]∈ Note that under the above morphism, aS ei1...ik A X is mapped to ei1...ik Ak (A ), but the equivalence relations are not the same on both sides. The following special case of Corollary 3.3 can be compared with Proposition 1.1. A 2 Corollary 3.4 Suppose is a line arrangement in K with d lines and nk points ≥ 2(A ) = ( − ) with multiplicity k, for k 2. Then rank R A k≥2 nk k 1 .

2(A ) = 2(A ) = ( ( ) − ) Proof One has rank R A p rank R A p p mult p 1 , where the sum is over all the multiple points p in A , the ﬁrst equality follows by Corollary 3.3 and the second follows from Example 3.2. In the case of supersolvable arrangements, we have the following decomposition theorem for the Orlik–Solomon algebra, see [180, Theorem 3.81]. Theorem 3.3 Let A be a supersolvable arrangement with a maximal chain of = < < ···< = (A ). B = A \ A modular ﬂats V X0 X1 Xr C Let i Xi Xi−1 and let 1 1 Bi ⊂ A (A ) be the R-linear span of the elements aH ∈ A (A ) for the hyperplanes H ∈ Bi for i = 1,...,r. Then the R-linear map

(R + B1) ⊗···⊗(R + Br ) → A(A ) induced by the multiplication in A(A ) is an isomorphism of graded R-modules. In particular, π(A , t) = (1 + b1t)(1 + b2t) ···(1 + br t), where bi =|Bi | for i = 1,...,r. Note that the final factorization result, which is just our previous Theorem 2.4, follows from the first claim in the above theorem, in view of Corollary 3.2. Finally, we define the Orlik–Solomon algebra for projective arrangements. A projective arrangement A in Pn−1 corresponds to a central arrangement A = n {H0,...,Hd } in K . Note that now |A |=d + 1. We already know how to define A∗(A ), namely A∗(A ) = E∗/I , with I = I (A ) as above. Lemma 3.2 The R-derivation ∂ : E∗ → E∗ introduced in (3.1) satisfies ∂(I ) ⊂ I, hence it induces an R-derivation ∂ : A∗(A ) → A∗(A ). Proof This is a direct consequence of ∂2 = 0 and of the form of the generators for the ideal I . Note that when A is not central, the above result is no longer true. One can see this for the line arrangement in Example 3.4. Using this new derivation, we have the following analog of Proposition 3.1. Proposition 3.2 For any non-empty central arrangement A , the complex of free R-modules ∂ ∂ ∂ 0 → An(A ) → An−1(A ) → ...→ A0(A ) → 0 is acyclic.

k Proof If a ∈ A (A ) satisﬁes ∂a = 0, then ∂[e1a]=a. 52 3 Orlik–Solomon Algebras and de Rham Cohomology

The projective arrangement A in Pn−1 has the same complement as the afﬁne a arrangement A0 = (A , H0) , namely

(A ) = Pn−1 \∪d = (Pn−1 \ ) \ (∪d \ ). M j=0 H j H0 j=1 H j H0

∗ Note that for the affine arrangement A0, the Orlik–Solomon algebra A (A0) has already been defined. But in this approach there is a choice of the hyperplane at infinity H0. We would like to have a description of the Orlik–Solomon algebra of A without such a choice. Define the following R-subalgebra in A∗(A )

K ∗(A ) = ker(∂ : A∗(A ) → A∗(A )).

Theorem 3.4 If R is a ﬁeld or a principal ideal domain, then there is an isomorphism of R-algebras ∗ ∼ ∗ A (A0) = K (A ).

Proof Since R is a ﬁeld or a PID, it follows that each component K p(A ) is a free R-module of ﬁnite type, since it is a submodule of the free module A p(A ),see[147]. Proposition 3.2 then yields the following equality of Poincaré polynomials

P(A∗(A ), t) = P(K ∗(A ), t) · (1 + t).

We leave it as an exercise for the reader to show that we also have the similar equality

∗ ∗ P(A (A ), t) = P(A (A0), t)(1 + t).

p p We conclude that the components K (A ) and A (A0) are free modules of the same ∗ rank for any p. Hence it is enough to construct a surjective morphism f : A (A0) → K ∗(A ) of graded algebras. To construct f , ﬁrst note that K ∗(A ) is generated as = − = , ,..., an algebra by the elements a j a j a0, with j 1 2 d. To see this, use Proposition 3.2 and the obvious equality

∂(a a ···a ) = a ···a . 0 j1 jp j1 jp

∗ ∗ It follows that there is a surjective algebra morphism g : E (A0) → K (A ) sending e j to a j . It remains to show that the morphism g is compatible with the relations used ∗ to deﬁne the afﬁne Orlik–Solomon algebra A (A0) and hence induces the required morphism f . The relations among the generators e j are of two types. Firstly, if an intersection ∩···∩ < H1 Hp is non-empty and has codimension p, then the corresponding rela- ∂( ... ) ( ∩ ∩···∩ )< tion is e1 ep . If this is the case, then clearly codim H0 H1 Hp + = ∂( ... ) = ... p 1. This in turn implies 0 a0a1 ap a1 ap. In other words, (∂( ... )) = ∂ ( ... ) = ∂( ... ) = , g e1 ep g e1 ep a1 ap 0 3.1 Orlik–Solomon Algebras for Hyperplane Arrangements 53 so g is compatible with this kind of relation. The second type of relation occurs when ∩···∩ ... H1 Hp is empty, and the corresponding relation is e1 ep. This happens when H1 ∩···∩ Hp ⊂ H0 and then the family (H0, H1,...,Hp) is dependent as ( ... ) = ... = , above. Then the above computation shows that g e1 ep a1 ap 0 and hence g is compatible with the second type of relation as well. This completes our proof.

3.2 The Arnold–Brieskorn and Orlik–Solomon Theorems

V.I. Arnold started the study of the cohomology of complex hyperplane arrangement complements with the case of the braid arrangement. Then E. Brieskorn continued with more cases, including the arrangements related to finite groups of reflections in Cn,see[30]. Finally, P. Orlik and L. Solomon discovered in [179] the beautiful combinatorial description of the cohomology algebra H ∗(M(A )) in the general case, and their result is described below. n Let A ={H1,...,Hd } be an affine arrangement in C and (A , A , A ) be the triple of hyperplane arrangements associated to the distinguished hyperplane Hd .Let (M, M, M) be the corresponding complements. We recall first a general result in this special setting, see also [74], pp. 46–47.

Proposition 3.3 The complement M is a smooth hypersurface in the complex manifold M and M = M \ M. There is a long exact sequence, called the Gysin exact sequence in cohomology with integral or complex coefﬁcients, given by

i ∗ R δ ···→ H k (M) → H k (M) → H k−1(M) → H k+1(M) →··· , where i ∗ is the induced map by the inclusion i : M → M, R is the Poincaré–Leray residue, and δ is the connecting homomorphism.

Proof Here we give the main idea of the proof, and refer to [126] for more details. Start with the long exact sequence of the pair (M, M).LetT be a tubular neighborhood of M in M. Consider the composition u of the following two isomorphisms:

u H i (M, M) → H i (T, T \ M) → H i−2(M), where the ﬁrst map comes from the excision theorem and the second map u is the well-known Thom isomorphism (applied to the vector bundle T → M), see [130, 211]. Then the morphism R is deﬁned to be the composition

δk u H k (M) → H k+1(M, M) → H k−1(M) and δ is the composition 54 3 Orlik–Solomon Algebras and de Rham Cohomology

u−1 H k−1(M) → H k+1(M, M) → H k+1(M).

Remark 3.1 If M has a global deﬁning equation f = 0insideM, with f a holomorphic function on M, then d f R α + ∧ β =[2πi(β| )], f M where α, β are holomorphic forms on M with dα = dβ = 0.

Example 3.6 Consider the case M = C and M : z = 0. Then

R : H 1(C∗, C) → H 0({0}, C)

( dz ) = π 1 dz is an isomorphism. Note that R z 2 i and hence 2πi z corresponds to 1 in the cohomology group H 0({0}, Z) with integral coefﬁcients.

Return now to our arrangement A . Suppose H j ∈ A is deﬁned by a linear equation n ∗ j = 0. Then the induced maps j : C \ H j → C are homotopy equivalences. In ∗ : 1(C∗) → 1(Cn \ ) particular, the induced morphisms j H H H j are isomorphisms. We introduce the following family of 1-forms in Ω1(M) ω := ∗ 1 dz = 1 d j . j j 2πi z 2πi j

ω ω Ω1( ) Ω1( ) Denote by j and j the corresponding forms in M (resp. in M ) asso- ∩ < ∗(ω ) = ω < ciated to H j (resp. to H j Hd )for j d. Note that i j j , for all j d, and in the Gysin sequence in Proposition 3.3, one has i ∗(ω ...ω ) = ω ...ω , j1 jp j1 jp where the cup product is denoted by juxtaposition and is preserved by i ∗. In addition, we have R(ω ...ω ) = 0 and R(ω ...ω ω ) = (−1)pω ...ω ,ifall j1 jp j1 jp d j1 jp jk ∈{1,...,d − 1}. = ( ,..., ) ω = ω ...ω = ω For an ordered set S H j1 H jp ,weset S j1 jp j1... jp ∈ Ω p(M) and deﬁne

p k−1 p−1 ∂ω := (− ) ω ∈ Ω ( ), S 1 j1... jk ... jp M k=1 in analogy with the formula (3.1) for the derivation ∂.

Proposition 3.4 If the set S is dependent, then ∂ωS = 0. Moreover, the relation ∩S =∅implies ωS = 0. 3.2 The Arnold–Brieskorn and Orlik–Solomon Theorems 55

p−1 = ∈ C Proof Suppose ﬁrst that S is dependent, e.g. jp ck jk ,forsomeck .This = k 1 d j implies ω = p = c jk ω . Then by direct computation, we get jp k jk jp k

∩ =∅ ={ ,..., } Suppose now that S and assume that S1 H j1 H jk is a maximal subset ∩ =∅ ∩ =∅ = ∪{ } in S such that S1 . Then the condition S , with S S1 H jm and > m k, implies that d jm (which is the linear part of the linear form jm ) is a linear ,..., combination of d j1 d jk . This clearly implies

ω ...ω ω = , j1 jk jm 0 which yields our second claim.

Corollary 3.5 For any afﬁne hyperplane arrangement A , there exists a morphism ∗ ∗ of graded Z-algebras φ : A (A , Z) → H (M(A ), Z) sending a j to ω j for any j = 1,...,d.

The main result of this chapter is the following. Theorem 3.5 (Orlik–Solomon Theorem) For any afﬁne hyperplane arrangement A , the above morphism φ : A∗(A , Z) → H ∗(M(A ), Z) is a Z-algebra isomorphism.

Proof We prove this claim by induction on |A |. We have the following diagram

/ i / j / − / 0 A p(A ) A p(A ) A p 1(A ) 0

φ φ φ

∗ δ / i / R / − δ / + ··· H p(M ) H p(M) H p 1(M ) H p 1(M ) →··· where the top row comes from Theorem 3.1 and the bottom row comes from Propo- sition 3.3. These two results insure that the diagram is commutative up to sign and the two rows are exact. Assume the claim in the theorem is true for A and A , and hence the morphisms φ and φ are all isomorphisms. Then Rφ =±φ j is onto, since both morphisms φ and j are onto. Hence R is onto as well, showing that the morphisms i ∗ in the Gysin sequence are injective, i.e. the bottom row consists of several short exact sequences as well. Then the 5-lemma implies that φ is an isomorphism. 56 3 Orlik–Solomon Algebras and de Rham Cohomology

Corollary 3.6 The Betti numbers of the complex complement M(A ) are determined by the intersection lattice L(A ). More precisely, one has

B(M(A ), t) = π(A , t) = P(A∗(A ), t).

In particular, the Euler characteristic of the complement M(A ) is given by the formula χ(M(A )) = π(A , −1) = χ(A , 1).

Note that the ﬁrst equality above follows from Corollary 3.2. Moreover, for a supersolvable arrangement A of rank r, one has

B(M(A ), t) = (1 + b1t)(1 + b2t) ···(1 + br t), (3.4) where the integers bi are those introduced in Theorems 2.4 and 3.3. The Universal Coefﬁcient Theorem for cohomology, see [130, 211], yields the following. Corollary 3.7 The cohomology algebra with R-coefﬁcients of the complex complement M(A ) is determined by the combinatorics, for any unitary commutative ring R. More precisely, one has an isomorphism of graded R-algebras

A∗(A , R) = H ∗(M(A ), R).

The main information needed from the intersection lattice L(A ) to compute the Betti numbers b j (M(A )) are the values (dim X,μ(X)) for any flat X ∈ L(A ).This explains why two arrangements with non-isomorphic intersection lattices may have the same Betti numbers for their complex complements. An example of this situation A A ( ) is given by the two realizations 1 and 2 of the 93 configuration in Example ∗( (A ), C) 2.15. Note, however, that the corresponding cohomology algebras H M 1 ∗( (A ), C) and H M 2 are not isomorphic, see Exercise 6.5. The question of whether the intersection lattice L(A ) determines finer invariants of the complement M(A ) has attracted much attention, see for instance Remark 4.8 as well as the papers [13, 139, 184, 196]. The following result shows that for the cohomology algebra H ∗(M(A ), C), thought of as the de Rham cohomology of M(A ), one can choose representatives for the cohomology classes such that the representative of the cup product of two classes is represented by the cup product of the representatives. Note that this is not possible in general: for instance, it is well known that the cup product of two harmonic forms may no longer be harmonic. Corollary 3.8 (Arnold–Brieskorn Theorem) Let Ω∗(M) be the algebra of holomorphic or rational differential forms on M. Let R∗(M) be the Z-subalgebra generated ∗ by the forms ω j ∈ Ω (M) for j = 1,...d. Then there are two isomorphisms u and v of graded Z-algebras

u v A∗(A , Z) → R∗(M) → H ∗(M(A ), Z), 3.2 The Arnold–Brieskorn and Orlik–Solomon Theorems 57 given by u : a j → ω j and v : ω j →[ω j ] for j = 1,...,d.

Proof In view of Proposition 3.4, there is a well-deﬁned surjective morphism u deﬁned as above. Since the composition φ = vu is an isomorphism by Theorem 3.5, it follows that both u and v are isomorphisms.

We also have the following direct consequence of Theorem 3.2. Corollary 3.9 (Brieskorn decomposition for cohomology) Let A be an afﬁne n arrangement in C .ForX ∈ L(A ), consider the inclusion i X : M(A ) → M(AX ). Then the induced morphism

⊕ ∗ :⊕ k ( (A )) → k ( (A )) X∈Lk (A )i X X∈Lk (A ) H M X H M is an isomorphism for any integer k.

Corollary 3.10 For any central arrangement A =∅, one has

χ(M(A )) = π(A , −1) = χ(A , 1) = 0.

Proof We give two proofs of this claim. The ﬁrst proof is topological. Proposition 2.1 yields a homeomorphism M(A ) =∼ M(A ) × C∗, where A is the projective arrangement associated to A . Applying the Künneth formula and Corollary 3.6,we get π(A , t) = B(A , t) · (1 + t), which implies our claim, via Exercise 2.6. The second proof is an algebraic proof. Recall that we have, by Exercise 3.1, the following relation

∗ ∗ P(A (A ), t) = P(A (A0), t)(1 + t),

where A0 is the afﬁne arrangement associated to the projective arrangement A .In view of Corollary 3.2, this again proves our claim.

Deﬁnition 3.5 With the above notation, the number χ(M(A )) is called the Beta invariant of the central arrangement in A , and is denoted by β(A ).

Now we say a few words about the complement of the product of two central arrangements, recall the discussion before Deﬁnition 2.5.

Proposition 3.5 Suppose A1 =∅and A2 =∅are two central hyperplane arrange- A = A × A A A A ments and deﬁne 1 2. Denote by 1 , 2 and the corresponding projective arrangements. Then one has the following. 58 3 Orlik–Solomon Algebras and de Rham Cohomology

(i) M(A ) = M(A1) × M(A2); (ii) π(A , t) = π(A1, t) × π(A2, t); (A ) = (A ) × (A ) × C∗; (iii) M M 1 M 2 (iv) β(A ) = 0.

Proof The claim (i) is obvious, and (ii) follows by the Künneth formula and Corollary 3.6. To prove (iii), one can use the map ( ) (A ) → (A ) × (A ) × C∗, [( , )] → [ ], [ ], 1 x , M M 1 M 2 x y x y 2(y) where x ∈ M(A1), y ∈ M(A2) and H1 : 1 = 0 (resp. H2 : 2 = 0) is any hyper- ∗ plane in A1 (resp. in A2). The claim (iv) follows from (iii), since χ(C ) = 0.

This result gives one implication in the following numerical characterization of indecomposable arrangements. Recall Deﬁnition 2.5 and see [183, Theorem 3.3.4] for a proof. Theorem 3.6 A central arrangement A is indecomposable if and only if its Beta invariant satisﬁes β(A ) = 0.

Example 3.7 For a non-empty central line arrangement A in C2, A is indecomposable if and only if |A | = 2.

For the following related result we refer to [80]. Theorem 3.7 (Factorization Theorem) Let A be a central, essential hyperplane arrangement in Cn = V . Then there is an essentially unique factorization A = A1 × ...× Aq of A as a product of indecomposable essential arrangements A j , where the integers q, d1 =|A1|,...,dq =|Aq | are determined by the combinatorics, i.e. by the intersection lattice L(A ).

Remark 3.2 The complement of an arrangement of linear subspaces of higher codimension in an affine real space (in particular, its Betti numbers and cohomology algebra) has also been studied, see for instance [27, 64, 65, 118, 123]. More precisely, the reader may find an excellent introduction to this important subject in [27], and the relation to the stratified Morse Theory for non-proper mappings is explored in [123]. The study of subspace arrangements is very important for applications as well. Indeed, many applications to real-life involve the analysis of large amounts of high-dimensional data. Such data are rather often mixed, as different parts of the data usually have different structures which cannot be described accurately by a single model. The subspace arrangements have been successfully used for modeling such mixed data: each subspace in the arrangement models just a homogeneous subset of the data. For more details on such applications of the theory of subspace arrangements, see [163]. 3.3 Exercises 59

3.3 Exercises

Exercise 3.1 With the notation of Theorem 3.4, show that one has

∗ ∗ P(A (A ), t) = P(A (A0), t)(1 + t).

Exercise 3.2 Let A be an afﬁne general position arrangement in K n consisting of d hyperplanes. Determine its Orlik–Solomon algebra A∗(A ). Exercise 3.3 Let A be a projective general position arrangement in Pn consisting of d hyperplanes. Determine its Orlik–Solomon algebra A∗(A ). Exercise 3.4 Show that for a line arrangement A in P2, the second Betti number of the complement M(A ) is given by b2(M(A )) = nk (k − 1) − d + 1, k≥2

where nk is the number of k-fold intersection points in A and d is the number of lines in A . Exercise 3.5 Let A be a hyperplane arrangement in V = K n and deﬁne, for any integer k,0≤ k ≤ n,thek-th truncation poset

L(A )≤k ={X ∈ L(A ) : codim(X) ≤ k} of the intersection poset L(A ) and the k-th truncation algebra

A≤k (A ) = A∗(A )/(Ak+1(A ) ⊕···⊕ An(A )) of the Orlik–Solomon algebra A∗(A ) with arbitrary coefﬁcients R.

(i) Show that the k-th truncation poset L(A )≤k determines the k-th truncation algebra A≤k (A ), both for a central arrangement A and for an afﬁne arrangement A . (ii) Let A be a central arrangement in V and let E ⊂ V be a generic linear subspace such that 0 ∈ E and dim E = k + 1. Deﬁne the restriction A |E of the arrangement A to E to be the central hyperplane arrangement in E whose hyperplanes are H ∩ E for H ∈ A . Show that we have an isomorphism of graded algebras

A≤k (A |E) = A≤k (A ).

Looking at the case n = 3 and k = 1, conclude that this result is best possible, i.e. in general A≤k+1(A |E) and A≤k+1(A ) are not isomorphic. (iii) Let A be an affine arrangement in V and let E ⊂ V be a generic affine subspace such that dim E = k. Define the restriction A |E of the arrangement A to E as above and show that 60 3 Orlik–Solomon Algebras and de Rham Cohomology

A≤k (A |E) = A≤k (A ).

Since Ak+1(A |E) = 0 and in general Ak+1(A ) = 0, this result is also best possible. This result should be compared with Remark 4.6 in Chap. 4 concerning Zariski’s theorem of Lefschetz type. (iv) Show that in the above statements one can replace ‘generic’ by the following n transversality property: for any X ∈ L(A )≤k , the codimension of X in K coincides with the codimension of X ∩ E in E. Use this to relate Exercises 3.2, 3.3, 3.5 via Exercise 2.21.

Exercise 3.6 Deﬁne in a natural way the direct product of two lattices. Then show that, for any two non-empty central arrangements A1 and A2, one has a lattice isomorphism ∼ L(A1 × A2) = L(A1) × L(A2).

Use this isomorphism to give a combinatorial proof of the equality

π(A1 × A2, t) = π(A1, t) × π(A2, t).

Exercise 3.7 Recall that if the multiplicative group C∗ acts on a complex algebraic ∗ ∗ variety X, then χ(X) = χ(X C ), where X C denotes the set of ﬁxed points of the action, see [25]. Use this result to give another proof of Corollary 3.10. Chapter 4 On the Topology of the Complement M(A )

Abstract In this chapter we discuss the minimality of the complement M(A ) of a hyperplane arrangement A and its relation to the degree of the gradient mapping of the defining equation for A . We mention two results of June Huh, the first on the log-concavity of the coefficients of the Poincaré polynomial, and the second on the relation between the degree of the gradient mapping of a projective hypersurface and the multiplicities of its singularities. We also discuss the fundamental group of the complement M(A ) and the arrangements for which M(A ) is a K (π, 1)-space, stating the deep results of Deligne and Bessis, on simplicial real arrangements, and respectively complex reflection arrangements. Then the fiber type arrangements are introduced, and their relation to supersolvable arrangements is explored.

4.1 Complements of Projective Hypersurfaces

We compare first the topology of projective hyperplane arrangement complements and of projective hypersurface complements, to see what is specific for the hyperplane arrangements. Assume that A : Q = 0 is a hyperplane arrangement in Pn, n ≥ 2, and hence M(A ) ⊂ Pn.LetV : h = 0 be a hypersurface in Pn defined by a reduced homogeneous polynomial h and let M(h) = Pn \ V be the corresponding complement. With this notation we clearly have M(A ) = M(Q).

Proposition 4.1 (i) The cohomology groups H ∗(M(A ); Z) and the homology groups H∗(M(A ); Z) of the hyperplane arrangement complement M(A ) have no torsion. (ii) If V : h = 0 is an irreducible hypersurface of degree d, then ∼ H1(M(h); Z) = Z/dZ.

Proof The ﬁrst claim is a direct consequence of Corollaries 8.14 and 3.7, and of the Universal Coefﬁcient Theorem in Algebraic Topology, see [130]. For the second claim, we refer to [74, Proposition 4.1.3].

Proposition 4.2 (i) The cohomology algebra H ∗(M(A ), Z) is generated in degree 1. (ii) If V is a smooth hypersurface of degree d > 2, then H ∗(M(h), Q) is not generated in degree 1.

Proof The ﬁrst claim is a direct consequence of Corollary 3.7. For the second claim, note that H 1(M(h), Q) = 0 by Proposition 4.1,butH n(M(h), Q) = 0, having the n−1( , Q) same dimension as the primitive middle cohomology group H0 V of V , which is quite large, see [74], pp. 151–152, when d > 2. Indeed, the exact sequence for the pair (Pn, V ) has the form

i k ···→ H k (Pn, V ) → H k (Pn) → H k (V ) → H k+1(Pn, V ) →···

and i k is injective in the middle dimensions. This implies that

n−1( , Q) := ( n−1) =∼ n(Pn, ; Q) =∼ ( ( ), Q) H0 V coker i H V Hn M h

where the last isomorphism comes from Alexander duality, see [211, Theorem 19, p. 297].

The following result can be regarded as a consequence of Proposition 4.2 (i).

Theorem 4.1 Let A be an afﬁne hyperplane arrangement in Cn and let G = Homeo(Cn, A ) be the group of homeomorphisms h : Cn → Cn such that h(N(A )) = N(A ), where N(A ) =∪H∈A H. Then, for any positive integer k, the obvious morphism ρk : G → Aut(H k (M(A ), Z) has a ﬁnite image. More precisely, one has

|ρk (G)|≤2d d!,

where d =|A |.

Proof If Hi : i = 0fori = 1, ··· , d are the hyperplanes in A , the generators of H 1(M(A ), Z) can be taken to be

ω = ∗(α ), i i i

1 ∗ ∗ with αi a generator of H (C , Z) and i : M(A ) → C . Alternatively, ωi is one 1 n 1 generator for the image of the morphism H (C \ Hi , Z) → H (M(A ), Z) induced by the obvious inclusion. Note that for a point p ∈ Hi , the local fundamental group

π1(Bp \ (Bp ∩ M(A )),

Cn ∈ 0 where Bp is a small ball in centered at p, is inﬁnite cyclic if and only if p Hi , 0 = \∪ ∈ ( 0) = 0 with Hi Hi j=i H j . It follows that any h G satisﬁes h Hi Hi ,forsome

i . By taking closures, we see that this implies h(Hi ) = Hi , in other words h permutes 4.1 Complements of Projective Hypersurfaces 63 the hyperplanes Hi in A . Using the above description of the cohomology class ωi , it follows that ∗ h (ωi ) =±ωi .

This proves our claim for k = 1. Note that the cohomology algebra H ∗(M(A ), Z) is generated in degree 1, just as above, by Corollary 3.7. It follows that the action of h∗ on H k (M(A ), Z) is determined by its action on H 1(M(A ), Z), and this completes our proof.

n Example 4.1 If Bn is the Boolean arrangement in C , one can consider the homeo- n morphisms of C obtained by permuting the coordinates x1, ..., xn and then replacing some of them by their complex conjugates. This shows, in view of Example 1.4, that the above bound for |ρ1(G)| can be attained. In the projective setting we have a similar result. Corollary 4.1 Let A be a hyperplane arrangement in Pn and consider the group G = Homeo(Pn, A ) of homeomorphisms h : Pn → Pn such that h(N(A )) = N(A ), where N(A ) =∪H∈A H. Then, for any positive integer k, the obvious morphism ρk : G → Aut(H k (M(A ), Z)) has a ﬁnite image. Proof As above, one can show that any homeomorphism h ∈ G induces a permutation σ, i.e. h(Hi ) = Hσ(i) for i = 1, ..., d. It remains to notice that the morphisms induced by the obvious inclusions yield the following two isomorphisms

1 1 n H (M(A ), Z) =⊕j>1 H (P \ (H1 ∩ H j ), Z) and 1 1 n H (M(A ), Z) =⊕j>1 H (P \ (Hσ(1) ∩ Hσ(j)), Z).

∗ 1 n The morphism h maps each term H (P \ (Hσ(1) ∩ Hσ(j)), Z) = Z isomorphically 1 n to the term H (P \ (H1 ∩ H j ), Z) = Z, and hence it is just multiplication by ±1. For the deﬁnition and basic facts on higher homotopy groups and the Hurewicz morphism we refer to [130, 211]. Corollary 4.2 If X = M(A ) and A =∅and k ≥ 2, then the Hurewicz morphism αX : πk (X) → Hk (X, Z) is trivial.

Proof Recall that an element [g]∈πk (X) is represented by a continuous map g : (Sk , p) → (X, x). This map g then induces a homomorphism in homology k k k k g∗ : Hk (S , Z) → Hk (X, Z) and αX ([g]) = g∗([S ]), where [S ]∈Hk (S , Z) is the k fundamental class of S . Since Hk (X) are torsion free, it sufﬁces to show that g∗ : H k (X) → H k (Sk ) are trivial. But this is obvious because H ∗(X) is generated in degree 1 and H 1(Sk ) = 0.

n n n ∼ Remark 4.1 If A =∅, X = P , then αX : π2(P ) → H2(P , Z) = Z is an isomorphism, by Hurewicz’s Theorem, since Pn is simply-connected. 64 4 On the Topology of the Complement M(A )

Next we recall a basic topological property of complex algebraic varieties. Theorem 4.2 Let X be a smooth quasi-projective complex variety of dimension n. Then X is a smooth real manifold of dimension 2n and X is homotopy equivalent to a finite CW-complex (or a finite simplicial complex) K X of real dimension dim K X ≤ 2n. If X is affine (e.g. X = M(A )), then dim K X ≤ n. Proof The idea of the proof is as follows: the variety X can be compactified to a smooth projective variety X¯ such that D = X¯ \ X is a normal crossing divisor. Then it can be shown that the pair (X¯ , D) can be triangulated, i.e. there is a homeomorphism ( ¯ , ) ∼ ( , ) \ X D = K X¯ K D , where K X¯ , K D are finite simplicial complexes. Moreover, K X¯ = \ ( ) ( ) K D is homotopy equivalent to K1 K X¯ T D , where T D is an open tubular neighborhood of D in K X¯ constructed using the second barycentric subdivision. By construction the complex K1 is finite, and this completes the proof of the first claim. When X is affine, we have to apply a general result due to H. Hamm, see [129]or[74, Theorem 1.6.8], which actually holds for more general spaces, i.e. possibly singular Stein varieties.

By considering the cellular chain complex (C∗(K X , Z), ∂) associated to the ﬁnite CW-complex K X and computing its homology, see [130] for details, one gets the following.

Corollary 4.3 If N p(K X ) is the number of p-cells (or p-dimensional simplices) in K X , then one has p χ(X) = (−1) N p(K X ). p≥0

Moreover, for any p ≥ 0, one has

bp(X) = rankHp(X, Z) ≤ N p(K X ).

Remark 4.2 After all the results on the topology of the complement M(A ) of an afﬁne arrangement A : Q = 0inCn obtained in the previous chapter, it is time to have a quick look at the corresponding zero-set N(A ) ={x ∈ Cn : Q(x) = 0}.If the arrangement A is central, then Q is a homogeneous polynomial, and the space N(A ) is contractible. When A is an essential afﬁne arrangement, then it is known that the space N(A ) is homotopy equivalent to a bouquet of (n − 1)-dimensional spheres, see [76, Corollary 2.1]. The number of spheres in this bouquet is given by

n bn−1(N(A )) = (−1) χ(M(A )), which follows from the additivity property of the Euler characteristic, namely

χ(M(A )) + χ(N(A )) = χ(Cn) = 1.

In particular, when the arrangement A is real, one has bn−1(N(A )) = b(A ) by Theorem 2.8. In other words, each bounded region of the real complement M(A )R 4.1 Complements of Projective Hypersurfaces 65 contributes by exactly one (n − 1)-dimensional sphere to the homotopy type of the complex complement M(A )C. This result should be compared with Theorem 2.9 (iii). For more on the topology of the union of hyperplanes N(A ), and for the corresponding results in the case of a subspace arrangement, see [27, 123].

On the other hand, let A : Q = 0 be a central arrangement in Cn+1 and consider the projective arrangement A associated to A . Then the projective hypersurface N(A ) given by Q = 0inPn has in general a rich cohomology, as the exact sequence of cohomology with compact supports of the pair (Pn, N(A ) )

···→ k( (A ), Z) → k(Pn, Z) → k( (A ), Z) → k+1( (A ), Z) →··· Hc M H H N Hc M

k ( (A ), Z) → k (Pn−1, Z) easily shows. In fact, the morphism Hc M H is dual to the morphism H 2n−k (Pn, Z) → H 2n−k (M(A ), Z) induced by the inclusion, which is known to be trivial, see [74, Exercise 5.2.16]. It follows that the above long exact sequence splits into short exact sequences

→ k (Pn, Z) → k ( (A ), Z) → k+1( (A ), Z) → . 0 H H N Hc M 0 (4.1)

If one deﬁnes the primitive cohomology of the variety N(A ) by the equality

k ( (A ), Z) = { k (Pn, Z) → k ( (A ), Z)}= k+1( (A ), Z), H0 N coker H H N Hc M then the above considerations combined with Corollary 4.1 yield the following result.

Corollary 4.4 Let A be a hyperplane arrangement in Pn and consider the group G = Homeo(Pn, A ) of homeomorphisms h : Pn → Pn such that h(N(A )) = N(A ), where N(A ) =∪H∈A H. Then, for any positive integer k, the obvious morphism ρk : → ( k ( (A ), Z)) G Aut H0 N has a ﬁnite image. One may ask under which conditions the topology of the complement M(A ) does not change when we deform the arrangement. One very useful answer to this question is the following, see [193].

Deﬁnition 4.1 (i) Let I be an open interval of R containing [0, 1] and d > 0an integer. A smooth 1-parameter family of central arrangements (At )t∈I of d hyperplanes in Cn is the data consisting of d distinct linear forms

i (x, t) = ai1(t)x1 +···+ain(t)xn

for each t ∈ I , such that the coefﬁcients aij(t) : I → C are smooth functions. We set Qt (x) = 1(x, t) ·····d (x, t). Then the arrangement At : Qt = 0is the family of d hyperplanes Hi,t : i (x, t) = 0fori = 1, 2, ..., d. (ii) We say that the 1-parameter family of central arrangements (At )t∈I is a lattice isotopy if, for any J ⊂{1, 2, ..., d}, the dimension dim ∩i∈J Hi,t is independent 66 4 On the Topology of the Complement M(A )

of t ∈ I . In such a case we say that A0 : Q0 = 0 and A1 : Q1 = 0 are lattice- isotopic arrangements. (iii) Two projective hyperplane arrangements A0 : Q0 = 0 and A1 : Q1 = 0are A c : = lattice-isotopic if the corresponding central arrangements 0 Q0 0 and A c : = 1 Q1 0 are lattice-isotopic. Example 4.2 Consider two line arrangements A : Q = 0 and A : Q = 0inP2 such that |A |=|A | and both A and A have only nodes. Then the line arrangements A : Q = 0 and A : Q = 0 are lattice-isotopic. To prove this, recall the smooth irreducible variety X(d) introduced in Remark 2.7 and parametrizing the 2 arrangements in P consisting of d lines. The set X(d)n of nodal arrangements in X(d) is clearly a Zariski open subset in X(d), and hence it is smooth and connected. d Let Y (d)n be the cone in (S1) over X(d)n, which is again smooth and connected.

Take now d =|A |=|A |. It follows that the two points p, p ∈ Y (d)n corresponding to the arrangements A : Q = 0 and A : Q = 0 can be joined by a smooth path inside Y (d)n. This shows that A : Q = 0 and A : Q = 0 are lattice-isotopic.

The notion of lattice-isotopy enters into the following result, see [193].

Theorem 4.3 If A0 : Q0 = 0 and A1 : Q1 = 0 are central lattice-isotopic arrangements, then the following hold.

(i) The complements M(A0) and M(A1) are diffeomorphic. (A ) (A ) (ii) The complements M 0 and M 1 of the associated projective arrange- A A ments 0 and 1 are diffeomorphic. (A ) (A ) (A ) (iii) The unions of hyperplanes N 0 and N 1 , and respectively N 0 and (A ) N 1 , are homeomorphic.

In particular, the fundamental groups π1(M(A0)) and π1(M(A1)), and respectively π ( (A )) π ( (A )) 1 M 0 and 1 M 1 , are isomorphic.

Remark 4.3 If A0 : Q0 = 0 and A1 : Q1 = 0 are central lattice-isotopic arrange- n ments in C , then clearly the intersection lattices L(A0) and L(A1) are isomorphic, i.e. the arrangements A0 and A1 are combinatorially equivalent. The converse implication does not hold in general, even for n = 3, see Remark 4.8. In fact, the converse implication is related to the connectivity of the moduli spaces A(L) and A(L)/G discussed in Remark 2.7 in the case n = 3.

4.2 Minimality of M(A ) and the Degree of the Gradient Map

Deﬁnition 4.2 A topological space X, which is homotopy equivalent to a ﬁnite CW complex K X , is minimal if the corresponding Betti numbers equal the number of cells in each dimension, that is, bp(X) = N p(K X ), for all p ≥ 0. 4.2 Minimality of M(A ) and the Degree of the Gradient Map 67

Example 4.3 (i) The n-dimensional torus (S1)n is minimal. Indeed, the case n = 1 is obvious, as the simplest CW-structure of the circle has N0 = N1 = 1. Using the product CW structure on (S1)n yields the result. (ii) A smooth projective complex curve of genus g is minimal, with N0 = N2 = 1 and N1 = 2g. (iii) The complex projective space Pn is minimal for any n ≥ 0, with

N0 = N2 =···= N2n = 1.

Remark 4.4 If X is minimal, then H∗(X, Z) has no torsion, as the differentials in the chain complex (C∗(K X , Z), ∂) have to be all zero. Note that the converse is not true, see Exercise 4.3.

The above remark and Proposition 4.1 show that in general a hypersurface complement M(h) is not minimal. However, for hyperplane arrangements we have the following fundamental result, see [89, 195].

Theorem 4.4 For any hyperplane arrangement A in Pn, the complement M(A ) is a minimal space.

We discuss the ideas and techniques used in [89] to prove this result. For any hypersurface V ⊂ Pn deﬁned by a homogeneous polynomial h of degree d, we consider the gradient mapping n n ψh = grad(h) : P P .

It is a rational map, deﬁned by ∂h ∂h x → (x) : ... : (x) . ∂x0 ∂xn

(ψ ) =|ψ−1( )| This map has a degree deﬁned by deg h h y , where y is a generic point in Pn. When the hypersurface V has only isolated singularities, it is known that n deg(ψh) = (d − 1) − μ(V, p), (4.2) p∈V see [89], p. 487. Here μ(V, p) denotes the Milnor number of the hypersurface singularity (V, p), recall Deﬁnition 2.17. For an example, if C is a plane curve and if p ∈ C is an ordinary point of multiplicity m, then one has

μ(C, p) = (m − 1)2. (4.3)

In order to get the topological meaning of this degree in general, one can use afﬁne Lefschetz theory and the theory of polar curves. In this way we get the following result, see [89]. 68 4 On the Topology of the Complement M(A )

Theorem 4.5 For a generic hyperplane H ⊂ Pn, the complement M(h) is obtained from the generic hyperplane section M(h) ∩ H by attaching Nn cells of dimension n, where n Nn = (−1) [χ(M(h)) − χ(M(h) ∩ H)].

Moreover, Nn = deg(ψh).

Proof of Theorem 4.4. Using the above result, we can prove Theorem 4.4.Let n n A0 be the affine arrangement in C = P \ H0, for a fixed hyperplane H0 ∈ A induced by the arrangement A . Consider the arrangement B in the generic hyperplane H obtained by taking all the intersections H ∩ H1 for H1 ∈ A and let B0 n−1 be the corresponding affine arrangement in C = H \ (H ∩ H0). We know that M(A ) = M(A0), M(B) = M(B0) and hence χ(M(A )) = π(A0, −1) = μ(X),

X∈L(A0)

and similarly, χ(M(A ) ∩ H) = χ(M(B)) = μ(Y ).

Y ∈L(B0) ∼ When H is generic, then we have a lattice isomorphism L(A0)≤n−1 = L(B0),given by X → Y = X ∩ H, where L(A0)≤n−1 is obtained from L(A0) by omitting the 0-dimensional edges in L(A0), see also Exercise 3.5. It follows that n deg(ψQ) = χ(M(A )) − χ(M(A ) ∩ H) = μ(X) = (−1) bn(M(A )),

X∈Ln (A0) (4.4) where A : Q = 0 and the last equality comes from the formula (3.3). In other words, M(A ) is obtained from M(A ) ∩ H by attaching exactly bn(A ) n-cells, where bn(A ) is the n-th Betti number. This clearly proves, by induction on n, the claim in Theorem 4.4.

Remark 4.5 (i) A completely different approach to minimality questions, applying also to 2-real subspace arrangements, i.e. finite collections of codimension 2 subspaces Ei in a real vector space V such that any non-empty intersection of the Ei ’s has an even codimension in V , can be found in [4]. This approach involves the combinatorial stratifications introduced in [28] and combinatorial Morse theory introduced in [119]. See also [27, 123]. (ii) It is a very interesting question whether the minimal CW-complex structure of the arrangement complement M(A ) can be described explicitly. A positive answer would allow a lot of very useful computations, e.g. the computation of the twisted cohomology of M(A ) with rank one local system coefficients, which is discussed in the next chapter. For this problem, see [202, 232]. Both classical and combinatorial Morse theory are used in these papers. 4.2 Minimality of M(A ) and the Degree of the Gradient Map 69

Remark 4.6 Consider j : M(h) ∩ H → M(h) and the induced morphisms

j p : H p(M(h)) → H p(M(h) ∩ H).

It follows from Theorem 4.5 that j p is an isomorphism for p ≤ n − 2 and is a monomorphism for p = n − 1, for generic H. In fact, this is a special case of Zariski’s theorem of Lefschetz type, see [74, Theorem 1.6.5]. A particular case of this result also occurs below in Theorem 4.9. The above proof shows that for a hyperplane arrangement complement M(A ), the morphism j p is an isomorphism for p = n − 1 as well. In particular, one has bn−1(A ) = bn−1(A ∩ H). Note also that bn(A )>0 if and only if A is essential, recall Deﬁnition 2.6. When A is regarded as an afﬁne arrangement by taking the c decone of A , this condition is equivalent to Ln(A ) =∅. These two facts yield the following result, to be compared with Exercise 2.5.

n Corollary 4.5 Let A be a projective arrangement in P such that ∩H∈A H =∅. Then all the Betti numbers b j (M(A )) are strictly positive integers for 0 ≤ j ≤ n.

Next we present some related results of June Huh, see [136, 137]. Recall that , ..., ≤ 2 a sequence of positive numbers a0 an is log concave if ai−1ai+1 ai , for all 0 < i < n.

Theorem 4.6 For a projective arrangement A ∈ Pn, the coefﬁcients of its Poincaré polynomial

n π(A , t) = a0 + a1t + ... + ant , in other words the Betti numbers a j = b j (M(A )), form a log concave sequence.

This result was conjectured around 1968 for the coefficients of the chromatic polynomial χG (t) associated to a finite graph G, introduced in Definition 2.21.Infact,toany such graph G, one can associate in a natural way a graphic arrangement AG , recall Definition 2.20, satisfying χG (t) = π(AG , t),see[213]. Hence the above result for hyperplane arrangements implies the corresponding result for graphs. I. Dolgachev showed that it is possible for n = 2 to classify the reduced plane curves C : h(x, y, z) = 0 such that deg(ψh) = 1. More precisely, such a curve C is called homaloidal and must be one of the following: a generic arrangement of three lines, a conic, or a conic with a tangent line [104]. Later S. Papadima and the author conjectured that, for n ≥ 3, d ≥ 3, the condition deg(ψh) = 1 implies that V : h = 0 has non-isolated singularities, see p. 487 in [89] and also [75]. This conjecture was proved by J. Huh in [137], after a large number of results in special cases, see [5, 44]. A key role in the proof is played by the following result.

Theorem 4.7 Let V be a hypersurface of degree d in the complex projective space Pn, with n ≥ 2, having only isolated singularities. Let m be the multiplicity of V at one of its points x. Then 70 4 On the Topology of the Complement M(A ) n n−1 deg(ψh) = (d − 1) − μ(V, p) ≥ (m − 1) , p∈V unless V is a cone with apex x.

Here we give a more precise result in the case of a line arrangement A : Q = 0in the plane P2. Denote by C = N(A ) the union of all the lines L ∈ A .

Theorem 4.8 For any line L0 ∈ A , consider the induced afﬁne line arrangement a 2 2 A0 = (A , L0) in C = P \ L0. Then 2 deg(ψQ) = (d − 1) − μ(C, p) = (mq − 1).

p∈C q∈(C\L0)

In particular, the last sum is independent of the choice of the line L0.

Proof We give an elementary proof, without using the formula (4.4). In fact, the ﬁrst part of the proof essentially recovers this formula for a plane curve. The long exact sequence of the pair (P2, C) in cohomology with Q coefﬁcients contains the following sequence

H 1(P2) → H 1(C) → H 2(P2, C) → H 2(P2) → H 2(C).

Since H 1(P2) = 0 and the morphism H 2(P2) → H 2(C) is injective, it follows that we have an isomorphism H 1(C) = H 2(P2, C). Now we compute the Betti number b1(C) using the general formula χ(C) = 2 − (d − 1)(d − 2) + μ(C, p), (4.5) p∈C see for instance [74], Corollary (5.4.4), p. 162. One clearly has b0(C) = 1, b2(C) = d, and hence 2 b1(C) = (d − 1) − μ(C, p). p∈C

Using now the Lefschetz duality theorem, it follows that 2 2 2 dim H (P , C) = dim H2(P \ C) = dim H2(M(A0)) = (mq − 1),

q∈(C\L0) where the last equality follows from Corollary 3.4.

Corollary 4.6 For any integer d ≥ 3 consider the line arrangement in P2 given by the equation 4.2 Minimality of M(A ) and the Degree of the Gradient Map 71

Qk (x, y, z) = x(x + y) ···(x + (d − 2)y)z = 0.

(ψ ) = − Then deg Qk d 2.

Proof If we apply Theorem 4.8 for the line L0 : z = 0, the resulting arrangement A0 − : = has a unique intersection point of multiplicity d 1. Note that for the line L0 x 0, A − the corresponding arrangement 0 has d 2 nodes as intersection points. Hence we get (m p − 1) = (mq − 1) = d − 2. ∈( \ ) ∈( \ ) p C L0 q C L0

2 Proposition 4.3 Let A be a line arrangement in P and set d =|A |. We denote by (A ) A μ(A ) = μ( (A ), ) C the union of the lines in , and set p∈C(A ) C p . Then the following hold. (i) μ(A) ≥ d(d − 1)/2 and equality holds if and only if the line arrangement A has only double points. (ii) μ(A) ≤ (d − 1)2 and equality holds if and only if the line arrangement A has a point of multiplicity d, i.e. it is the union of d lines passing through one point.

Proof To prove the ﬁrst claim, recall the formula from Exercise 2.8. With the notation used there, one clearly has 2 μ(A ) = nk (k − 1) . k≥2

It remains to note that, for k ≥ 2, one has k ≤ (k − 1)2 2

with equality only for k = 2. The second claim follows from Theorem 4.7 or from Theorem 4.8.

4.3 The Fundamental Group of the Complement M(A )

We say a few words now on the fundamental groups of hypersurface complements. We start with the following basic result. For a more general statement, see [74, Theorem 1.6.5].

Theorem 4.9 (Zariski’s theorem of Lefschetz type) Suppose V : h = 0 is a hypersurface in Pn with complement M(h) and E ⊂ Pn is a generic k-dimensional linear subspace. Then the morphism induced by inclusion 72 4 On the Topology of the Complement M(A )

π1(E ∩ M(h)) → π1(M(h)) is surjective for k = 1, and it is an isomorphism for k ≥ 2.

Remark 4.7 A similar result holds for a hypersurface V : h = 0 in the afﬁne space n C . It is enough to apply Theorem 4.9 to the projective hypersurface V = V ∪ H0, n n where H0 = P \ C is the hyperplane at inﬁnity.

Corollary 4.7 (i) For the computation of π1(M(h)) it is enough to consider the case n = 2. (ii) The group π1(M(h)) has a natural set of d = deg(h) generators γ j satisfying the relation γ1...γd = 1. (iii) If the hypersurface V has k irreducible components V1, V2, ..., Vk , of degrees d1, d2, ..., dk respectively, then Z <σ, ..., σ > ab ∼ 1 k π1(M(h)) = H1(M(h), Z) = , < d1σ1 + ... + dk σk >

where σ j is an elementary loop around the component Vj ,forj= 1, ..., k. In particular, if A is a projective hyperplane arrangement with d =|A |, then Z <σ, ..., σ > ab ∼ 1 d π1(M(A )) = H1(M(A ), Z) = , <σ1 + ... + σd >

where σ j =[γ j ].

It is useful to note the following difference between the loops σ j and γ j . The loops γ j , being in the fundamental group, should all be based at one and the same point. On the other hand, the loops σ j , being cycles in homology, are free loops, not based at a particular point. In particular, one may choose the loop σ j inside a small tubular neighborhood of the hyperplane H j .

Proof The ﬁrst claim is clear by Theorem 4.9, as is the claim that the γ j give a set of generators. To get the relation among them, note that, for n = 2 and E = L a line, one ∼ 2 has π1(M(h) ∩ L) = π1(S −{d points}). If we represent the sphere by the Earth, imagine that one point of the intersection V ∩ L,saya1, is the North Pole, while the remaining d − 1 points, say a2, ..., ad , are located in the Southern Hemisphere. The generator γ j corresponds to a small loop in M(h), around the point a j , all loops being based at the same point b ∈ M(h), situated on the Equator, and turning in the same γ ...γ = γ −1 direction. It is clear that this implies 2 d 1 , with the right choice of loops. Indeed, both paths in this equality can be represented by the Equator, but travelling in opposite directions. The last claim follows from [74], Proposition 4.1.3, p. 102. Here we just notice that the line L intersects an irreducible component Vj in exactly d j distinct points, and the associated d j loops γm are conjugate in π1(M(h)), and hence the corresponding loops σm are equal in H1(M(h), Z). 4.3 The Fundamental Group of the Complement M(A ) 73

n Corollary 4.8 If A ={H1, ..., Hd } is an afﬁne arrangement in C , then

H1(M(A ), Z) = Z <σ1, ..., σd >, where the loop σ j =[γ j ] is as above, i.e. an elementary loop about the hyperplane H j for j = 1, ..., d.

Another basic result going back to Zariski is the following. For a proof refer to [63, 120].

Theorem 4.10 If V : h = 0 is a curve in P2 having only nodes as singularities, then π1(M(h)) is commutative. In particular, for an afﬁne line arrangement A having only points of multiplicity two and no parallel lines, one has

|A | π1(M(A )) = H1(M(A )) = Z .

In the case of a line arrangement A in P2 having only double points, the proof of the above result can be done by making use of the following simple idea. Let γi and γ j be elementary loops about two distinct lines Li and L j in the arrangement. Denote by P the intersection point of the lines Li and L j and let U be a small neighborhood of P. Then U is homeomorphic to (C∗)2, and hence has a commutative fundamental 2 group, i.e. π1(U) = (Z) . The loops γi and γ j can be deformed in M(A ) such that they differ only inside the open set U. The fact that π1(U) is abelian then implies that γi γ j = γ j γi in π1(M(A )). Since the loops of type γi generate π1(M(A )),it follows that this group is abelian. An alternative proof in the case of a line arrangement A in P2 having only double points can be given as follows. If d =|A |, then consider the hyperplane arrangement

B : · ... · = d x1 xd 0

d−1 in P , which is just the projective version of the Boolean arrangement Bd . Clearly π ( (B )) = Zd−1 = 1 M d and apply Zariski’s theorem of Lefschetz type 4.9 with k 2. ∩ (B ) = (C ) C = ∩ B Then E M d M , where E d is a nodal line arrangement in E = P2. We conclude using Example 4.2 and Theorem 4.3. The following result related to Theorem 4.10 was obtained by A.D. Choudary, S. Papadima and the author, see [43].

2 Theorem 4.11 Assume the afﬁne line arrangement A = A (m1, ..., mr ) in C has only nodes and consists of r families of parallel lines with cardinalities m1, ...mr , m j ≥ 1. Then π ( (A )) =∼ F × ... × F , 1 M m1 mr where Fm is the free group on m generators. 74 4 On the Topology of the Complement M(A )

Example 4.4 Consider the hyperplane arrangement B in Cr given by the equation

( , , ..., ) = ( ) ( ) ··· ( ) = , Q x1 x2 xr Pm1 x1 Pm2 x2 Pmr xr 0 where Pm (t) = (t − 1)(t − 2) ···(t − m), for some integers m1, ...mr , m j ≥ 1. Then M(B) is a product of spaces obtained from C by removing m j points, for j = 1, ..., r, and hence clearly

π ( (B)) =∼ F × ... × F . 1 M m1 mr

Let E be a generic 2-plane in Cr , and denote by A the line arrangement in E = C2 obtained by taking the traces of the hyperplanes in B. It is clear that A is an arrangement as in Theorem 4.11 and an application of Theorem 4.9 and Remark 4.7 shows that π( (A )) =∼ π ( (B)) =∼ F × ... × F . M 1 M m1 mr

Remark 4.8 In the previous two results the intersection lattice L(A ) determines the fundamental group π1(M(A )). But this is not the case in general, see [13, 196], where two line arrangements A and B are constructed, each having 13 lines and such that the intersection lattices L(A ) and L(B) are isomorphic, but the fundamental groups π1(M(A )) and π1(M(B)) are not isomorphic. In the case of projective line arrangements, Theorem 4.10 has the following converse. Proposition 4.4 If A is a line arrangement in P2 such that the fundamental group π1(M(A )) is abelian, then A has only double points. Proof Assume that there is a point p in A of multiplicity at least 3 and let B denote the set of lines in A passing through p. Then the fundamental group π1(M(B)) is free with at least two generators. On the other hand, there is an epimorphism

π1(M(A )) → π1(M(B)), since any loop in M(B) can be deformed to a loop in M(A ) without changing its class in π1(M(B)). This epimorphism implies that π1(M(B)) is commutative, a contradiction.

To end this chapter, we say a few words about K (π, 1)-arrangements. We refer to [130, 211] for general facts on K (π, 1)-spaces.

Deﬁnition 4.3 We call an arrangement A a K (π, 1)-arrangement if the complex complement M(A ) is a K (π, 1)-space, i.e. if πk (M(A )) = 0, for k ≥ 2.

Example 4.5 The Boolean arrangement Bn is a K (π, 1)-arrangement. Indeed, this follows from the homotopy equivalence M(A ) =∼ (S1)n, since the torus (S1)n has Rn as its universal covering space. Then of course 4.3 The Fundamental Group of the Complement M(A ) 75

1 n n πk (M(A )) = πk ((S ) ) = πk (R ) = 0 for k ≥ 2. Example 4.6 Every central arrangement A in C2 is a K (π, 1)-arrangement. Indeed, if A =∅, then we have a ﬁbration

C∗ → M(A ) → M(A ) and M(A ) has the homotopy type of a bouquet of circles. The claim follows using the homotopy long exact sequence of a fibration. In fact, this fibration is trivial by Proposition 2.1. Examples of non-central K (π, 1)-line arrangements are given in Remark 8.3, following M. Falk [114]. Example 4.7 Now we give examples of line arrangements which are not K (π, 1). Consider a line arrangement A in C2 having only nodes, no parallel lines and such that r =|A |≥3. Then, using either Theorem 4.10 or Theorem 4.11, it follows that r r π1(M(A )) = Z .AK (Z , 1)-space should be homotopic to the obvious candidate, 1 r 3 which is Tr = (S ) ,ther-dimensional real torus. Note that H (Tr , Q) = 0, in fact r Q this cohomology group has dimension 3 as a -vector space. On the other hand, H 3(M(A ), Q) = 0, since M(A ) is an affine surface, recall Theorem 4.2.More generally, any arrangement A = A (m1, ..., mr ) as in Theorem 4.11, with r ≥ 3, is not a K (π, 1)-space. Indeed, [89, Theorem 18 (ii)] implies that one has

π2(M(A )) = 0 in this case. Other examples of hyperplane arrangements which are not K (π, 1)- spaces can be found in [98] (the case of line arrangements), and especially in [89], where the structure of higher homotopy groups πk (M(A )) for k > 1 is considered in detail.

Example 4.8 Let Brn be the braid group on n-strings, for n ≥ 1. It has n − 1 generators ai for i = 1, ..., n − 1, corresponding to the switching of the i-th and i + 1- st strings. The relations are given by ai a j = a j ai for |i − j| > 2, and ai ai+1ai = ai+1ai ai+1,for1≤ i ≤ n − 2. For some examples, note that Br1 = 1, the trivial group, Br2 = Z and Br3 is isomorphic to the group of the trefoil knot. Indeed, for n = 3 we have two generators a1 and a2 for the braid group Br3 satisfying one relation

a1a2a1 = a2a1a2.

If we set β = a2a1 and α = a1β, we see that the braid group Br3 is also generated by α and β, which satisfy the well-known relation deﬁning the group of the trefoil knot, namely α2 = β3.

Clearly we have a group homomorphism h : Brn → Sn from the braid group Brn to the symmetric group Sn, sending a braid to its ordered set of n endpoints. The 76 4 On the Topology of the Complement M(A ) kernel of h is called the pure braid group on n-strings and it is denoted by PBrn. n If Br n is the braid arrangement in C , then M(Br n) is a K (PBrn, 1)-space. The proof uses induction on n and the following observation: the projection on the ﬁrst (n − 1)-coordinates Cn → Cn−1 induces a map

M(Br n) → M(Br n−1) which is a locally trivial fibration, with fiber F = C \{n − 1 points}. Then the homotopy long exact sequence of a fibration yields the sequence

... → πk (F) → πk (M(Br n)) → πk (M(Br n−1)) → πk−1(F) → ... showing that πk (M(Br n)) = 0fork > 1. The isomorphism π1(M(Br n)) = PBrn essentially follows from another ﬁbration, namely

n Sn → M(Br n) → M(Br n)/Sn = C \ Δ, where Δ is the discriminant hypersurface associated to a general polynomial of degree n n in one variable, see for more details [74], pp. 113–115. Note that π1(C \ Δ) = Brn almost by deﬁnition, and the morphism h : Brn → Sn discussed above is precisely the n connecting homomorphism Brn = π1(C \ Δ) → π0(Sn) = Sn from the homotopy long exact sequence of the above ﬁbration.

Example 4.8 can be regarded as a special case of the following very general result due to P. Deligne, see [61] and also [189] for an alternative proof. We recall that an open cone C in a real vector space V of dimension n is simplicial if there is a basis e1, ..., en of V such that x = x1e1 +···+xnen ∈ C if and only if xi > 0for i = 1, ··· , n.

Theorem 4.12 Let V be a finite-dimensional real vector space and A a finite collection of hyperplanes of V such that each connected component of M = M(A ) = V \H∈A H is an open simplicial cone. Then AC, the complexification of the real arrangement A ,isaK(π, 1)-arrangement.

In particular, let V be as above, and let G ⊂ GL(V ) be a finite group generated by reflections (and hence a Coxeter group). Suppose also that only the zero vector is fixed by all elements of G.LetA (G) be the collection of hyperplanes corresponding to the reflections in G. It is well known that V and A (G) satisfy the hypotheses of the theorem above, and hence A (G)C, the complexification of the Coxeter arrangement A (G),isaK (π, 1)-arrangement. The case of complex reflection groups is due to D. Bessis, see [24].

Theorem 4.13 Let G be a finite complex reflection group and let A (G) be the corresponding reflection arrangement in the complex vector space V . Then the complement M(A (G)) is a K (π, 1)-space. 4.3 The Fundamental Group of the Complement M(A ) 77

For the braid groups and their generalizations, the Artin groups, one may read [30, 36, 59, 66]. A very good survey on this topic is [190]. There are also a number of interesting results on afﬁne Artin groups, see [37, 38]. A detailed description of the fundamental group of a complement of the com- plexiﬁcation of a real hyperplane arrangement is due to M. Salvetti, see [201]. In particular, the next result follows from the constructions in Salvetti’s paper. See also [180], p. 173.

Theorem 4.14 For a real hyperplane arrangement A in Rn, the homotopy type of the complement M(AC) of the complexiﬁed arrangement is determined by the oriented matroid G (A ).

To prove such a deep result, Salvetti in [201] and respectively Orlik and Terao in [180, Sect. 5.2] follow the main idea of Deligne’s paper [61] and construct a simplicial complex K , now called the Salvetti complex of A , inside the complement M(AC), respectively M(A ), such that K has the same homotopy type as this complement. Paris has constructed the universal covering Kˆ of the Salvetti complex K in [189]. These complexes K yield presentations of the corresponding fundamental groups using the 2-skeleton of the complex K . On the other hand, such complexes K are far from being minimal, and their very large number of simplices prevents their use in doing explicit computations. In the case of a real hyperplane arrangement, one way of obtaining a minimal CW-complex is by applying discrete Morse theory to the Salvetti complex, as explained in [202]. A similar construction of a finite regular cell complex of the homotopy type of M(A ) was given in [28]. The study of higher homotopy groups of hypersurface complements, in particular of M(A ), is very interesting as well, as can be seen from the papers [89, 156, 185, 201]. There is another possible generalization of Example 4.8, namely the arrangements of fiber type which we now define inductively.

Definition 4.4 Let A be an affine hyperplane arrangement in Cn. (i) We say that A is strictly linearly fibered if, up to a linear change of coordinates, n n−1 the projection πn : C → C on the first n − 1 coordinates induces a fiber n−1 bundle Fn → M(A ) → M(B), where B is a hyperplane arrangement in C , and the fiber Fn is obtained from C by deleting finitely many points. (ii) Any arrangement A in C is fiber type. (iii) For n ≥ 2, an affine hyperplane arrangement A in Cn is fiber type if A is strictly linearly fibered as above, with base space M(B), for a fiber type arrangement B in Cn−1.

This deﬁnition and induction on n clearly imply the following result.

Theorem 4.15 Any ﬁber type arrangement is a K (π, 1)-arrangement.

A surprising deep connection between topology and combinatorics in the theory of hyperplane arrangements is the following result, see [180, Theorem 5.113]. 78 4 On the Topology of the Complement M(A )

Theorem 4.16 Let A be a central, essential complex hyperplane arrangement. Then A is supersolvable if and only if A is ﬁber type.

n Note that for an arrangement A of fiber type in C , one can set An = A , An−1 = B, the arrangement in Cn−1 from Definition 4.4 (iii). It follows that one has a new fiber bundle Fn−1 → M(An−1) → M(An−2) and so on. The collection of fibers Fn, Fn−1, ..., F1 obtained in this way enters into the following result, see [116], [180, Theorem 5.13].

Theorem 4.17 If A is a ﬁber type arrangement in Cn, then one has an isomorphism of graded Z-modules

∗ ∗ ∗ H (M(A ), Z) = H (F1, Z) ⊗···⊗ H (Fn, Z).

Example 4.9 We illustrate the above result in the case of the central arrangement 3 in C given by the deleted B3-arrangement A , and its projective version A in P2, recall Example 2.8. In the projective setting, a linear projection C3 → C2 as 2 in Definition 4.4 corresponds to a linear projection πp from a point p ∈ P onto a line L ⊂ P2 such that p ∈/ L. Choose p = (1 : 1 : 0) and L : x + y − z = 0. Then the 4 lines through p in A , namely L0 : z = 0, L3, L6 and L7 in the notation from Example 2.8, give rise to 4 points on the line L, denoted by q0 = L0 ∩ L, q3 = L3 ∩ L, q6 = L6 ∩ L and q7 = L7 ∩ L.IfwesetB ={q0, q3, q6, q7}, then 1 the projection πp induces a fiber bundle M(A ) → M(B ), with fiber F3 = P \ {5 points}. Here the 5 points correspond to the point p and the 4 intersection points with the remaining lines L1, L2, L4 and L5. This construction gives rise to a linear π : C3 → C2 = C · projection X2 , with kernel X2 p, the line corresponding to p as in Example 2.9 (iv), and the image equal to the plane x + y − z = 0, identified with C2. One can take, for instance,

π ( , , ) = ( − , − , ), X2 x y z x a y a z

1 where a = (x + y − z)/2. Continuing in this way, we get F2 = P \{4 points} and 1 F1 = P \{2 points}. This result is clearly compatible with the formula for π(A , t) given in Example 2.13. If we are interested in the projective complement M(A ), then we can stop at the ﬁrst step, and regard the corresponding ﬁbration F3 → M(A ) →

M(B ) = F2 as giving the expected result

∗ ∗ ∗ H (M(A ), Z) = H (F2, Z) ⊗ H (F3, Z).

The previous results shed new light on Theorems 2.4 and 3.3. See also Theorem 7.12 further on in this book.

n Remark 4.9 If Br n denotes the braid arrangement in C , we have seen in Example 4.8 that the symmetric group Sn acts freely on the complement M(Br n). Combining Example 2.14 and Exercise 2.6, or using Theorem 2.4, we get 4.3 The Fundamental Group of the Complement M(A ) 79

π(Br n, t) = (t + 1)(t + 2) ···(t + n − 1).

B B Let r n denote the projective arrangement associated to the braid arrangement r n. (B ) = (B ) × C∗ Using Corollary 3.6 and the product M r n M r n , it follows that the (B ) Betti polynomial of M r n is given by ( (B ), ) = ( + ) ···( + − ). B M r n t t 2 t n 1 χ( (B )) = ( (B ), − ) = ( − )!. In particular, we get M r n B M r n 1 n 2 Note that the sym- (B ) metric group Sn acts on the complement M r n , but the action is no longer free !=| | χ( (B )) (otherwise n Sn would have divided M r n ). Indeed, it is easy to see that the points of the form (1 : ξ : ξ 2 :···:ξ n−1), with ξ a primitive n-th root of unity, have non-trivial isotropy groups.

Remark 4.10 Let G be a reﬂection group in Cn and let A be the set of reﬂecting hyperplanes of G. Then G acts on the arrangement complement M(A ) and it is a very interesting problem to describe the cohomology groups H m(M(A ), C) as G-modules, see for instance [87, 148, 150].

For an example of a similar type, consider the line arrangement in P2 given by

A : Q(x, y, z) = xyz(x − y)(x − z)(y − z) = 0.

The corresponding afﬁne picture is given in Fig. 4.1. In fact, it follows from Exercises 2.11 and 2.12 that this arrangement is the essential version of the braid arrangement Br 4. The symmetric group S3 acts on the m complement M(A ) and hence the cohomology groups H (M(A ), C) become S3- modules. Recall that there are three irreducible S3-representations, namely the trivial

Fig. 4.1 The arrangement B

L5 L1 L2 80 4 On the Topology of the Complement M(A ) one, denoted by 1, the sign representation, denoted by , and the 2-dimensional irreducible representation, denoted by ρ.

Proposition 4.5 With this notation, one has the following

H 0(M(A ), C) = 1, H 1(M(A ), C) = 1 + 2ρ and H 2(M(A ), C) = 3ρ.

Proof To perform this computation, we replace the arrangement A by its decone 2 B consisting of the following 5 lines in C : L1 : x = 0, L2 : x − 1 = 0, L3 : y = 0, L4 : y − 1 = 0 and L5 : x − y = 0. Since H m(M(A ), C) = H m (M(B), C) = Am (B, C), for any integer m,itis enough to determine the Orlik–Solomon algebra A∗(B, C) of B and the corre- 0 1 sponding S3-action on this algebra. It is easy to see that A (B, C) = C, A (B, C) 2 has a basis given by a j ,for j = 1, ..., 5 and A (B, C) has a basis given by a13, a14, a15, a23, a24 and a25. Indeed, the two pairs of parallel lines L1, L2 and L3, L4 yield the relations a12 = a34 = 0, and the two triple points (0, 0) and (1, 1) imply that we have

a35 = a15 − a13 and a45 = a25 − a24.

The symmetric group S3 is generated by the following two involutions

σz : (x : y : z) → (y : x : z) and σy : (x : y : z) → (z : y : x).

Indeed, the composition σx = σz ◦ σy ◦ σz acts by the formula

σx : (x : y : z) → (x : z : y), and hence the third involution σx is in the group spanned by the ﬁrst two involutions 2 σy and σz. The involutions σy and σz, acting on P , correspond to the following involutions on M(B) ⊂ C2 1 y τ : (x, y) → (y, x) and τ : (x, y) → , . z y x x

To determine the action of these new involutions on the vector space A1(B, C),itis enough to recall that a j , the element corresponding to the line L j , is represented in the de Rham cohomology by d j / j , with L j : j = 0, and then use the pull-back of differential forms under regular mappings. In this way we get the following actions

τ ∗ : → , → , → , → , → z a1 a3 a2 a4 a3 a1 a4 a2 a5 a5 and 4.3 The Fundamental Group of the Complement M(A ) 81

τ ∗ : →− , → , → − , → − , → − . y a1 a1 a2 a2 a3 a3 a1 a4 a5 a1 a5 a4 a1 τ ∗ τ ∗ It follows that both z and y have trace equal to 1. A direct computation shows that the ﬁxed part under the resulting S3-action is 1-dimensional, spanned by the vector

−2(a1 + a3) + 3(a2 + a4 + a5).

This gives the claim H 1(M(A ), C) = 1 + 2ρ. To determine the action on the vector space A2(B, C), it is enough to recall that aij = ai a j and use the relations deﬁning the Orlik–Solomon algebra. It follows that τ ∗ τ ∗ z (and respectively y ) acts on the generators in the following way a13 →−a13, a14 →−a23, a15 → a15 − a13, a23 →−a14, a24 →−a24, a25 → a25 − a24 and respectively

a13 →−a13, a14 →−a15, a15 →−a14, a23 → a23, a24 → a25, a25 → a24.

τ ∗ τ ∗ It follows that both z and y have trace equal to 0 and a direct computation shows that the ﬁxed part under the new S3-action is trivial. This clearly gives us the claim H 2(M(A ), C) = 3ρ. Since the claim H 0(M(A ), C) = 1 is obvious, the proof is complete.

Remark 4.11 The fundamental groups of line arrangement complements give many interesting examples of finitely generated groups with various finiteness or non- finiteness properties. Consider the group G = F2 × F2 × F2, where F2 is the free group on two generators, and the group morphism

χ : G → Z sending each generator to 1. If N = ker χ, then Stallings in [214] has shown that N is ﬁnitely presented, but H3(N, Z) is not ﬁnitely generated. This group is in fact the fundamental group of the following line arrangement in C2

A : xy(x − 1)(y + 1)(2x + y) = 0, see [218, Remark 12.4] for all the details. A larger class of groups, with similar properties, are the Bestvina–Brady groups NΓ ,see[92] where hyperplane arrangements are used to decide when such a group is quasi-projective, i.e. it is isomorphic to the fundamental group of a smooth quasi-projective variety X,seeRemark5.3.As soon as such a group is quasi-projective, it is shown in [15] that the smooth quasi- projective variety X above can be chosen to be the complement M(A ) of a very explicit hyperplane arrangement. 82 4 On the Topology of the Complement M(A )

Remark 4.12 There are several types of Lie algebras which can be naturally associated with a hyperplane arrangement, carrying useful information on the fundamental group G of the arrangement complement: the graded Lie algebra coming from the lower central series of G, the holonomy Lie algebra, and the Malcev Lie algebra. The reader can ﬁnd more information about this very important topic in the monograph [50], as well as in a multitude of research papers, see for instance [52, 67, 116, 143, 144, 161, 186, 206].

4.4 Exercises

Exercise 4.1 Consider two line arrangements A : f = 0 and A : f = 0inP2 such that |A |=|A | and both A and A have only nodes, except for a single triple point. Show that A : f = 0 and A : f = 0 are lattice-isotopic.

Exercise 4.2 Consider the line arrangement A : f = 0inP2, where

f = xy(3x + y − 3z)(3x + 2y − 6z)(2x + 3y − 6z)(x + 2y − 3z)

(2x + y − 2z)(3x + 4y − 6z)(5y − 6z) = 0.

Fig. 4.2 The afﬁne Pappus (93)1-arrangement 4.4 Exercises 83

A picture of the afﬁne line arrangement obtained by setting z = 1 is given in Fig. 4.2.It A : = ( ) is a new version of the Pappus line arrangement 1 Q1 0 of type 93 1 described A : = A : = in Example 2.15. Show that f 0 and 1 Q1 0 are lattice-isotopic. If needed, have a look at the paper [231]. Exercise 4.3 Consider the Poincaré homology sphere Σ3 = SU(2)/(BI), where BI is the binary icosahedral group of order 120. It is known that

3 3 Hm (Σ , Z) = Hm(S , Z),

3 for any integer m. In fact, Σ is the link of the E8-simple surface singularity

x2 + y3 + z5 = 0,

and the intersection matrix of the minimal resolution of this singularity is known to be unimodular, see for more details [74]. 3 (i) Show that the fundamental group π1(Σ ) is isomorphic to the binary icosahedral group BI, and hence it is non-trivial. (ii) Deduce that Σ3 is not a minimal topological space, even though its homology is torsion free. Exercise 4.4 Let A : Q = 0 be a non-empty hyperplane arrangement in the com- n plex projective space P ,letH0 ∈ A be a hyperplane and denote by A0 the corre- a sponding afﬁne arrangement (A , H0) . Show that n deg(ψQ) = (−1) μ(X),

X∈Ln (A0)

where μ is the Möbius function of the poset L(A0). Exercise 4.5 Let A be a central hyperplane arrangement in C4 and let A be the corresponding arrangement in the complex projective space P3. Assume that A is essential and supersolvable and hence Theorem 3.3 holds. Assume that the integers bi occurring in this theorem are ordered by b1 ≤ b2 ≤ b3 ≤ b4.

(i) Show that b1 = 1.

(ii) Show that B(M(A ), t) = (1 + b2t)(1 + b3t)(1 + b4t).

(iii) Replacing A by the afﬁne arrangement A0 as in Theorem 3.4, show that

B(M(A ), t) = B(M(A0), t) = π(M(A0), t). (iii) Check Theorem 4.6 in this case. Exercise 4.6 Let A : Q = 0 be a non-empty line arrangement in the complex pro- 2 jective plane P . Show that Q is homaloidal, i.e. deg(ψQ) = 1, if and only if up to a linear change of coordinates Q = xyz. Exercise 4.7 Let A : Q = 0 be a line arrangement in the complex projective plane P2 such that |A |=5. Find all the possibilities for the global Milnor number μ(A ) deﬁned in Proposition 4.3. 84 4 On the Topology of the Complement M(A )

Exercise 4.8 Let A : Q = 0 be a line arrangement in the complex projective plane P2 such that |A |=d and A has only double and triple points. Show that d − 1 b (M(A )) = − n (A ), 2 2 3 where n3(A ) is the number of triple points in A . Hint: use the formulas (4.2) and (4.4) and Exercise 2.8.

Exercise 4.9 Show that any hyperplane arrangement A in the complex projective space Pn, with n ≥ 2, is a subarrangement in a K (π, 1)-arrangement B by following the next steps.

(i) Choose a point p on a hyperplane in A and consider the central projection σp from p to Pn−1, represented by some hyperplane in Pn not in A . Show that there is a hyperplane arrangement A in Pn−1 such that the induced projection

n−1 σp : M(A ) → P

is locally trivial over M(A ). (ii) Show that if A ⊂ B and B is a K (π, 1)-arrangement, then the arrangement B A σ −1( ) ∈ B obtained from by adding all the hyperplanes p H for H gives rise to a locally trivial ﬁbration

σp : M(B) → M(B ),

showing that B is also a K (π, 1)-arrangement. (iii) Prove the main claim by induction on n, starting with the case n = 2. Chapter 5 Milnor Fibers and Local Systems

Abstract In this chapter we start our detailed discussion of the Milnor fiber F associated to a central hyperplane arrangement and of the monodromy action on the cohomology H ∗(F). We recall the definition and some basic properties of rank one local systems, and explain the relation between monodromy eigenspaces and the twisted cohomology of the complement M(A ), where A is the projective arrangement associated to A . Then we state a very general vanishing result for the twisted cohomology of the complement M(A ) with coefficients in a rank one local system. This result involves the total turn monodromy operators associated to dense edges in the arrangement.

5.1 Milnor Fibers and Monodromy

In this section, let A be a central hyperplane arrangement in Cn+1 defined by Q(x) = 0, with Q a homogeneous polynomial, and denote by A the arrangement in Pn corresponding to A . Definition 5.1 The Milnor fiber of the central arrangement A , or of the projective arrangement A , is the affine smooth hypersurface F = F(A ) given by the equation Q(x) = 1 in the affine space Cn+1. If p ∈ F is any point, then Euler’s relation applied to the homogeneous polynomial Q yields n ∂ Q p (p) = dQ(p) = d, j ∂x j=0 j

∂ where d is the degree of Q. This implies that one cannot have Q (p) = 0 for all ∂x j j, and hence F is indeed a smooth hypersurface. It follows from Theorem 4.2 that the Milnor ﬁber F is homotopic to a ﬁnite CW-complex of dimension less than or equal to n, in particular b j (F) = dimQ H j (F; Q) = 0for j > n. As mentioned in the Introduction, a major open problem is the following: are the Betti numbers b j (F),for j ≤ n, determined by the intersection lattice L(A )? The problem is not even solved for the case of b1(F) and when n = 2. © Springer International Publishing AG 2017 85 A. Dimca, Hyperplane Arrangements, Universitext, DOI 10.1007/978-3-319-56221-6_5 86 5 Milnor Fibers and Local Systems

The Milnor ﬁber F is endowed with a natural mapping. The geometric monodromy transformation h : F → F is given by h(x) = λx, for any x ∈ F, with

λ = exp(2πi/d). (5.1)

This is in fact the monodromy of the locally trivial ﬁbration

Q F → Cn+1 \ Q−1(0) = M(A ) → C∗, or, more precisely, of its restriction above the unit circle S1 ⊂ C∗. Moreover, as with any ﬁbration over S1, there is an associated Wang exact sequence

hq −Id ... → H q (M(A )) → H q (F) → H q (F) → H q+1(M(A )) → ... (5.2)

Here and in the sequel we use C-coefﬁcients for the (co)homology unless stated otherwise. Let < h > denote the group generated by h and note that it is isomorphic to the cyclic group Z/dZ. If a ﬁnite group G acts on a topological space X, then ρ : → ( ∗( )) → ( ∗)−1 there is a linear representation G Aut H X given by g hg , where hg takes x ∈ X to gx. This formula is chosen so that the following relation holds

ρ(g1 · g2) = ρ(g1) · ρ(g2),

∗ −1 for any elements g1, g2 ∈ G. Usually, in our setting, we denote (h ) by T ,but many results below do not change by using h∗ instead of T . For a detailed discussion of this point, see [96]. For a d-th root of unity η = 1, write

q q q H (F)η = ker(T − ηId : H (F) → H (F)).

When η = λk ,forλ as in (5.1) and k = 1,...,d − 1, some authors use the notation q q q H (F)η = H (F)k , and let H (F)0 denote the fixed part under the monodromy. However, we will not use such notations in the sequel, e.g. the fixed part under the q monodromy will be denoted by H (F)1. Note that the quotient topological space F/ is naturally identified with M(A ) and, more precisely, F → M(A ) is a Galois covering with deck transformation group < h >, a cyclic group of order d. It follows that q q H (F)1 = H (M(A )) is determined by the intersection lattice L(A ) for any integer q. It is also clear that 0 0 H (F) = H (F)1 = C as soon as the equation Q is reduced. Moreover, the Euler characteristic χ(F) = dχ(M(A )) = dβ(A ) (5.3) is also determined by L(A ). One has the following general result, see [87]. 5.1 Milnor Fibers and Monodromy 87

Proposition 5.1 Let X be a ﬁnite CW complex, with a free G-action, where G is a ﬁnite group. Then we have χ G ( ) = χ( ) X Y RegG in the Grothendieck representation ring R(G), with Y = X/G.

Recall that R(G) is the free Z-module generated by the isomorphism classes of complex, irreducible G-representations, endowed with the product

[V1][V2]=[V1 ⊗ V2].

Moreover, using the induced linear actions of G on the cohomology of X as described above, one deﬁnes the equivariant Euler characteristic

G n χ (X) =[H0(X)]−[H1(X)]+[H2(X)]−···+(−1) [Hn(X)]∈R(G), see for more details and examples [85]. If G = Z/dZ is ﬁnite cyclic, then all the irreducible G-representations are 1-dimensional, namely they are given by χ0,...,χd−1 ∗ ˆ k with χk : G → C , χk (1) = λ . More precisely, the map sending t to χ1 induces an isomorphism of Z-algebras Z[t] → R(G). (td − 1) =[ ]∈ ( ) C Recall also that RegG V R G , where V is a -vector space having as basis the elements g of G, and where the G-action is given by setting h ag g = ag(hg), g∈G g∈G for any h ∈ G and ag ∈ C.ForG = Z/dZ, one clearly has

= χ +···+χ . RegG 0 d−1

The following consequence of Proposition 5.1 reﬁnes the formula in (5.3).

Corollary 5.1 For any k = 0, 1,...,d − 1, one has j j (−1) dimH (F)λk = χ(M(A )). j=0,n

The classical way to express these relations is via the Alexander polynomials of the monodromy, see [74], pp. 106–109. Deﬁne the q-th Alexander polynomial of the monodromy h by

Δq (h, t) = det(t · Id − hq : H q (F) → H q (F)) (5.4) 88 5 Milnor Fibers and Local Systems and the corresponding zeta function

n q Z(h, t) = (Δq (h, t))(−1) . q=0

Then Corollary 5.1 can be restated as follows

Z(h, t) = (td − 1)χ(M(A )). (5.5)

Example 5.1 If n = 1, consider d lines through the origin in C2, e.g. choose for instance Q(x, y) = xd − yd . This is an isolated hypersurface singularity and it follows from Milnor’s work that the corresponding Milnor ﬁber F is homotopy equiv- 1 2 alent to a bouquet of circles ∨(d−1)2 S . In particular, b1(F) = (d − 1) ,seefor instance [74], pp. 78–79. The action of the monodromy in this case follows from 0 1 0 Corollary 5.1:dimH (F)1 = 1 and dim H (F)1 = d − 1, while dim H (F)η = 0 1 and dim H (F)η = d − 2 for any d-th root of unity η = 1.

Remark 5.1 In fact, one can deﬁne the Milnor ﬁber exactly as above for any projective hypersurface D : f = 0inPn, i.e. we set F : f = 1inCn+1. Many of the properties listed above for the monodromy remain true in this more general frame- work, e.g. formula (5.5). For more on this we refer to [74]. This extension is used in the last chapter of our book, where we discuss the monodromy, and in particular, the Alexander polynomials, of some classes of projective plane curves.

Remark 5.2 We have seen above that the Milnor fiber F of a central hyperplane arrangement A can be regarded as the total space of a Galois covering of the projective complement M(A ), with structure group Z/dZ. On the other hand, the Milnor fiber F can be regarded as the total space of an infinite cyclic covering of the affine complement M(A ). Indeed, the homotopy exact sequence of the fibration ∗ F → M(A ) → C shows that π1(F) can be identified with the kernel K of the mor- ∗ phism π1(M(A )) → π1(C ) = Z, sending each of the standard generators γ j to 1. If X → M(A ) denotes the infinite cyclic covering of the complement M(A ) corresponding to the normal subgroup K , then the inclusion morphism i : F → M(A )) has a lifting to a map i˜ : F → X,see[130, 211]. It is easy to see that i˜ induces isomorphisms between the homotopy groups of F and X. Since both F and X have the homotopy type of CW-complexes, such a map i˜ is a homotopy equivalence, see [130, 211]. This homotopy equivalence is compatible with the monodromy action on F and the action of a well-chosen generator of the deck transformation group Z on X. Note that in general, for a given finite CW-complex Y , an infinite cyclic covering X of Y may no longer have the homotopy type of a finite CW-complex, e.g. when X = S1 ∨ S2.

Remark 5.3 When A : Q = 0 is an affine arrangement in Cn+1, the polynomial Q is no longer homogeneous, and the mapping Q : M(A ) → C∗ is no longer a locally trivial fibration. All the smooth fibers F of the mapping Q are homeomorphic among 5.1 Milnor Fibers and Monodromy 89 them, but there are also singular fibers Fsing coming from the isolated singularities of the polynomial Q on M(A ),see[76]. Any singular fiber Fsing has the homotopy type of a space obtained from the smooth fiber F by attaching finitely many (n + 1)- k dimensional cells. In particular, it follows that the constructible sheaf R Q∗QM(A ) is a local system on C∗ for k < n, i.e. we can speak of a Milnor fiber and a monodromy operator in this range, see [76]. Moreover, for n ≥ 2, it follows that π1(F) = π(Fsing), and this implies that there is an exact sequence of fundamental groups, as if the mapping Q : M(A ) → C∗ were a locally trivial fibration. More precisely, one has in this case the exact sequence

∗ 1 → π1(F) → π1(M(A )) → π1(C ) → 0, see [92] for a proof and an application to the study of Bestvina–Brady groups. The following result shows that the Milnor ﬁbration is stable by lattice-isotopy. Recall Deﬁnition 4.1 for the notation, and see [194] for the proof of the following.

Theorem 5.1 If A0 : Q0 = 0 and A1 : Q1 = 0 are central lattice-isotopic arrangements, then the corresponding Milnor ﬁbrations are smoothly equivalent. ∗ ∗ This means that there are diffeomorphisms φ : M(A0) → M(A1) and ψ : C → C such that the diagram φ M(A0) M(A1) (5.6)

Q0 Q1 ψ C∗ C∗ is commutative. Here is a brief description of Randell’s proof of this result. Let D be the closed unit ball in Cn centered at the origin and consider the map F : D × I → C × I given by (x, t) → (Qt (x), t), Qt being the homotopy between Q0 and Q1 as in Deﬁnition 4.1.LetB be a large closed disk in C, centered at the origin, such that f (D × I ) ⊂ (Int B) × I . Hence we get a proper map

F : D × I → B × I, which we stratify as follows. We first stratify B using the strata ∂ B,IntB \{0}, and {0}, then we stratify I using just the stratum I , and consider the product stratification on B × I . Finally, we stratify D × I in a way compatible with the family of hyperplane arrangements At , and use Thom’s second isotopy lemma to conclude that F is locally trivial over I . This implies that there are stratified homeomorphisms φ and ψ so that the diagram φ D ×{0} D ×{1} (5.7)

Q0 Q1 ψ B ×{0} B ×{1} 90 5 Milnor Fibers and Local Systems is commutative and the restrictions of φ and ψ to any stratum are smooth mappings. The stratification on D × I gives by restriction a stratification on any ball D ×{t}, whose strata are essentially the intersection of the ball with the flats in At . Since the only open stratum in D = D ×{0} (resp. D = D ×{1})isD ∩ M(A0) (resp. D ∩ M(A1)), the claim in Theorem 5.1 follows. Indeed, the global Milnor fibration of a homogeneous polynomial can be localized at the origin, as explained in [74, Exercise 3.1.13]. In the situation considered in Theorem 5.1, the corresponding monodromy diffeomorphisms hi : F(Ai ) → F(Ai ) for i = 0, 1 can be chosen so that the diagram

φ F(A0) F(A1) (5.8)

h0 h1 φ F(A0) F(A1) is commutative. This implies the following.

Corollary 5.2 If A0 : Q0 = 0 and A1 : Q1 = 0 are central lattice-isotopic arrangements, then the corresponding Milnor ﬁbrations have diffeomorphic Milnor ﬁbers F(A0) and F(A1), and the same Alexander polynomials, i.e.

q q Δ (h0, t) = Δ (h1, t), for all integers q. Another way of obtaining this corollary from Theorem 4.3 is by using Remark 5.2, i.e. by regarding F(Ai ) as the inﬁnite cyclic covering of the complement M(Ai ) for i = 0, 1.

5.2 Monodromy Eigenspaces and Twisted Cohomology

It turns out that the monodromy eigenspaces can be described using the twisted cohomology of the projective arrangement complement with coefficients in rank one local systems. We briefly recall here the basic properties of local systems as discussed, for instance, in [77], pp. 47–57. Definition 5.2 Let X be a good topological space (e.g. connected smooth manifold or connected CW-complex). A rank k local system L of C-vector spaces on X is a rank k locally free sheaf of C-vector spaces on X. That is, every point in X has an L Ck open neighborhood U on which is the constant sheaf U . Proposition 5.2 The isomorphism classes of rank k local systems on the topological space X are in one-to-one correspondence to the isomorphism classes of rank k linear representations of the fundamental group π1(X). 5.2 Monodromy Eigenspaces and Twisted Cohomology 91

We have many natural examples of local systems coming from topology, differential geometry, and complex analytic geometry. In topology, Steenrod found in the 50s that if p : E → B is a locally trivial ﬁbration, then for any positive integer m we m m get a local system R p∗(CE ) by putting together the cohomology groups H (Fb, C) −1 of all the ﬁbers Fb = p (b) of the mapping p. From the differential geometry point of view, let V → X be a smooth vector bundle with connection ∇.LetV be the sheaf of local sections of V . Then the connection yields morphisms

∇:V ⊗ Ωm → V ⊗ Ωm+1, X X for any integer m ≥ 0. The connection ∇ is said to be integrable if ∇2 = 0. If the connection ∇ is integrable, then L = ker∇, the sheaf of (local) horizontal sections of ∇, is a local system on X of rank k = rank V . Let us consider a basic example in complex analytic geometry, which plays a d key role in the sequel. Let X = C \{a1,...,ad }, α = (α1,...,αd ) ∈ C and λ = ∗ (λ1,...,λd ) with λ j = exp(−2πiα j ) ∈ C . Then we take V = OX , the sheaf of holomorphic functions on X and deﬁne

d α j dz ∇αu = du + u . (5.9) z − a j=1 j

If we set Lα = ker∇α, then the rank one local system Lα can also be expressed via ∗ ∗ a representation ρ : π1(X) → C = G1(C). Since C is a commutative group, the morphism ρ factors through the homology group H1(X, Z), yielding a morphism ∗ ρ¯ : H1(X, Z) → C . Suppose σ1,...,σd are the Z-basis of H1(X, Z), given by the elementary loops about the points a1,...,ad as in Corollary 4.8. Then the local system Lα corresponds to the representation ρ¯ sending σi to λi . If X is a topological space and L a local system on X, then H m (X, L ),the twisted cohomology of X with coefficient in the local system L , can be defined using sheaf cohomology, twisted de Rham complexes, or cellular complexes. If X is a smooth manifold, resp. a Stein or affine complex variety, then

ker{∇ : (X, V ⊗ Ωm) → (X, V ⊗ Ωm+1)} H m (X, L ) = X X , {∇ : ( , V ⊗ Ωm−1) → ( , V ⊗ Ωm )} im X X X X Ω∗ where X denotes the sheaf of smooth (resp. holomorphic or regular algebraic) differential forms on X. Using the cellular complex C∗(X, C) with a modified differential map depending on L to define the twisted cohomology groups H m(X, L ), one can see that the corresponding Euler characteristic satisfies the relation χ(X, L ) = (−1) j dim H j (X, L ) = χ(X), (5.10) j 92 5 Milnor Fibers and Local Systems

if rank(L ) = 1. A very useful result when dealing with twisted cohomology is the following.

f Theorem 5.2 (Leray spectral sequence) If F → E → B is a locally trivial ﬁbration, then there is an E2-spectral sequence:

p,q = p( , q L ) ⇒ p+q ( , L ). E2 H B R f∗ H E

q For any local system L on E, R f∗L is the local system on B whose stalk at the ∈ q ( , L ) point b B is given by H Fb |Fb . We will mainly apply this theorem in the following two situations. First, consider the (trivial) ﬁbration

C∗ → M(A ) → M(A ),

where A is a central hyperplane arrangement and A is the associated projective arrangement. For any rank one local system L on M(A ), there is a nonzero complex number ∗ T (L ) = ρ(σa) ∈ C ,

called the total turn of monodromy of L , where σa is just a positively oriented loop about the origin in the line C · a spanned by a point a ∈ M(A ). This number is independent of the choice of the point a, since the class [σa]∈H1(M(A ), Z) is clearly equal to [σ1]+···+[σd ] if d =|A |, in the notation of Corollary 4.8.Ifthe local system L corresponds to the collection of local monodromies λ = (λ1,...,λd ) with λ j =¯ρ(σj ), then it follows that

T (L ) = λ1 ...λd . (5.11)

Note that we have an exact sequence

∗ f# π1(C , a) → π1(M(A ), a) −→ π1(M(A )) → 1.

Using this sequence and the triviality of the ﬁbration under consideration, it follows that [σa] generates ker( f#). These remarks yield the following result.

Proposition 5.3 With the above notation, one of the following two disjoint cases occurs. (i) T (L ) = 1. This happens if and only if there exists a rank one local system L on M(A ) such that f ∗L = L , and then

H ∗(M(A ), L ) =∼ H ∗(M(A ), L ) ⊗ H ∗(C∗).

(ii) T (L ) = 1, and then H ∗(M(A ), L ) = 0. 5.2 Monodromy Eigenspaces and Twisted Cohomology 93

Proof In the ﬁrst case, use the Künneth Theorem and Proposition 2.1. In fact, if the rank one local system L on the projective complement M(A ) corresponds to a ∗ ∗ morphism ρ : π1(M(A )) → C , then the pull-back local system f L corresponds ∗ to the morphism ρ ◦ f# : π1(M(A )) → C . In the second case, note that T (L ) = 1 q ∗ implies R f∗L = 0 for any q, which in turn yields H (M(A ), L ) = 0.

The second situation when we use the Leray spectral sequence is the Galois covering p : F → M(A ), with p−1(b) a set of d points, which one may identify with the group of d-th roots of unity μd .LetγH be the loop around H in M(A ) based at b, for any H ∈ A . Then the monodromy of p with respect to γH is multiplication −1 0 by λ , with λ = exp(2πi/d),see[46]. Moreover, the stalk (R p∗CF )b can be regarded as the C-vector space of functions on μd , and these two facts imply the following.

Proposition 5.4 With the above notation, one has the following. 0 (i) R p∗CF = L0 ⊕ L1 ⊕···⊕Ld−1, with Lk the rank one local system on k M(A ) associated to the representation γH → λ for any H ∈ A . m m (ii) For any integers m and k, one has dim H (F)λk = dim H (M(A ), Lk ).

Note that the equalities in Proposition 5.4 (ii) and (5.10) give a new proof for Corollary 5.1. To study a non-proper, smooth, complex algebraic variety Y , for instance our complement M(A ), it is usual in Algebraic Geometry to use Hironaka’s embedded resolution of singularities and to construct a good compactiﬁcation (X, D) for Y , where X is a smooth, proper, algebraic variety and D is a normal crossing divisor in X such that X \ D is isomorphic to Y . For convenience, we recall here the deﬁnition of a (strict) normal crossing divisor.

Deﬁnition 5.3 A normal crossing divisor (for short NCD) D =∪i∈I Di in a complex manifold X of dimension n is a union of smooth irreducible hypersurfaces Di such that at any point p ∈ D, there exists a local system of coordinates (x1,...,xn) for X at p with D : x1 ···xk = 0forsomek ≤ n.

Example 5.2 To obtain a good compactiﬁcation for the complement M(A ), where A is a line arrangement in P2, it is enough to blow up once the points (edges) with multiplicities greater than or equal to 3. Note that these points correspond exactly to those edges X ∈ L(A ) such that the corresponding central arrangement AX is indecomposable, see Example 3.7.

To explain how to construct a good compactification for a projective hyperplane arrangement complement M(A ) in general, we first introduce a definition.

Deﬁnition 5.4 Let A be a central arrangement in Cn+1 with associated projective arrangement A in Pn. An edge X ∈ L(A ), resp. the associated edge X ∈ L(A ) for X = 0, is called dense if the central arrangement AX is indecomposable. 94 5 Milnor Fibers and Local Systems

To construct a good compactiﬁcation for the projective hyperplane arrangement complement M(A ), we proceed as follows. First we blow-up all the dense 0- ∈ (A ) : → Pn dimensional edges X0 L and get a morphism p1 Z1 . Then blow-up all ∈ (A ) the proper transforms under p1 of the dense 1-dimensional edges X1 L , and get a new morphism p2 : Z2 → Z1. We continue in this way until we get to the variety Zn−1, obtained by blowing-up the proper transforms in Zn−2 of the dense (n − 2)- ∈ (A ) : → . dimensional edges Xn−2 L , and get a new morphism pn−1 Zn−1 Zn−2 n We set X = Zn−1 and p : X → P given by the composition p1 · p2 · ...· pn−1. −1 Then D = p (∪H ∈A H ) is an NCD in X and p induces an isomorphism

X \ D =∼ M(A ). (5.12)

For L a rank one local system on M(A ) and for each dense edge Y in A ,we have a corresponding irreducible component DY of D (obtained by blowing-up Y as explained above) and a total turn monodromy TY (L ) associated to it. Indeed, let σY be an oriented loop around DY in X, then p(σY ) is an oriented loop in M(A ). We deﬁne ∗ ∗ TY (L ) =¯ρ(p([σY ])) ∈ C , where ρ¯ : H1(M(A ), Z) → C is the representation induced by the local system L .

Example 5.3 Assume that the projective line arrangement A is given by xyz(x + y) = 0. Then the mapping p constructed above is just the blow up of the unique point O = (0 : 0 : 1) ∈ P2 of multiplicity >2, i.e. in local coordinates p(u, v) = (u, uv). If σ is the oriented loop around the exceptional divisor E = p−1(O) : u = 0given by σ(t) = (exp(2πit), 1), then its image p(σ) is given by

p(σ)(t) = (exp(2πit), exp(2πit)).

This loop coincides with σa,fora = (1, 1), used in the general deﬁnition of the total turn monodromy in formula (5.11) applied to the local central arrangement AO in C2 consisting in the 3 lines passing through O. This implies that

TE (L ) = λ1λ2λ3, where λ1, resp. λ2, resp. λ3 are the monodromies of L about the lines x = 0, resp. y = 0, resp. x + y = 0.

These notions enter into the following very general vanishing result inspired by [159], see [49] for the proof involving perverse sheaves.

Theorem 5.3 Let L be a rank one local system on M(A ). Assume there is a hyperplane H ∈ A such that the corresponding local monodromy of L satisﬁes λH = 1 and for any dense edge Y ⊂ H, the total turn monodromy TY (L ) = 1. Then H m(M(A ), L ) = 0,form = n and dim H n(M(A ), L ) =|χ(M(A ))|.

Corollary 5.3 Let A be a line arrangement in P2 and let η = 1 be an eigenvalue of 1 1 1 1 the monodromy operator h : H (F) → H (F), in other words H (F)η = 0. Then 5.2 Monodromy Eigenspaces and Twisted Cohomology 95 the order o(η) divides d =|A |, and for any line L ∈ A , there is at least a point x ∈ L whose multiplicity in A is mx ≥ 3, with mx a multiple of o(η).

Proof This result follows from Theorem 5.3, since an intersection point x of A is a k dense edge if and only if mx ≥ 3. Then η = λ for some k = 1,...,d − 1, and the corresponding total turn monodromy is given by

k mx mx Tx (Lk ) = (λ ) = (η) .

Remark 5.4 Let A be a line arrangement in P2 deﬁned over R. Then η = 1 and 1 H (F)η = 0 imply that for any line L ∈ A , there are at least two points x, y ∈ L whose multiplicities in A are mx ≥ 3 and m y ≥ 3, with both mx and m y multiples of o(η). The proof of this interesting result, discovered by M. Yoshinaga and given in [233], follows completely new ideas, different from those used in the proof of Theorem 5.3.

Corollary 5.4 Let A be a line arrangement in P2. Assume there exists a line in A which contains only double points. Then the corresponding Milnor ﬁber F satisﬁes the equalities 1 1 1 H (F) = H (F)1 = H (M(A )).

Corollary 5.5 Let A be an arrangement of d lines in P2 having only double points and triple points as singularities. If 3 d, then h∗ = Id on H 1(F). Otherwise, the only possible eigenvalues of h∗ on H 1(F) are the cubic roots of unity: 1, ε and ε2, where ε = exp(2πi/3).

In the case of line arrangements in P2 having only double points and triple points, it was shown by S. Papadima and A. Suciu that the Betti numbers b j (F) and the monodromy action are determined by the intersection lattice L(A ), see Theorem 7.5.

Corollary 5.6 Let A be a general position arrangement in Pn. Then there are no n ∗ m dense edges in A and D =∪H∈A H is an NCD in P . Therefore h = Id on H (F) for m < n and − n d 2 dim H (F)η =|χ(M(A ))|= , n for any d-th root of unity η = 1.

Proof For the case m < n, we apply Theorem 5.3 directly. To obtain the last equality, we have to replace the projective arrangement A by the associated afﬁne arrangement in Cn having d − 1 hyperplanes and then use Corollary 3.6 and Example 2.19. For an alternative proof of this result, see [46].

Example 5.4 Consider the monomial line arrangement in P2 introduced in Example 2.24, also known under the name of the Ceva arrangement C(m), and given by 96 5 Milnor Fibers and Local Systems

A (m, m, 3) : Q = (xm − ym )(xm − zm)(ym − zm) = 0 for m ≥ 2. Then d =|A (m, m, 3)|=3m and any line L ∈ A (m, m, 3) contains one point p of multiplicity m p = m and in addition m triple points. It follows from 1 Corollary 5.3 that η = 1 and H (F)η = 0 imply that o(η) divides either 3 or m.Itis known that the cubic roots of unity occur as eigenvalues, see [165], and they are the only eigenvalues η = 1 for this arrangement, see Theorem 8.18. Note that Remark 1 5.4 does not imply that H (F)η = 0foro(η) > 3 a divisor of m, since the deﬁning equations of C(m) do not split into linear factors with real coefﬁcients. Indeed, for m ≥ 3, there are no double points in the line arrangement, and one may use the Sylvester-Gallai property in Theorem 1.5 to conclude.

Remark 5.5 Note that the Ceva arrangement C(3), introduced already in Example 1.6, is not supersolvable when considered as a central arrangement in C3. Indeed, the 1-ﬂats in the central arrangement correspond to the multiple points in the projective arrangement, which are the following 12 triple points: a group of nine points of the form (1 : α : β) with α3 = β3 = 1, which are in fact the base points of the corresponding pencil (see Deﬁnition 6.3 and the discussion following it), and a second group, formed by following three points

(1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1).

Let X be a 1-flat corresponding to a point in the first group, say to p = (1 : 1 : 1).If X corresponds to p = (1 : α : α2) for α = 1, then X + X ∈/ L(C(3)). Indeed, the line in P2 determined by p and p is not in C(3). Hence X is not modular. The same type of argument shows that none of the nine 1-flats associated to the nine points in the first group is modular. Let now Y be a 1-flat corresponding to a point in the second group, say to q = (1 : 0 : 0).IfY corresponds to q = (0 : 1 : 0), then the line determined by q and q is the line z = 0, and hence again it is not a line in C(3). In this way one shows that none of the three 1-flats associated to the three points in the second group are modular.

Example 5.5 Consider the full monomial line arrangement in P2 introduced in Exam- ple 2.23 and given by

A (m, 1, 3) : Q = xyz(xm − ym )(xm − zm)(ym − zm) = 0 for m ≥ 2. Then d =|A (m, 1, 3)|=3m + 3 and any line L ∈ A (m, m, 3), distinct from the coordinate axes x = 0, y = 0 and z = 0, contains one point p of multiplicity m p = m + 2 and m triple points. One of the coordinate axes, say x = 0, contains two points of multiplicity m + 2, namely (0 : 0 : 1) and (0 : 1 : 0), and m nodes, namely the points (0 : 1 : η) with ηm = 1. It follows from Corollary 5.3 that η = 1 1 and H (F)η = 0 imply that η is a cubic root of unity. It is known that the cubic roots of unity occur as eigenvalues for m ≡ 1 (mod 3),see[165] as well as Example 6.12 below. 5.2 Monodromy Eigenspaces and Twisted Cohomology 97

Many properties of the Milnor ﬁber F = F(A ) are different from the properties of the complement M(A ). We mention the following two. The ﬁrst one is due to G. Denham and A. Suciu, see [68].

Theorem 5.4 There are examples of Milnor fibers F with Tors(H∗(F, Z)) = 0.In particular, such Milnor fibers are not minimal. Another difference between hyperplane arrangement complements and their Mil- nor fibers is the following. Using Corollary 3.8 one can prove that the differential graded algebras (E ∗(M(A )), d) of complex-valued differential forms on M(A ) with the exterior differential of forms d and (H ∗(M(A ), C), d = 0) have the same minimal model and hence M(A ) is a formal space in the sense of Sullivan’s rational homotopy theory, see [90] for a brief discussion on these notions and references. On the other hand, it was shown by H. Zuber that the Milnor fiber of the monomial arrangement A (3, 3, 3) in Example 5.4 does not have such a formality property, see [241]. Remark 5.6 There are other very interesting and useful approaches to the compactification of an arrangement complement, working even in the more general case of subspace arrangements and leading to the so-called wonderful compactifications or wonderful models, see [56, 58, 237].

5.3 Exercises

n+1 Exercise 5.1 Let A : Q(x0,...,xn) = 0 be a central arrangement in C such that dim C(A ) = c, where C(A ) is the center of the arrangement A , as in Definition 2.2. Using the split formula (2.1), show that we can suppose, up to a linear change n+1 of coordinates on C , that Q does not depend on xn−c+1,...,xn. Deduce that c the Milnor fiber F : Q = 1 of the arrangement A has a product structure F1 × C , n−c+1 where F1 is the Milnor fiber of a central hyperplane arrangement in C , and this decomposition is compatible with the monodromy actions on F and F1.

Exercise 5.2 Show that for the Boolean arrangement A = Bn+1, the action of the monodromy on the cohomology, h∗ : H ∗(F) → H ∗(F), is the identity. Exercise 5.3 Let A be a central, essential, indecomposable arrangement in Cn+1, for some n ≥ 1. Show that the action of the monodromy on the cohomology, h∗ : H ∗(F) → H ∗(F), cannot be the identity. Exercise 5.4 Using Theorems 4.3 and 5.1, show the following. (i) Consider the central arrangement A in C3 deﬁned by xd + yd = 0. Let B be any central arrangement in C3 such that the intersection lattice L(B) is isomorphic to the intersection lattice L(A ). Show that the corresponding complements M(A ) and M(B) (resp. the corresponding Milnor ﬁbers F(A ) and F(B))are diffeomorphic. Compute the Alexander polynomials for the monodromy operator of the arrangement B. 98 5 Milnor Fibers and Local Systems

(ii) Consider the same question as above when the central arrangement A in C3 is deﬁned by (xd−1 + yd−1)z = 0.

Exercise 5.5 The Hessian arrangement is given by the equation A : Q = xyz (x3 + y3 + z3)3 − 27x3 y3z3 = 0.

(i) Show that Q is a product of linear forms, namely Q = xyz (ε j x + εk y + z), j,k=0,1,2

where ε = exp(2πi/3). (ii) Deduce that any line L ∈ A has two double points of A and three 4-tuple points of A . 1 (iii) Show that η = 1 and H (F)η = 0 imply that η is a root of unity of order 2 or 4. It is known that all these roots of unity occur as eigenvalues, see [32, Remark 3.3 (iii)] and Exercise 6.3 in this book. Chapter 6 Characteristic Varieties and Resonance Varieties

Abstract Characteristic varieties (resp. resonance varieties) are jumping loci for some cohomology groups which are topologically (resp. algebraically) defined. We explain the relation between the characteristic varieties and the homology of finite abelian covers. The polynomial periodicity properties of the first Betti numbers of such covers, and the smooth surfaces obtained as coverings of P2 ramified over a line arrangement, are also discussed. The main results in this chapter are the Tangent Cone Theorem explaining the close relation between the two types of jumping loci and the relation with the multinet structures introduced by M. Falk and S. Yuzvinsky. After a brief discussion of the translated components of the characteristic varieties, we treat in great detail the deleted B3-line arrangement.

6.1 Topological and Algebraic Jumping Loci

If X is a topological space and G = π1(X) denotes its fundamental group, then it is well known that H1(X, Z) = G/G , where G is the commutator subgroup of G.This homology group is much easier to compute than G itself, e.g. when X = M(A ) is a hyperplane arrangement complement. To study a better approximation of G, namely the quotient G/G, where G =[G, G] is the second derived subgroup of G, one introduces the characteristic varieties of X, which are determined by this quotient, see [94, Corollary2.5]. Resonance varieties are algebraic jumping loci, while characteristic varieties are topological jumping loci. The former are in a sense linear algebra approximations for the latter. Let A be an afﬁne arrangement in V = Cn with |A |=d. As we have seen in the previous chapter, a rank one local system L on M(A ) can be represented either ∗ by a representation ρ : π1(M(A )) → C , or by the induced group homomorphism ∗ ρ¯ : H1(M(A ), Z) → C .LetT (A ) be the set of isomorphism classes of rank one local systems on M(A ), which is equal to

∗ ∼ 1 ∗ ∼ ∗ d Hom(H1(M(A ), Z), C ) = H (M(A ), C ) = (C ) , an afﬁne d-dimensional torus.

j (A ) Deﬁnition 6.1 The (cohomological) characteristic variety Vk is the subset of the torus T (A ) given by

j (A ) ={L ∈ (A )| j ( (A ), L ) ≥ }. Vk T dimC H M k

j (A ) To be more precise, one may call Vk the characteristic variety of the arrangement A of degree j and depth k. There is also a homological version of these varieties, j where H (M(A ), L ) is replaced by H j (M(A ), L ). However, the vector space isomorphism ∨ ∼ j ∨ H j (M(A ), L ) = H (M(A ), L ) , (6.1) see for instance [77], p. 50, implies that the two types of characteristic varieties are transformed into one another by the involution of the torus T (A ) given by L → L ∨ = L −1, and hence enjoy similar properties. j (A ) (A ) Proposition 6.1 Vk is a closed algebraic subset of T . The idea of the proof given below goes back to Green and Lazarsfeld, see [125]. For a more general version of the result one can refer to [188, Lemma2.3].

Proof The regular functions on the afﬁne variety T (A ) are given by the Laurent C[ ±1,..., ±1] = (A ) polynomials in t1 td . To simplify the notation, set X M and consider the abelian universal covering of X, X˜ → X, corresponding to the kernel of ∼ d {π1(X) → H1(X, Z) = Z }. Then the group of deck transformations of this cover- Zd = C[Zd ]=C[ ±1,..., ±1] ing is and hence, the associated group ring R t1 td acts ˜ ˜ on the cellular chain complex C∗(X, C) of X with C-coefﬁcients. ˜ Note that in fact, Cm (X) = Cm(X) ⊗C R, because for each m-cell α in X, there are Zd cells α˜ in X˜ covering α. It follows that the differential ˜ ˜ ∂m : Cm(X) → Cm−1(X) is an R-linear map, given by a matrix Dm(t) = (aij(t)) with aij(t) ∈ R some Laurent polynomials. If L corresponds to a point λ = (λ1,...,λd ) ∈ T (A ), then

ker Dm(λ) Hm (X, L ) = , imDm+1(λ) where the linear map Dm (λ) : Cm (X) → Cm−1(X) is obtained from the matrix Dm(t) by replacing t by λ. It is clear that

dim Hm(X, L ) = dim Cm (X) − rank(Dm(λ)) − rank(Dm+1(λ)), and therefore,

dimC Hm (X, L ) ≥ k ⇔ rank(Dm(λ)) + rank(Dm+1(λ)) ≤ dim Cm (X) − k. 6.1 Topological and Algebraic Jumping Loci 101

m(A ) It follows that the homological characteristic variety Vk can be described as

m(A ) =∪ ∩ Vk a,b≥0,a+b≤dim Cm (X)−k Em,a Em+1,b where E p,q ={λ|rank(Dp(λ)) ≤ q}. It is clear that E p,q is given by the vanishing of all the q + 1 minors in the matrix Dp(t), and hence it is a closed algebraic subset of the afﬁne torus T (A ). Therefore the statement is proved for the homological characteristic varieties, and in view of our discussion above, see formula (6.1), for the cohomological characteristic varieties as well. Indeed, any regular involution takes algebraic sets to algebraic sets.

Remark 6.1 For any topological space X, having the homotopy type of a finite, connected CW-complex, one can define exactly as above the group T (X) of its characters, alias isomorphism classes of rank one local systems on X, which is ( ( , Z), C∗) =∼ 1( , C∗) m( ) equal to Hom H1 X H X and the characteristic varieties Vk X . The only difference occurs when there is torsion in H1(X, Z) since then T (X) has several connected components. The component T 0(X) containing the unit element 0 of this group is an affine torus of rank given by b1(X) and the quotient T (X)/T (X) is isomorphic to the torsion part of H1(X, Z). Many of the results discussed in this chapter hold in this more general setting, e.g. Proposition 6.1 holds true with essentially the same proof. Note that the proof of Proposition6.1 implies that two characters ρ and ρ of the space X which differ by an automorphism φ of C over Q, in the sense that ρ = φ ◦ρ, have the following property. The characters ρ and ρ have the same position with ρ ∈ m( ) ρ ∈ m( ) respect to the characteristic varieties, i.e. Vk X if and only if Vk X ,for any m and k.

Remark 6.2 The computation of the twisted homology using the cellular chain complex described in the proof above may be used in practice if the CW complex X, homotopy equivalent to the complement M(A ), has not too many cells and if the corresponding boundary maps are known. This approach for a complexiﬁed afﬁne arrangement in C2 is described in [121], where the complex X is obtained by applying discrete Morse theory to the Salvetti complex.

Remark 6.3 When a ﬁnite presentation of the fundamental group G = π1(X) is 1( ) given, one may compute the characteristic varieties Vj X using the Fox free differ- 1( ) ential calculus. In fact, the characteristic varieties Vj X may be identiﬁed with the determinantal varieties of the Alexander matrix AG of the group G. For details on this important relation, allowing many explicit computations, see [134, 157, 216].

Example 6.1 For any topological space X having the homotopy type of a ﬁnite, 0( ) ={ } L connected CW-complex, V1 X 1 , the trivial local system on X. Indeed, if is a rank one local system on X, then L is trivial if and only if H 0(X, L ) = 0. 0( ) =∅ ≥ Moreover, one clearly has Vk X if k 2. 102 6 Characteristic Varieties and Resonance Varieties

Example 6.2 In the case of a hyperplane arrangement A in Cn, when n = 1, then (A ) ∨ 1 1(A ) = M is homotopy equivalent to a bouquet of circles d S . It follows that V0 ...= 1 (A ) = (A ) Vd−1 T . Indeed, one has

dim H 0(M(A ), L ) − dim H 1(M(A ), L ) = χ(M(A )) = 1 − d, hence dim H 1(M(A ), L ) = dim H 0(M(A ), L ) + d − 1. It follows that in this 1(A ) ={ } 1(A ) =∅ > case one has Vd 1 and Vk if k d. Example 6.3 Let n = 2 and assume the line arrangement A is central. Then we have a homeomorphism M(A ) =∼ M(A ) × C∗, and any L ∈ T (M(A )) has a total turn monodromy T (L ) = λ1 ...λd . By Proposition5.3,ifT (L ) = 1, then H m(L ) = 0 for any m, while for T (L ) = 1, there exists a rank one local system L on M(A ) such that

H j (M(A ), L ) = H j (M(A ), L ) ⊕ H j−1(M(A ), L ) by Künneth’s theorem. It follows that

1(A ) ={λ ∈ (A )|λ ...λ = } =∼ (C∗)d−1, V1 T 1 d 1 a subtorus of T (A ). Moreover, exactly as in the previous example, one has

1(A ) = 1(A ) = ...= 1 (A ) =∼ (C∗)d−1, V1 V2 Vd−2

1 (A ) = 1(A ) = 1(A ) =∅ > Vd−1 Vd 1 and Vk for k d. Remark 6.4 Assume X has the homotopy type of a ﬁnite, 2-dimensional CW- complex, e.g. X = M(A ), with A a line arrangement in P2. If the Euler characteristic χ( ) 1( ) X is known, then the knowledge of all the characteristic varieties Vk X in degree 2( ) 1 yields complete information on the characteristic varieties Vj X in degree 2. To see this, recall the formula (5.10) and the fact that, for X connected, H 0(X, L ) = 0 for any rank one local system L = CX . We explain now the relation between the monodromy action on the Milnor ﬁber F : Q = 1 of a central hyperplane arrangement A : Q = 0inCn+1 and the j (A ) A characteristic varieties Vk of the corresponding projective arrangement in n ∗ P .Letd = deg Q =|A |, λ = exp(2πi/d) and let ρk : π1(M(A )) → C denote k the character sending any elementary loop γH around a hyperplane H ∈ A to λ . Using Proposition5.4, we see that

m m dim H (F, C)λk = depth (ρk ) (6.2) where by deﬁnition the degree m depth of a character η is given by

m(η) = { : η ∈ m( (A ))}. depth max s Vs M (6.3) 6.1 Topological and Algebraic Jumping Loci 103

Note that this relation can be restated in the following way, regarding H m(F, C) as a R = C[t]/(td − 1)-module, as in the discussion following Proposition 5.1. One has an isomorphism of R-modules

m m depth (ρd/k ) H (F, C) =⊕k|d (C[t]/Φk(t)) , (6.4) where Φk (t) denotes the k-th cyclotomic polynomial. It follows that to determine the monodromy action on H m (F, C) is equivalent to determining the position of the characters ρd/k with respect to the degree m characteristic varieties of M(A ). Now we pass to the definition of the resonance varieties, see [115]. Recall that by Theorem 3.5 we have an isomorphism of graded K -algebras H ∗(M(A ), K ) =∼ ∗ (A ) ω ∈ 1 (A ) AK , where K is any field. Let AK and define the corresponding Aomoto complex

ω∧ ω∧ ω∧ ( ∗ (A ), ω) : → 0 −→ 1 −→ ...−→ n (A ) → . AK 0 AK AK AK 0

Definition 6.2 The resonance varieties of the arrangement A over a field K are defined by

j (A ) ={ω ∈ 1( (A ), ) = 1 (A )| j ( ∗ (A ), ω∧) ≥ }. Rm K H M K AK dimK H AK m

j To be more precise, one may call Rm (A ) the resonance variety of the arrangement j A of degree j and depth m. Note that the resonance variety Rm (A )K is easy to compute in principle because it is determined by the intersection lattice L(A ).The following result is analogous to Proposition6.1 and has a similar proof.

j Proposition 6.2 The resonance varieties Rm (A )K are algebraic subsets in the affine 1 (A ) = d ω ∈ space AK K . Moreover, they are conical subsets, that is, for any j j Rm(A )K and any a ∈ K , one has aω ∈ Rm(A )K . j (A ) =∅ > j (A ) ∈ j (A ) Note also that clearly Rm if m dim AK and 0 Rm for all ≤ j (A ) m dim AK . Remark 6.5 For any topological space X having the homotopy type of a finite, connected CW-complex, one can define exactly as above the corresponding res- m ( ) onance varieties Rk X K , by replacing in the definition of the Aomoto complex ∗ ∗ the Orlik–Solomon algebra A (A )K by the cohomology algebra H (X; K ) with K -coefficients. In view of Theorem 3.5, this definition agrees with the previous one = (A ) 1 when X M . Note that the topological (resp. algebraic) jumping loci Vm (resp. 1 Rm) for a topological space of the considered type actually depend only on the fundamental group G = π1(X). This comes from the fact that, assuming for simplicity that X is a finite CW complex itself, the classifying space K (G, 1) is obtained from X by adding cells of dimension ≥ 3. It follows that the inclusion X → K (G, 1) induces isomorphisms H 1(X, C) = H 1(K (G, 1), C) and H 1(X, C∗) = H 1(K (G, 1), C∗) as well as an injection H 2(K (G, 1), C) → H 2(X, C). Moreover, for any rank 104 6 Characteristic Varieties and Resonance Varieties one local system L on X, there is a corresponding rank one local system L on K (G, 1) such that H 1(X, L ) = H 1(K (G, 1), L ). The direct consequence for 1 1 us of this observation is that, to study Vm and Rm of an arrangement complement M(A ), one can use Exercise 3.5 and Zariski’s Theorem 4.9 to consider only the case dim M(A ) = 2. This is done below, for instance in the discussion of multiarrange- ments.

Example 6.4 If n = 1, then H 0(M(A ), K ) = K , H 1(M(A ), K ) = K d and all cup products are trivial in H 1(M(A ), K ). It follows that one has the following for d ≥ 2 1(A ) = ...= 1 (A ) = 1( (A ), ) = d , R1 Rd−1 H M K K

1(A ) = 1 (A ) =∅ > Rd 0 and Rm ,form d. Example 6.5 Let n = 2 and assume A is central. Use Example3.2 to identify d d−1 the corresponding Aomoto complex with 0 → K → K → K → 0. Let ω = d λ { } 1 η = β ∈/ ω ω ∧ η = i=1 i ai , where ai is a basis of A .Let i ai K with 0. Then 0 = ∂(ω ∧ η) = λi η − βi ω, which implies λi = 0. For d ≥ 3, this implies 1(A ) = ...= 1 (A ) ={ω ∈ 1(A ) | λ = }, R1 Rd−2 A i 0

1 (A ) = 1(A ) = 1 (A ) =∅ > Rd−1 Rd 0, and Rm for m d. The following result explains the close relationship between characteristic and resonance varieties, and plays a central role in this theory.

Theorem 6.1 (Tangent Cone Theorem for M(A )) If K = C, and A is an arrangement in Cn, then the following hold for any ﬁxed integers j and m. j (i) The irreducible components W of the characteristic varieties Vm (A ) passing through 1 ∈ T (A ) are afﬁne subtori in the character torus T (A ). j (ii) The irreducible components E of the resonance varieties Rm(A ) are linear subspaces in H 1(M(A ), C). Moreover, for j = 1, if E and E are two such irreducible components, then either E = E or E ∩ E = 0. (iii) The exponential map

exp : Cd = H 1(M(A ), C) → (C∗)d = T (A ) = H 1(M(A ), C∗),

taking (α1,...,αd ) to (exp(−2πiα1),...,exp(−2παd )), sends each irreducible component E into an irreducible component W = exp(E) and induces a 1-1 correspondence between the two sets of irreducible components listed above. 6.1 Topological and Algebraic Jumping Loci 105

(iv) If W = exp(E) and L ∈ W, one has

dim H 1(M(A ), L ) ≥ m = dim E − 1 = dim W − 1

and this inequality is actually an equality, for all L ∈ W, except ﬁnitely many local systems.

Example 6.6 Let n =2 and assume the hyperplane arrangement A is central. Then ={ α = } 1( (A ), C) the hyperplane E i 0 in H M is taken, by the exponential map, 1 ∗ to the subtorus W ={ λi = 1} in H (M(A ), C ). Compare this with Examples 6.3 and 6.5. The first proof of Theorem6.1 was given in [48]. The multinet structures as in The- orem 6.6 or the more general Theorem 6.7 below can be used to obtain the result for 1( (A )) 1( (A )) V1 M and R1 M . The latter approach also shows that the claim (i) holds for any quasi-projective smooth variety X when j = 1, see [91] for a proof in the case j > 1. Theorem 6.1 (ii) does not apply for a smooth non-proper algebraic variety X,e.g. for the configuration space X of n ≥ 3 points on an elliptic curve C,see[93]. In fact, 1( ) in this case, the resonance variety R1 X is irreducible but not linear. In general, the 1( ) 1( ) tangent cone to V1 X is contained in R1 X for a smooth algebraic variety X,as proved by A. Libgober [158]. See also Remark6.12 (ii) below. Another proof of claim (iii) in Theorem 6.1 is given below in Theorem 7.3. Similar results hold for compact Kähler manifolds and for a quasi-projective smooth variety X such that H 1(X, Q) is a pure Hodge structure of type (1, 1),see[93]. For an arbitrary quasi-projective smooth variety X, the result continues to hold if we modify the definition of the resonance varieties, as explained in [91]. Finally in this section we briefly discuss the relation between characteristic varieties and the homology of finite abelian covers in some special cases. We refer to Libgober [155, 157], Hironaka [134], Sakuma [200] and Suciu [216] for the general case, more details and proofs. Consider a space X having the homotopy type of a b finite CW-complex and assume in the sequel that H1(X, Z) = Z for some b > 0. Let p be a prime number and let μp denote the cyclic group of p-th roots of unity in C. A Galois regular cover X → X with deck transformation group μp is determined by the kernel of a surjective group morphism ρ : π1(X) → μp,see[130, 211]. Such a morphism gives rise to a torsion element of order p, denoted again by ρ,inthe character group T (X). Then one has the following, see [167]. Proposition 6.3 With the above notation, one has

b1(X ) = b + (p − 1) depth(ρ),

(ρ) = { : ρ ∈ 1( )} where depth max s Vs X . Example 6.7 Consider a central line arrangement A in C2.IfwesetX = M(A ), then using Example 6.3 we conclude that for any Galois cover X → X of order p (a prime number), one has 106 6 Characteristic Varieties and Resonance Varieties

b1(X ) = d + (p − 1)(d − 2), where d =|A |.

Example 6.8 Consider a central line arrangement A in C3 such that p =|A | is a prime number. If we set X = M(A ), where A is the associated line arrangement in P2, then the Milnor ﬁber F of the arrangement A is a Galois cover of X of order p, corresponding to an obvious character ρ. Proposition6.3 yields in this case

b1(F) = p − 1 + (p − 1) depth(ρ), which is equivalent to the formula (6.4)form = 1 and d = p. Indeed, in this formula, the integer k can take only two values: k = p and then ρ1 = ρ, and k = 1 when ρp = 1, the trivial character. The claim follows using the fact that deg Φp = p − 1.

The congruence cover X N of a space X as above is the Galois covering with structure group (Z/NZ)b determined by the kernel of the composition

π1(X) → H1(X, Z) → H1(X, Z/NZ).

For this important class of ﬁnite abelian covers, we have the following result, see [134, 155, 157, 200]. Theorem 6.2 With the above notation, one has b1(X N ) = b + depth(ρ),

ρ∈TN (X)

N where TN (X) denotes the set of characters ρ = 1 in T (X) such that (ρ ) = 1. Example 6.9 Consider a central line arrangement A in C2.IfwesetX = M(A ), then using Example 6.3 we conclude that for any N > 0 one has

d−1 b1(X N ) = d + (d − 2)(N − 1), where d =|A |.

One important property of the sequence of ﬁrst Betti numbers b1(X N ) of congruence covers of X is polynomial periodicity, see [200, 203]. More precisely, the following holds.

Theorem 6.3 For every ﬁnite CW-complex X, there exists an integer T > 0 and polynomials P1(t),...,PT (t) ∈ Q[t] such that one has

b1(X N ) = Pi (N) if N ≡ imodT. 6.1 Topological and Algebraic Jumping Loci 107

For an interesting example of the computation of the period T and of the polynomials Pi in the case of a line arrangement complement, see Corollary6.5 and further on in this chapter. Let now A be a line arrangement in P2 and consider its complement M(A ).If d−1 d =|A |, then H1(M(A ), Z) = Z by Proposition4.1.LetMN (A ) → M(A ) be the N-th congruence cover of M(A ), and extend it to the associated branched ˆ 2 2 ˆ cover MN (A ) → P of P . Then MN (A ) is a normal surface, and by definition the Hirzebruch, or Hirzebruch–Kummer, covering surface SN (A ) is the minimal ˆ 2 desingularization of MN (A ). This surface comes with a covering map SN (A ) → P , which is ramified of order d about each line in the line arrangement A ,see[135]. d−1 Recall that H1(M(A ), Z) = Z is generated by the loops σi , which are the meridians associated to the lines Li in the arrangement, for i = 1, 2,...,d.Fora character ρ ∈ T (M(A )), we define a subarrangement A (ρ) ⊂ A by setting

A (ρ) ={Li ∈ A :˜ρ(σi ) = 1},

∗ where ρ˜ : H1(M(A ), Z) → C is the morphism induced by ρ. The obvious inclusion M(A ) → M(A (ρ)) gives rise to an epimorphism

π1(M(A )) → π1(M(A (ρ))).

Using the deﬁnition of the subarrangement A (ρ), it follows that the character ρ admits a unique extension to a character ρ[A (ρ)]∈T (M(A (ρ))). This construction enters into the following result, giving a formula for b1(SN (A )),see[200]. For the second claim, see [132, 200].

Theorem 6.4 With the above notation, one has b1(SN (A )) = depth(ρ[A (ρ)]).

ρ∈TN (M(A ))

Moreover, the sequence b1(SN (A )) is polynomially periodic, for any line arrangement A in P2.

One example of computation using this formula is given in Corollary6.6. Besides the Hirzebruch covering surfaces SN (A ), there are other interesting surfaces constructed as coverings of P2 branched over a line arrangement, for instance the Burniat surfaces. For more on this very beautiful area we refer to [21, 22, 35, 42, 133, 146, 166, 220]. For the study of the Milnor ﬁber F associated to a line arrangement A in P2, one considers cyclic coverings of P2 which contain the Milnor ﬁber as a Zariski open subset, see the proof of Theorem 7.7 and Remark7.4 further on in this book, as well as Zhenjian Wang thesis [228].

Remark 6.6 For a free abelian cover X → X coming from an epimorphism r π1(X) → Z for some r > 0, one can express the ﬁniteness of the dimension of the k-th truncated homology 108 6 Characteristic Varieties and Resonance Varieties H≤k (X , C) = Hi (X , C) i=0,k

∪ i ( ) of X in terms of the union of characteristic varieties i=0,k V1 X . See [218]for more details and a lot of other applications.

Remark 6.7 Cyclic coverings of Pn ramiﬁed over a hyperplane arrangement in general position have been considered in [122] to construct interesting Calabi– Yau threefolds, and in [230] to construct families of algebraic varieties with large monodromy groups.

6.2 Jumping Loci and Pencils of Plane Curves

There is a geometric method to obtain more information on the irreducible compo- 1 1 = A nents of the jumping loci Vm and Rm . Assume n 3 and is a central arrangement in C3, with associated line arrangement A in P2. Suppose given a regular mapping

f : M(A ) → P1 \{k points}=:S,

−1 with k ≥ 3, with connected generic ﬁber Fb = f (b). Lemma 6.1 For any a ∈ Fb, the morphism f# : π1(M(A ), a) → π1(S, b) is surjective.

Proof Since f is a locally trivial fibration except over finitely many points in S,we may assume that b is not such a bifurcation point. Then we can represent any loop in π1(S, b) by a loop σ in S avoiding these bifurcation points. Such a loop can be lifted to M(A ) and yields a path σ˜ whose starting point, say a, and ending point, say a , are in the fiber Fb. Since this fiber is path-connected, we can extend σ˜ to get a closed path in π1(M(A ), a) just by picking any path in Fb from a to a. Corollary 6.1 With the above notation, one has the following. (i) The induced morphism f∗ : H1(M(A ), Z) → H1(S, Z) is surjective. (ii) The induced morphism f ∗ : H 1(S, C) → H 1(M(A ), C) is injective and

= ∗( 1( , C)) ⊂ 1 (A ) E f f H S Rk−2

1 is a linear subspace with dim E f = dim H (S, C) = k − 1. (iii) The induced morphism f ∗ : H 1(S, C∗) → H 1(M(A ), C∗) injective, and = ∗( ( )) ⊂ 1 (A ) ( − ) W f f T S Vk−2 is a k 1 -dimensional subtorus. ⊂ 1 (A ) Proof The only claim needing some justiﬁcation is the inclusion E f Rk−2 . This comes from the fact that the restriction of the cup product of H 1(M(A ), C) to 1 E f can be identiﬁed with the cup product on H (S, C), which is trivial. 6.2 Jumping Loci and Pencils of Plane Curves 109

Remark 6.8 To compare the dimensions of H 1(S, L ) and H 1(M(A ), f ∗(L )) for a local system L ∈ T (S) = H 1(S, C∗) is a very interesting and subtle question, see [78] as well as our discussion below in Theorem6.9.

The main question is, how do we obtain such maps f : M(A ) → S?Noticethat ∈ A ≥ ( , ) any point p of multiplicity k 3 gives rise to a pair E f p W f p as above, by 2 1 taking f p to be the restriction to M(A ) of the central projection πp : P \{p}→P of center p. To generalize this observation, Falk and Yuzvinski introduced multinets, deﬁned as follows, see [117]. Deﬁnition 6.3 A multiarrangement is a pair (A , m), where A is a line arrangement 2 in P and m : A → Z>0 is a function, H → m H , the multiplicity of H.A (k, d)-multinet is a multiarrangement (A , m) such that there is a partition A = A ∪ ...∪ A ≥ A 1 k , with k 3, and a set X of multiple points in , satisfying the following conditions. (i) The sum ∈A m H = d is independent of i, i.e. all subarrangements A have H i i the same degree, denoted by d. ∈ A ∈ A = ∩ ∈ (ii) If H i , H j with i j, then H H X. ∈ = (iii) For each p X,thesum H∈A ,p∈H m H n p is independent of i, i.e. the i A point p has the same multiplicity in all the subarrangements i . = ,..., A (iv) For any i 1 k and any two lines H, H in i , there is a chain

H = H1,...,Hs = H

A H ∩ H + ∈/ X j (∪ ∈A H) \ X of lines in i such that j j 1 for all , i.e. the curve H i is connected. (v) G.C.D.(m H )H∈A = 1. (This is a minimality condition, which may be fulﬁlled by dividing by G.C.D.(m H )H∈A if necessary.) (vi) If all m H = 1, then the (k, d)-multinet is said to be reduced. If, furthermore, n p = 1 for any point p ∈ X,the(k, d)-multinet is called a (k, d)-net. m H Let Qi = ∈A l be a homogeneous polynomial in x, y, z of degree d,for H i H i = 1,...,k, deﬁning the multiarrangement Ai . Consider the linear span

P =< Q1, Q2,...,Qk >,

A = A = and assume it has dimension 2. Since we suppose i j for i j,itfollows that any pair (Qi , Q j ) can be chosen as a basis of P. In particular, one can write

Q j = u j Q1 + v j Q2 with u j , v j ∈ C for j = 3,...,k. For any such pencil P, we get a regular map

1 fP : M(A ) → S = P \{k points},(x : y : z) → (Q1(x, y, z) : Q2(x, y, z)), 110 6 Characteristic Varieties and Resonance Varieties where the deleted points are exactly the points (1 : 0), (0 : 1) and (v j :−u j ) for j = 3,...,k. Note that the set X in Definition6.3 is exactly the base locus of the 2 associated pencil, i.e. it is defined by the equations Q1 = Q2 = 0inP . The geometric meaning of a multinet is revealed in the following result of Falk and Yuzvinsky, see [117]. (A , ) = (A ∪...∪A , ) Theorem 6.5 The multiarrangement m 1 k m with a partition satisfying (i) in Definition6.3 is a (k, d)-multinet if and only if the linear span P of the degree d polynomials Q1,...,Qk is two-dimensional and the generic fiber of the associated mapping fP is connected. (A , ) = (A ∪ ...∪ A , ) ( , ) In other words, m 1 k m is a k d -multinet if and only A P if the k subarrangements i occur as special fibers in a pencil of plane curves of degree d, up to some multiplicities, and the generic member of the pencil P is irreducible. (A , ) = (A ∪ ...∪ A , ) Remark 6.9 Consider a multinet m 1 k m and the associated pencil P of plane curves as above, with base locus X. Then the following conditions are clearly equivalent. (A , ) = (A ∪ ...∪ A , ) (i) The multinet m 1 k m is a net. (ii) The generic member of the associated pencil P is smooth. (iii) Any member of the associated pencil P is smooth at any base point p ∈ X.

Example 6.10 The following example of a reduced (3, 4)-multinet which is not a net was given in [117]. Consider the line arrangements in P2 given by the following equations

A : = ( + )( + )( − + )( + − ) = , 1 Q1 x 4z x 2z y x 3z y x 3z 0 A : = ( − )( − )( − − )( + + ) = 2 Q2 x 4z x 2z y x 3z y x 3z 0 and A : = ( + )( − ) = . 3 Q3 xz y z y z 0

Then it is easy to check that Q1 − Q2 = 12Q3, and hence we have indeed a reduced multinet (or reduced pencil) by taking the union of all these 12 lines. But n p = 2 for p = (0 : 1 : 0) ∈ X, hence this is not a net. The afﬁne picture of this multinet A is represented in Fig. 6.1, where the lines in 1 are drawn as continuous lines, those A A = in 2 as dotted lines and those in 3 as dashed lines, with the line at inﬁnity z 0 not drawn. This arrangement has 12 double points, 13 triple points and one point of multiplicity six.

The motivation for the introduction of the (k, d)-multinets is the following key result, see [117]. 6.2 Jumping Loci and Pencils of Plane Curves 111

Fig. 6.1 A reduced multinet which is not a net

1(A ) Theorem 6.6 Any irreducible component of R1 is obtained as a pull-back as in Corollary6.1 (ii) by using either a map f p coming from a point p ∈ A of multiplicity k ≥ 3 (the corresponding components are called local components) or a map fP coming from a multinet structure on a subarrangement of A (and the corresponding components are then called global components).

Example 6.11 Consider the monomial (alias Ceva) line arrangement in P2 given by

A (m, m, 3) = C(m) : Q = (xm − ym )(xm − zm )(ym − zm) = 0

m m m m for m ≥ 2. This is clearly a (3, m)-net, by choosing Q1 = x − y , Q2 = x − z m m and Q3 = y − z . Note that the set X in Deﬁnition6.3 is exactly the base locus of the associated pencil P. The arrangement A (2, 2, 3) is pictured in Fig. 2.6, where the pair of lines belonging to the same member of the pencil are drawn in the same style.

Example 6.12 Consider the full monomial line arrangement in P2 given by

A (m, 1, 3) : Q = xyz(xm − ym )(xm − zm)(ym − zm) = 0

m m m for m ≥ 2. This is clearly a (3, 2m)-multinet if we choose Q1 = z (x − y ), m m m m m m Q2 = y (x − z ) and Q3 = x (y − z ). Consider the associated morphism f : M(A (m, 1, 3)) → S, where S = P1 \{(1 : 0), (0 : 1), (1 : 1)}, given by

(x : y : z) → (zm(xm − ym ) : ym (xm − zm)).

Let L1 be the rank one local system on S with monodromy ε = exp(2πi/3) around ∗ any of the three punctures. Then L = f (L1) is a rank one local system on M(A (m, 1, 3)) with monodromy εm about the lines x = 0, y = 0 and z = 0, 112 6 Characteristic Varieties and Resonance Varieties and ε about the other lines. It follows that for m ≡ 1 (mod 3), the local system L coincides with the local system Lm+1 introduced in Proposition5.4, and hence

1 1 dim H (F, C)ε = dim H (M(A (m, 1, 3) ), Lm+1) = 0, by Proposition5.4 and Corollary6.1. Example 6.13 The Hessian arrangement is given by the equation A : Q = xyz (x3 + y3 + z3)3 − 27x3 y3z3 = 0.

3 3 3 j This is a (4, 3)-net if we choose Q0 = xyz and Q j = x + y + z − 3θ xyz,for j = 1, 2, 3, and θ = 1 a cubic root of unity, e.g. θ = ε = exp(2πi/3) as above. Remark 6.10 (i) It is known that a pencil of plane curves can have at most four completely reducible fibers, i.e. fibers corresponding to line arrangements in P2, possibly with some multiplicities. In fact, the Hessian arrangement is the only known (and conjecturally the unique) such pencil with four completely reducible fibers, see [191, 238]. (ii) A classification of projective line arrangements of low degree coming from pencils can be found in [84]. Remark 6.11 The same line arrangement A can admit several multinet (or even net) structures, giving rise to distinct irreducible components of the resonance variety 1(A ) (A ) R1 . An interesting upper bound for the number Essk of essential irreducible 1(A ) components of the resonance variety R1 coming from k-net structures on a line arrangement A ,fork = 3, 4, is given in [184, Theorem 1.5]. Example 6.14 The Ceva arrangement C(3) = A (3, 3, 3) from Example6.11 has the following four 3-nets structures:

[(L1, L2, L3), (L4, L5, L6), (L7, L8, L9)], [(L1, L4, L7), (L2, L5, L8), (L3, L6, L9)],

[(L1, L5, L9), (L2, L6, L7), (L3, L4, L8)], [(L1, L6, L8), (L2, L4, L9), (L3, L5, L7)].

Here we use the following notation for the lines in this arrangement L1 : x − y = 0, 2 2 L2 : x −ey = 0, L3 : x −e y = 0, L4 : y−z = 0, L5 : y−ez = 0, L6 : y−e z = 0, 2 L7 : z − x = 0, L8 : z − ex = 0 and ﬁnally L9 : z − e x = 0, with e = exp(2πi/3). This example is also discussed in [219, Example2.12]. An interesting question is how to ﬁnd all the multinets for a given line arrangement A . Here is one possible approach.

Deﬁnition 6.4 Let V1 and V2 be two vector spaces over a ﬁeld K with a bilinear mapping ·, · : V1 × V1 → V2 which is alternating. Let W be a subspace of V1. Then W is said to be isotropic with respect to ·, · if

v, w=0, ∀ v, w ∈ W. 6.2 Jumping Loci and Pencils of Plane Curves 113

W is called a maximal isotropic subspace if it is isotropic and not strictly contained in another isotropic subspace.

1(A ) Corollary 6.2 The irreducible components of R1 are precisely the maximal isotropic subspaces E ⊂ H 1(M(A ), C), with respect to the cup product

∪:H 1(M(A ), C) × H 1(M(A ), C) → H 2(M(A ), C) and such that dim E ≥ 2.

1(A ) Proof Let E be an irreducible component of R1 . By Theorem 6.6 we can write = ∗( 1( , C)) : (A ) → E fE H S , where fE M S is a regular mapping as in Corollary6.1. It follows that E is isotropic with respect to the cup product, since the cup product on H 1(S, C) is trivial. Maximality of E comes from the fact that E is an irreducible 1(A ) ≥ ≥ component of R1 . The restriction dim E 2 comes from the condition k 3 in Theorem 6.6. A mapping fE as above is said to be associated to the subspace E. Conversely, if E ⊂ H 1(M(A ), C) is a maximal isotropic subspace with respect ≥ ⊂ 1(A ) to the cup product and such that dim E 2, it clearly follows that E R1 . 1(A ) Since the irreducible components of R1 are isotropic subspaces by the previous discussion, it follows from the maximality of E that E itself is such an irreducible component.

Remark 6.12 (i) By Theorem 6.6, any maximal isotropic linear subspaces E ⊂ H 1(M, C) is rationally deﬁned, i.e. there is a linear subspace

∗ 1 1 EQ = f (H (S, Q)) ⊂ H (M, Q)

1 1 such that E = EQ ⊗Q C under the identiﬁcation H (M, C) = H (M, Q) ⊗Q C. (ii) Corollary6.2 implies the second part of claim (ii) in Theorem6.1. Indeed, if v ∈ E ∩ E and v = 0, then it follows that v ∧ w = 0 for any w = u + u with ∈ ∈ + ⊂ 1(A ) u E and u E . It follows that E E R1 , and the maximality of E and E implies E = E + E = E.

Note that this property fails for j > 1 as the following example shows. Consider the central hyperplane arrangement A in C4 given by

Q = xyzw(x + y + z)(y − z + w) = 0. (6.5)

Then by a direct computation using the Orlik–Solomon algebra A∗(A ),itfollows R2(A ) that the resonance variety 1 consists of two 3-dimensional components

E1 : x1 + x2 + x3 + x6 = x4 = x5 = 0 and 114 6 Characteristic Varieties and Resonance Varieties

E2 : x2 + x3 + x4 + x5 = x1 = x6 = 0

(the hyperplanes are numbered according to the position of the corresponding factor 1 in the product (6.5) and x j is the coordinate on H (M(A ), C) associated with the hyperplane H j ). It follows that the intersection E1 ∩ E2 is 1-dimensional.

Remark 6.13 If E is a maximal isotropic subspace in H 1(M, C) of dimension at least 1(A ) two, then it has a unique associated component WE in V1 corresponding to E under the bijection in Theorem 6.1, and such that 1 ∈ WE .IfE is a rationally deﬁned maximal isotropic subspace in H 1(M, C) of dimension 1, then there is no associated 1(A ) ∈ component WE in V1 with 1 WE . But such a subspace may be associated to a translated component of the characteristic variety, as discussed below.

Remark 6.14 If B ⊂ A is a proper subarrangement, the inclusion M(A ) → M(B) induces a morphism H ∗(M(B), C) → H ∗(M(A ), which is injective when ∗= 1 (B) → 1 (A ) 1. This yields an embedding Rm Rm . An irreducible component of 1 (A ) Rm which is in the image of such an embedding is called non-essential. The 1 (A ) other irreducible components of Rm are called essential.

6.3 Translated Components in the Characteristic Varieties

We discuss in this section some properties of the characteristic varieties in a more general context, namely for a connected smooth quasi-projective complex variety M, essentially following [78].

Deﬁnition 6.5 Let M be a connected quasi-projective smooth variety. An irreducible m( ) ∈/ component W of a characteristic variety Vk M is said to be translated if 1 W. The following result is due to several authors, depending on the setting: projective or quasi-projective, algebraic or Kähler, see (Arapura [11], Bauer [20], Beauville [23], Budur and Wang [34], Campana [39] and Simpson [210]). Theorem 6.7 Let W be an irreducible component in the characteristic variety 1( ) > ρ ∈ ( ) V1 M with dim W 0. Then there exist a torsion character T M and a surjective morphism f : M → S, with connected generic ﬁber F, such that W = ρ · f ∗(T (S)). Here S is a smooth connected algebraic curve, and W is a translated component if and only if ρ is not induced by a character in T (S). In spite of some progress, see for instance [14], the following question seems to be still quite open.

1( ) Question Find a (similar) geometric description for the isolated points in V1 M , i.e. for the 0-dimensional irreducible components. 6.3 Translated Components in the Characteristic Varieties 115

The relation between f and ρ in the previous theorem can be further analysed. Let C( f ) be the minimal bifurcation set for f , i.e. the minimal subset of the curve S such that F → M → S is a topological locally trivial fibration, with fiber F, base space S = S \ C( f ) and total space M = f −1(S). Then C( f ) is a finite set and we have the following short exact sequence π1(F) → π1(M ) → π1(S ) → 0, coming from the long exact sequence of homotopy groups of a fibration, see [130, 211]. This yields, by looking at the corresponding abelianizations, a new exact sequence for the integral homology groups

H1(F, Z) → H1(M , Z) → H1(S , Z) → 0.

However, the sequence of integral homology groups

i∗ f∗ H1(F, Z) −→ H1(M, Z) −→ H1(S, Z) → 0 may not be exact. The group T ( f ) = ker( f∗)/im(i∗) measures how far the above sequence is from being exact and it is computable in terms of the multiple fibers of f , as we explain now. If c ∈ C( f ), then one can write the following equality of divisors q −1 Fc = f (c) = ai Ci , i=1 where the ai ’s are strictly positive integers and the Ci ’s are some irreducible hypersurfaces in M. Denote G.C.D.(a1,...,aq ) by mc. The fiber Fc is said to be multiple if mc > 1. The following result is due to Serrano [207], with a correction in the case when S is affine given in [78]. Theorem 6.8 Consider a surjective morphism f : M → S from the connected quasi-projective smooth variety M onto the smooth curve S, with connected generic fiber F. Then the following hold. (i) If the curve S is proper, then

⊕ ∈ ( )Z/m Z T ( f ) = c C f c , (1¯,...,1¯) ¯ where 1 denotes the class of 1 in the various cyclic groups Z/mcZ. 116 6 Characteristic Varieties and Resonance Varieties

(ii) If the curve S is not proper, then T ( f ) =⊕c∈C( f )Z/mcZ. Therefore, we have the following exact sequence of abelian groups, regarded as Z-modules f∗ 0 → T ( f ) → H1(M, Z)/Im(i∗) −→ H1(S, Z) → 0.

∗ Applying the functor Hom(−, C ) and using the fact that H1(S, Z) is a free Z- module, we get an exact sequence

f ∗ 1 → T (S) −→ T (M)F → T ( f ) → 1. (6.6)

∗ If we set T ( f )F := Hom(H1(M, Z)/Im(i∗), C ), then, for a rank one local system ∗ L , one has L ∈ T ( f )F if and only if L ∈ T (M) = Hom(H1(M, Z), C ) and the restriction L|F of L to the generic ﬁber F is the trivial local system CF .For the following result, we refer to [78], while the reader unfamiliar with constructible sheaves can have a look at [77].

Theorem 6.9 Let L1 ∈ T (M) be a rank one local system and consider the con- 0 structible sheaf F = R f∗(L1) on S. Let = (F ) be the singular support of F (i.e. the set of those points of S in the neighborhood of which F is not a local system). Then either ∼ (i) L1|F = CF , F |S\ is a rank one local system and Fs = 0 if and only if s ∈ , or (ii) L1|F CF and F = 0.

Moreover, if S is afﬁne, then for any rank one local system L2 ∈ T (S), the sequence

1 1 ∗ 0 1 0 → H (S, F ⊗ L2) → H (M, L1 ⊗ f L2) → H (S, R f∗L1 ⊗ L2) → 0

is exact and the last group is trivial for all L2 ∈ T (S), with ﬁnitely many exceptions. ∼ Corollary 6.3 If L ∈ T (M) satisﬁes L |F = CF , then

1 ∗ dim H (M, L ⊗ f L2) ≥−χ(S) +||,

with equality for all rank one local systems L2 ∈ T (S), except for ﬁnitely many. 1( ) Moreover, any strictly positive dimensional component W of V1 M is obtained in this way.

Remark 6.15 Note that exceptional local systems L2, for which the inequality above is strict, may exist even when =∅. See Example7.7 below for the case when L = CM and f : M → S is a locally trivial ﬁbration, i.e. the bifurcation set C( f ) is empty. In fact, in this example, M = M(A ) where A is the Ceva line arrangement A (3, 3, 3) in P2. A similar situation occurs for any monomial line arrangement A (m, m, 3), with m a multiple of 3, in view of Theorem 8.16 below. The fact that f : M → S is a locally trivial ﬁbration, which clearly implies that =∅for 6.3 Translated Components in the Characteristic Varieties 117

0 F = R f∗(CM ), can be proved for the line arrangements above using the following two remarks. (i) All the singular members in the corresponding pencil are in the arrangement. (ii) Any two members in the corresponding pencil meet transversally at any point in the base locus. For more details we refer to [117].

Note that a locally trivial ﬁbration f : M → S, with M a surface, implies that πi (M) = 0 for all i > 1, in other words in such a case M is a classifying space K (G, 1). The above discussion implies the following, in view of Theorem4.16 and Remark5.5.

Example 6.15 The Ceva line arrangement A = C(3), regarded as a central arrangement in C3, is not supersolvable, and hence not of ﬁber type. Nevertheless, A = C(3) is a K (π, 1)-arrangement.

Recall the exact sequence of abelian groups (6.6). It follows that the irreducible ∗ component W = ρ · f (T (S)) is determined by the class ρˆ of ρ ∈ T (M)F in the group T( f ).

Deﬁnition 6.6 For a character class ρˆ ∈ T( f ), deﬁne the support supp(ρˆ)ofρˆ to 0 be the singular support of the constructible sheaf F = R f∗(Lρ ), where ρ is any representative of the class ρˆ.

For the following result, we refer to [78]. Theorem 6.10 With the above notation supp(ρ)ˆ =∅if and only if ρˆ = 1.In particular, for a translated component W associated to f : M → S, one has

dim H 1(M, L )>−χ(S),

for all L ∈ W.

Corollary 6.4 Let f : M → S be surjective, with connected generic ﬁber. Then one has the following. χ( )< 1( ) (i) If S 0, then the irreducible components of V1 M associated to f are in bijection to T( f ). (ii) If χ(S) = 0, e.g. S is C∗ or a genus one projective curve, then the irreducible 1( ) ( ) \{ } components of V1 M associated to f are in bijection to T f 1 . Finally, we mention a key result of Artal Bartolo, Cogolludo, and Matei, [14, Proposition6.9]. 118 6 Characteristic Varieties and Resonance Varieties

Theorem 6.11 Let X be a smooth, quasi-projective variety. Suppose V and W are 1( ) 1( ) two distinct, positive-dimensional irreducible components of Vp X and Vq X , L ∈ ∩ L 1 ( ) respectively. If V W, then is a torsion local system in Vp+q X . The torsion part of the claim also follows from [175].

6.4 The Deleted B3-Line Arrangement

We end our discussion with a rather long but very instructive example, as it illustrates well all the notions introduced in this chapter.

Example 6.16 Consider the deleted B3-arrangement dA in Example2.8. To determine the corresponding resonance variety, we ﬁrst need to identify the cup product on H 1(M(dA ), C). This is the same as the cup product on A1(dA , C), which is given in Example3.5, see Table3.1. To calculate all the maximal isotropic subspaces inside H 1(M(dA , C) = A1(dA , C), we proceed as follows. Let v be a non-zero vector 1 from A (dA , C) and set v = x1a1 + x2a2e2 +···+x7a7. Consider its orthogonal space ⊥ 1 v ={w = y1a1 +···+y7a7 ∈ A (dA , C) : v ∧ w = 0}.

Now, using Table3.1, it follows that

v ∧ w = (x1 y2 − x2 y1 − x2 y3 + x3 y2)a12 + (x1 y3 − x3 y1 + x2 y3 − x3 y2)a13

+ (x1 y5 −x5 y1 −x5 y7 +x7 y5)a15 +(x1 y6 −x6 y1)a16 +(x1 y7 −x7 y1 −x7 y5 +x5 y7)a17

+ (x2 y4 −x4 y2 −x4 y6 +x6 y4)a24 +(x2 y6 −x6 y2 −x6 y4 +x4 y6)a26 +(x2 y7 −x7 y2)a27

+ (x3 y4 −x4 y3 −x4 y5 +x5 y4)a34 +(x3 y5 −x5 y3 −x5 y4 +x4 y5)a35 +(x4 y7 −x7 y4)a47

+ (x5 y6 − x6 y5)a56.

Hence v ∧ w = 0 if and only if the following equations hold. x1 y2 − x2 y1 = x2 y3 − x3 y2 =−x1 y3 + x3 y1, x1 y5 − x5 y1 = x5 y7 − x7 y5 = −x1 y7 + x7 y1, x2 y4 − x4 y2 = x4 y6 − x6 y4 =−x2 y6 + x6 y2, x3 y4 − x4 y3 = x4 y5 − x5 y4 = −x3 y5 + x5 y3, x1 y6 = x6 y1, x2 y7 = x7 y2, x4 y7 = x7 y4, x5 y6 = x6 y5. Consider the above equations as a linear system in the coordinates yi , i.e. a system of the form D(x) · y = 0, with the matrix D(x) given by 6.4 The Deleted B3-Line Arrangement 119 ⎛ ⎞ −x2 x1 + x3 −x2 0000 ⎜ ⎟ ⎜ −x2 − x3 x1 x1 0000⎟ ⎜ ⎟ ⎜ −x5 000x1 + x7 0 −x5 ⎟ ⎜ ⎟ ⎜ −x5 − x7 000x1 0 x1 ⎟ ⎜ ⎟ ⎜ 0 −x4 0 x2 + x6 0 −x4 0 ⎟ ⎜ ⎟ ⎜ 0 −x4 − x6 0 x2 0 x2 0 ⎟ D(x) = ⎜ ⎟ ⎜ 00−x4 x3 + x5 −x4 00⎟ ⎜ ⎟ ⎜ 00−x4 − x5 x3 x3 00⎟ ⎜ ⎟ ⎜ −x6 0000x1 0 ⎟ ⎜ ⎟ ⎜ 0 −x7 0000x2 ⎟ ⎝ ⎠ 000−x7 00x4 0000−x6 x5 0

1(A ) ( ( )) ≤ Note that v belongs to R1 if and only if rank D x 5. To determine 1(A ) Singular the irreducible components of R1 , we compute using the software [55] the primary decomposition of the ideal generated by all the 6 × 6 minors of the matrix D(x). This gives us the following 12 maximal isotropic subspaces of H 1(M(dA , C) = A1(dA , C) having dimension ≥ 2. E1 =< a1 −a3, a2 −a3 >, E2 =< a3 −a5, a4 −a5 >, E3 =< a2 −a6, a4 −a6 >, E4 =< a1 − a7, a5 − a7 >, E5 =< a1, a4 >, E6 =< a2, a5 >, E7 =< a3, a6, a7 >, E8 =< a1 − a3 − a4 + a6, a2 − a3 − a4 >, E9 =< a2 + a3 − a4, a4 − a5 − a6 >, E10 =< a1 − a3 + a5, a2 − a3 + a4 >, E11 =< a1 + a3 − a5, a4 − a5 + a7 >, E12 =< a2 − a3 − a5 + a7, a1 − a3 − a5 >. Now, for each maximal isotropic subspace E ⊂ H 1(M, Q) of dim E ≥ 2, we construct the associated mapping fE : M → SE as explained in Proposition3.19 in =< ∗(ω ) : = , ··· , > {ω : = [79]. More precisely, one has E fE j j 1 k , where j j 1 1 1, ··· , k} is a basis for H (SE , C) with SE = P \ BE and BE ={(0 : 1), (1 : b j ), j = 1, ··· , k}. Moreover, for each j,

∗ d(Q − b j P) dP f (ω j ) = − , Q − b j P P where P and Q are the homogenous polynomials of the same degree, such that the map fE corresponds to the pencil < P, Q >. As an example, consider the maximal isotropic subspace E5 =< e1, e4 >, where a1 = dx/x and a4 = d(x − z)/(x − z) in the differential form notations. By the proof of Proposition3.19 in [79], we get after some computations B ={(0 : 1), (1 : 0), (1 : 1)}, x f (x : y : z) = . E5 z

In the afﬁne coordinates obtained by setting z = 1, we get S = C \{0, 1} and

( , ) = . fE5 x y x 120 6 Characteristic Varieties and Resonance Varieties

Table 6.1 The rational pencils associated to the components Ei E SE fE : M(dA ) → SE C \{ , } ( , ) → x E1 0 1 x y y C \{ , } ( , ) → x−1 E2 0 1 x y y−1 C \{ , } ( , ) → x−1 E3 0 1 x y y C \{ , } ( , ) → x E4 0 1 x y y−1 E5 C \{0, 1} (x, y) → x E6 C \{0, 1} (x, y) → y E7 C \{0, ±1} (x, y) → x − y C \{ , } ( , ) → x(x−y−1) E8 0 1 x y (x−y)(x−1) C \{ , } ( , ) → x−1 E9 0 1 x y y(x−y) C \{ , } ( , ) → x(y−1) E10 0 1 x y y(x−1) C \{ , } ( , ) → x(x−y) E11 0 1 x y (x−y+1)(x−1) C \{ , } ( , ) → x E12 0 1 x y y(x−y+1)

Reasoning in the same way, we get Table6.1. Moreover, the irreducible components Ei for i ={1,...,7} are local components, corresponding to the 6 triple points and the quadruple point in the line arrangement dA . The other five components exp(Ei ) of the resonance variety, for i satisfying 8 ≤ i ≤ 12, are global components, and A they correspond to braid subarrangements in d of type A4 consisting of 6 lines as in Exercise2.12. For each fE in the list above we can use Corollary6.4 to see that 1( A ) there is no translated component in V1 d associated to such an fE . Let Vi be the component of the characteristic variety V1(dA ) corresponding to each Ei above, for i = 1,...,12, i.e. Vi = exp(Ei ). It was discovered by A. Suciu 1( A ) that the characteristic variety V1 d has, in addition to these twelve components Vi , exactly one 1-dimensional translated component W. This component is associated to the mapping f : M(dA ) → C∗ defined as follows (in affine coordinates)

x(y − 1)(x − y − 1)2 f (x, y) = (x − 1)y(x − y + 1)2

∗ 7 with ρW = (1, −1, −1, −1, 1, 1, 1) ∈ (C ) ,see[216, 217]. In other words,

W = ρ ⊗{(t, t−1, 1, t−1, t, t2, t−2) | t ∈ C∗}.

Moreover, it can easily be checked that

W ∩ V8 ∩ V9 ∩ V10 = ρ and W ∩ V10 ∩ V11 ∩ V12 = ρ , 6.4 The Deleted B3-Line Arrangement 121

where ρ = (−1, 1, −1, 1, −1, 1, 1) ∈ (C∗)7,see[216, 217]. With the notation above, one can check that C( f ) ={1} and the ﬁber F1 is 2(L ∩ M(A )), where L : x+y−1 = 0. It follows that m1 = 2 and T ( f ) = Z/2Z. By Corollary6.4, there is ∗ exactly one translated component W f associated to f with dim W = dim T (C ) = 1. 1( A ) The description of the characteristic variety V1 d given above implies the following result, see also [216, Example10.6].

Corollary 6.5 Let MN (dA ) be the N-th congruence cover of the complement of the deleted B3-line arrangement. Then one has the following. 3 2 (i) b1(MN (dA )) = 2N + 11N + N − 10 for N even; 3 2 (ii) b1(MN (dA )) = 2N + 11N − 6 for N odd.

Proof We apply Theorem 6.2. It is clear that b = b1(M(dA )) = 7. Consider ﬁrst the case N odd. Then each of the 2-dimensional components Vi , i = 7, which are in 1 1 2 − V1 but not in V2 , contributes to the sum in Theorem 6.2 by N 1, the 3-dimensional 1 1 ( 3 − ) component V7, which is in V2 but not in V3 , contributes by 2 N 1 . The translated component W gives no contribution at all, since there are no elements of odd order in W. It follows that in this case one has

2 3 3 2 b1(MN (dA )) = 7 + 11(N − 1) + 2(N − 1) = 2N + 11N − 6.

ρ,ρ 1 1 Assume now that N is even and note that are in V2 but not in V3 , and W is 1 1 = ,..., in V1 but not in V2 . It follows that the components Vi for i 1 6 each give 2 3 a contribution of N − 1, the component V7 gives a contribution of 2(N − 1),the 2 components Vj for j = 8, 9, 11, 12 each give a contribution of N −2 as they contain 2 one of the two characters ρ,ρ , while V10 gives a contribution of N − 3, since it contains both characters ρ,ρ. Each character ρ,ρ gives a contribution of 2, while W contributes by N − 2, since it contains both characters ρ,ρ. It follows that in this case one has

2 3 2 2 b1(MN (dA )) = 7 + 6(N − 1) + 2(N − 1) + 4(N − 2) + (N − 3) + 4 + (N − 2)

= 2N 3 + 11N 2 + N − 10.

Using Theorem 6.4 and a discussion similar to the above proof, but requiring a lot more work, one can establish the following result, see also [216, Example10.6].

Corollary 6.6 Let SN (dA ) be the corresponding Hirzebruch covering surface asso- 2 ciated to the deleted B3-line arrangement and set P(N) = (N −1)(2N +9N −24). Then one has the following.

(i) b1(SN (dA )) = P(N) + N − 2 for N ≡ 0 mod 4; (ii) b1(SN (dA )) = P(N) + (N − 2)/2 for N ≡ 2 mod 4; (iii) b1(SN (dA )) = P(N) for N odd. 122 6 Characteristic Varieties and Resonance Varieties

1( A ) The description of the characteristic variety V1 d given above also implies the following result. Corollary 6.7 The monodromy operator h1 : H 1(F, C) → H 1(F, C) is the identity morphism for the Milnor ﬁber F of the deleted B3-line arrangement. Proof The claim follows using the formula (6.2) and the following observation: any element in an irreducible component Vi has at least one coordinate equal to 1, while an element in W has the third component equal to −1, but the ﬁrst 3 components cannot all be equal to −1. If we consider plane curve complements instead of line arrangement complements, then it is easy to construct examples with a lot of translated components, see [239].

6.5 Exercises

1 (A ) Exercise 6.1 Let E be an irreducible component of the resonance variety Rm . Show that dim H 1(A∗(A ), ω∧) = dim E − 1 for any nonzero 1-form ω ∈ E. Hint: use the Tangent Cone Theorem6.1 (ii) and Corollary6.2. Exercise 6.2 Show that for the Milnor ﬁber F of the monomial line arrangement 1 C(m) = A (m, m, 3), one has H (F, C)η = 0 for any cubic root of unity η. Exercise 6.3 Show that for the Milnor ﬁber F of the Hessian line arrangement, one 1 4 has H (F, C)η = 0 for any η ∈ C with η = 1.

Exercise 6.4 Show that the Pappus line arrangement (93)1 given by

A : ( − )( − )( − − )( + + )( + − )( − + ) = 1 xyz x y y z x y z 2x y z 2x y z 2x 5y z 0 is combinatorially equivalent to the subarrangement of the Hessian arrangement given by the equation

A : (x3 + y3 + z3)3 − 27x3 y3z3 = 0.

1 Deduce that H (F, C)η = 0 for any cubic root of unity η, where F denotes the Milnor ﬁber of any of these two arrangements.

Exercise 6.5 Recall the two distinct realizations (93)1 and (93)2 of the conﬁguration (93) from Example2.15, namely

A : ( − )( − )( − − )( + + )( + − )( − + ) = 1 xyz x y y z x y z 2x y z 2x y z 2x 5y z 0 and

A : ( + )( + )( + )( + + )( + + )( + + ) = . 2 xyz x y x 3z y z x 2y z x 2y 3z 2x 3y 3z 0 6.5 Exercises 123

(i) Show that the three cubic forms c1(x, y, z) = (x − y)(y − z)(2x + y + z), c2(x, y, z) = y(x − y − z)(2x + y − z) and c3(x, y, z) = xz(2x − 5y + z) (A ) → = P1 \{ } form a pencil giving a map M 1 S, where S 3 points . 1(A ) 1(A ) (ii) Determine the resonance varieties R1 1 and R1 2 . ∗( (A ), C) ∗( (A ), C) (iii) Conclude that the cohomology algebras H M 1 and H M 2 are not isomorphic.

Exercise 6.6 Show that the pencil P1 : uc1 +vc2 associated to the line arrangement A P : ( 3 + 3 + 3) + 1 as in Exercise6.5 (i) and the pencil 2 u x y z vxyz associated to the Hessian arrangement are not equivalent under the following two operations (i) a linear change of coordinates on P2, and (ii) a linear change of the basis elements of the pencils.

Hint: consider the intersections of these two pencils with the hypersurface in P(S3) = P9 corresponding to the set of singular cubic curves. The reader can also have a look at [16, Theorem 1.3] and the Pappus arrangement completed example in [224]. Chapter 7 Logarithmic Connections and Mixed Hodge Structures

Abstract In this chapter we prove a version of the Tangent Cone Theorem for smooth quasi-projective varieties. Then we discuss the mixed Hodge structure on the cohomology of the hyperplane complement M(A ) and of the Milnor fiber F. We define the corresponding spectrum, and state the key results of N. Budur and M. Saito stating that this spectrum is determined by the intersection lattice and giving an explicit formula in the case of a line arrangement in P2. Next we discuss the polynomial count property of algebraic varieties Y defined over the rationals Q. This property always holds when Y = M(A ), while in the case when Y is the Milnor fiber F of such an arrangement, this property is related to the triviality of the monodromy action on H ∗(F). A discussion of Hodge–Deligne polynomials completes this chapter.

7.1 Two Theorems of Pierre Deligne

In this chapter we introduce the logarithmic connections starting with three simple examples, next we state two fundamental results due to P. Deligne and then we use them to give a proof of (a version of) the Tangent Cone Theorem 6.1. Example 7.1 Let X = C \{0} be the punctured complex line, with coordinate x.Let V = OX be the trivial line bundle on X. For any α ∈ C, we consider the connection ∇ : O → Ω1 α X X given by dx αu ∇α(u) = du + αu = u + dx, x x where u is a local section of OX , that is a holomorphic function deﬁned on some open subset of X. Then Lα = ker(∇α) is a rank one local system on X. More precisely,

αu u ∈ Lα ⇔ u + = 0. x

Hence u(x) = c · x−α = c exp(−α log x),forsomec ∈ C, is a holomorphic function deﬁned locally on X (and even globally if α ∈ Z). Consider the oriented loop γ0 : © Springer International Publishing AG 2017 125 A. Dimca, Hyperplane Arrangements, Universitext, DOI 10.1007/978-3-319-56221-6_7 126 7 Logarithmic Connections and Mixed Hodge Structures

[0, 1]→X, γ0(t) = exp(2πit). Then u(γ0(t)) = c · exp(−2πiαt) and hence

u(γ0(1)) = c · exp(−2πiα) = λ · u(γ0(0)),

∗ where λ = exp(−2πiα) ∈ C is the local monodromy of Lα about the origin. In ∗ other words, Lα corresponds to the representation ρ : π1(X, 1) = Z ·[γ0]→C , taking [γ0] to λ. The number α = Res0(∇α) is called the residue of the connection 1 ∇α at 0. We can think of X as being P \{0, ∞}. Then by letting y = 1/x we get a coordinate in a neighborhood of the point ∞. In the coordinate y, the equation for ∇α has the form dy ∇α(u) = du − αu , y and hence Res∞∇α =−α.

Example 7.2 Consider now the more general situation when X = C \{a1,...,ad }. α = (α ,...,α ) ∈ Cd ∇ : O → Ω1 For 1 d , deﬁne a connection α X X by

d α j dx ∇α(u) = du + u . x − a j=1 j

Then Lα = ker ∇α is a rank one local system on X, with local monodromies λ j = exp(−2πiα j ) around the points a j ,for j = 1,...,d. Around the point ∞,usingthe same coordinate y as above, we get

d u α j dy ∇α(u) = du − , y 1 − α y j=1 j (∇ ) =− d α and hence Res∞ α j=1 j . Example 7.3 Let A be a hyperplane arrangement in Pn,forn ≥ 2, and L ⊂ Pn be a generic line. Let |A |=d and suppose each hyperplane H j in A is deﬁned by ∼ 1 an equation j = 0. Then L ∩ (∪H∈A H) = B is a ﬁnite set in L = P , consisting of d points, and we have an inclusion i : L \ B → M(A ). The induced map on homology is an isomorphism

∗ d−1 ∼ ∼ d−1 i : Z = H1(L \ B) → H1(M(A )) = Z , α = (α ,...,α ) ∈ Cd α = as noted in Remark 4.6. Consider 1 d such that j=1,d j 0. ∇ : O → Ω1 ∇ ( ) = + We can then define a connection α M(A ) M(A ) by setting α u du uωα, with d ω = α d j . α j j=1 j 7.1 Two Theorems of Pierre Deligne 127 α = ω The condition j=1,d j 0 is exactly the condition needed for the form α to be a ∗ well-defined 1-form on M(A ), see for instance [74, Chap. 6]. Note that i (∇α),the restriction of this connection to the punctured line L \ B, is a connection of the type studied in the previous Example 7.2. Recall the notion of a normal crossing divisor as introduced in Definition 5.3.

Deﬁnition 7.1 Suppose that D is an NCD in the smooth, complex variety X.The Ωm ( ) sheaf X log D of differential m-forms on X with logarithmic poles along D is the O Ωm = \ : → subsheaf of the sheaf of X -modules j∗ M with M X D and j M X the inclusion, described at the stalk level by

Ωm( ) ={ω ∈ ( Ωm ) | · ω ∈ Ωm · ( ω) ∈ Ωm+1}, X log D x j∗ M x f X,x and f d X,x with f = 0 a reduced local equation for D at x in X.

Recall that for any open set V ⊂ X, one has by deﬁnition the following description Ωm of sections of the sheaf j∗ M :

Γ( , Ωm ) = Γ( ∩ ,Ωm ). V j∗ M V M M

ω ∈ Γ( , Ωm ) · ω ∈ Γ( ,Ωm ) · ω Moreover, for V j∗ M , we write f V M if the form f , which is defined on V ∩ M, admits a holomorphic extension defined on V . A similar ω ∈ ( Ωm ) · ω ∈ Ωm convention is used above for a germ j∗ M x , when we write f X,x . : Ωm ( ) → Ωm+1( ) Note also that we get a well-defined differential d X log D X log D : Ωm → Ωm+1 induced by the exterior derivative d M M , for any positive integer m. Ωm( ) Proposition 7.1 The sheaves X log D of differential m-forms on X with logarithmic poles along D have the following properties. Ωm ( ) O (i) X log D are locally free X -modules on X for any m. Ωm ( ) =∧m Ω1 ( ) (ii) X log D X log D for any m. (iii) For p ∈ D and (x1,...,xn) a system of local coordinates at p such that the ··· = Ω1 ( ) germ of the hypersurface D at p is given by x1 xk 0, the stalk X log D p is a free OX,p-module with a basis given by

dx1/x1,...,dxk /xk , dxk+1,...,dxn.

: Ω1 ( ) → O Corollary 7.1 There are well-deﬁned residue maps R j X log D D j , where D j is any irreducible component of D. ω ∈ Ω1 ( ) (ω) Indeed, with the notation of Proposition 7.1 (iii), if X log D p, then R j p is, up to a sign, just the germ at p of the coefﬁcient of dx j /x j restricted to D j ,if j ≤ k, and it is zero otherwise.

Deﬁnition 7.2 A logarithmic connection (V , ∇) on (X, D) is a pair, with V a vector ∇:V → V ⊗ Ω1 ( ) bundle on X and X log D a connection. 128 7 Logarithmic Connections and Mixed Hodge Structures

For more details we refer to [77], pp. 74–79.

Example 7.4 Let V = OX be the trivial line bundle on X and choose a global loga- ω ∈ Γ( ,Ω1 ( )) ∇ = + · ω rithmic form X X log D . Then ωu du u is a logarithmic connection on X and Lω = ker ∇ω is a rank one local system on M. Moreover, the residues of the connection ∇ω, namely

α = (∇ ) := (ω), j ResD j ω ResD j are in C when the irreducible components D j are compact, since a regular function on an irreducible compact complex variety is necessarily constant. It follows exactly as in Example 7.1 that the monodromy of Lω about the irreducible component D j of D is given by λ j = exp(−2πiα j ). (7.1)

Deﬁnition 7.3 Let (V , ∇) be a logarithmic connection on (X, D). The complex of sheaves on X given by

(V , ∇) = (Ω∗ ( ) ⊗ V , ∇) DR X log D is called the logarithmic de Rham complex of the logarithmic connection (V , ∇).

Example 7.5 If V = OX and ∇=d, then DR(OX , d) is just the usual de Rham logarithmic complex

→ Ω0 ( ) −→d Ω1 ( ) −→···d −→d Ωn ( ) → , 0 X log D X log D X log D 0 where n = dim X. All the residues α j are zero in this case, since there are no logarithmic differentials dx j /x j involved, as we are in the setting of Example 7.4 with ω = 0.

One has the following fundamental result, due to P. Deligne, relating logarithmic de Rham complexes and twisted cohomology. See [60, 105, 111] for a more general version.

Theorem 7.1 Let V = OX be the trivial line bundle on the compact complex man- ∇ : O → Ω1 ( ) ifold X and consider the logarithmic connection ω X X log D given by

∇ωu = du + u · ω,

ω ∈ Γ( ,Ω1 ( )) α = (∇ ) ∇ with X X log D . If no residue j ResD j ω of the connection ω is in Z≥1, then m m H (M, Lω) = H (X, DR(V , ∇ω)), for any positive integer m, where Lω = ker ∇ω. 7.1 Two Theorems of Pierre Deligne 129

The general machinery relating the hypercohomology of a sheaf complex to some spectral sequences, see for instance [77], p. 25, yields the following. α = (∇ ) Z Corollary 7.2 If no residue j ResD j ω is in ≥1, then there exists an E1- spectral sequence

p,q = q ( ,Ωp ( )) ⇒ p+q ( , L ), E1 H X X log D H M ω where the differential p,q : p,q → p+1,q d1 E1 E1 is induced by the cup product by ω.

The second fundamental result we need is the relation between the previous spectral sequence in the case ω = 0 and the mixed Hodge structure (MHS for short) on the cohomology of algebraic varieties constructed by P. Deligne. See [62, 192, 226]for the general theory and Appendix C in [74] for a quick survey. Theorem 7.2 If ω = 0 and X is a projective manifold, then the above spectral m sequence degenerates at E1 and the induced ﬁltration on H (M, C) is the Hodge ﬁltration of the canonical MHS on H m(M, C). In particular, in this case we have ( ) = q ( ,Ωp ( )). bm M dim H X X log D p+q=m

Remark 7.1 In this remark we consider the case D =∅and hence M = X, and relate the above results to the classical Hodge theory of the projective manifold X.It Ω p ( ) = Ω p is clear that in this case X log D X for any integer p. Then the logarithmic de Rham complex from Example 7.5 is just the usual holomorphic de Rham complex (Ω∗ , ) ω = X d on X. Theorem 7.1 in the case 0 yields the isomorphism

m ( ) = Hm ( ,(Ω∗ , )) H X X X d between the cohomology of X with C-coefﬁcients and the hypercohomology of (Ω∗ , ) the complex X d . This is the holomorphic version of the well-known result that the cohomology of a real manifold can be computed using smooth differential global forms, recall de Rham’s Theorem 1.7. Finally, Theorem 7.2 is in this case a reformulation of the Hodge decomposition theorem

m ( ) =⊕ q ( ,Ωp ). H X p+q=m H X X

q ( ,Ωp ) p,q ( ) Indeed, the cohomology group H X X can be identified with the group H X of cohomology classes of type (p, q) via Dolbeault’s Theorem, see [192]. Recall that in this case the Hodge filtration F on H m (X) is defined by

k m p,q F H (X) =⊕p≥k H (X), 130 7 Logarithmic Connections and Mixed Hodge Structures and hence one has

F k H m(X) Grk H m(X) := = H k,m−k (X) = H m−k (X,Ωk ). (7.2) F F k+1 H m (X) X

In other words, the Hodge filtration F is just the filtration induced by the spectral sequence in Theorem 7.2 on its limit. Starting from this observation, Deligne defines the Hodge filtration on H m(M) in the general case, i.e. when D =∅, as being the filtration induced by the spectral sequence in Theorem 7.2 on its limit, and hence one has in general k m( ) = m−k ( ,Ωk ( )). GrF H M H X X log D (7.3)

For any complex algebraic variety M, the MHS on its cohomology H ∗(M, Q) consists of two major ingredients: the decreasing Hodge filtration F on H ∗(M) = H ∗(M, C) defined as above in the case M smooth, and an increasing weight filtration W on H ∗(M, Q), depending on the complexity of the divisor at infinity D when M is smooth. In particular, one can define the corresponding graded pieces

W H m (X, Q) W H m (M) = k , Grk m (7.4) Wk−1 H (X, Q)

W m ( ) and the Hodge filtration F induces a Hodge filtration on Grk H M , which transforms this graded piece into a pure Hodge structure of weight k, i.e. a structure similar to the k-th cohomology of a smooth projective (or compact Kähler) manifold. It is m known that Wk H (M) = 0 when k < m or k > 2m, for any smooth variety M.The mixed Hodge numbers of the variety M are defined by

h p,q (H m (M)) = dim H p,q (H m(M)) where p,q ( m( )) = p W m ( , C). H H M GrF Gr p+q H M

m m We say that H (M, Q) is a pure Hodge structure of weight k if Wk−1 H (X, Q) = 0 m m m and Wk H (X, Q) = H (X, Q) and we say that H (M, Q) is a pure Hodge structure of type (p, p) when H m(M, Q) has pure weight 2p and in addition F p H m (X) = H m(X) and F p+1 H m(X) = 0.

Now we come back to our hyperplane arrangement A in Pn and use the notation n from Example 7.3. Then Z =∪H j ⊂ P is an NCD if and only if the hyperplane arrangement A is in general position as in Definition 2.19. If this is not the case, then we have to consider π : (X, D) → (Pn, Z), the embedded resolution of Z, obtained by blowing up the dense edges Y ∈ L(A ) as explained after Definition 5.4. It is not difficult to check the following, where the connection ∇α and the differential form ωα were introduced in Example 7.3. 7.1 Two Theorems of Pierre Deligne 131

∗ ∗ Proposition 7.2 The pull-back connection π (∇α) = (OX ,π (ωα)) is a logarithmic connection on (X, D).

The following example illustrates very well the arguments, of a local nature, used in the general proof of the above result.

Example 7.6 Let n = 2 and suppose A consists of k ≥ 3 lines passing through the point O = (0 : 0 : 1) ∈ P2, plus the line at inﬁnity z = 0. Suppose the equations for the lines through O are H j : j = y − m j x = 0, for j = 1,...,k. Note that x, y are local coordinates at O. Then Y ={O} is a dense edge and the corresponding 2 2 blow-up map BO (P ) → P is described in local coordinates by π(u, v) = (u, uv). It follows that the exceptional divisor E is given by the equation u = 0 and the proper : = transform of the line H j is the line given by H j v m j . Moreover, since − d j = dy m j dx , j y − m j x it follows that ( − ) π ∗ d j = du + d v m j , j u v − m j

2 ∗ a logarithmic form on B (P ). Moreover, we have Res π (∇α) = Res ∇α = α O H j H j j = ,..., π ∗(∇ ) = α . for j 1 k and ResE α j=1,k j In the special case when M = X \ D = M(A ), it is known that H m(M, C) = F m H m(M, C), see Theorem 7.7,(i)or[149]. From this and (7.3) we obtain

q ( ,Ωp ( )) = H X X log D 0 if q > 0, and hence m( , C) =∼ 0( ,Ωm ( )) H M H X X log D for any m. This equality can also be regarded as a consequence of the Arnold–Brieskorn Theorem in Corollary 3.8. In view of these vanishings, if we take ω ∈ 0( ,Ω1 ( )) = 1( (A )) = 1(A ), H X X log D H M A we see that the spectral sequence from Corollary 7.2 becomes the complex

ω∧ ω∧ ω∧ 0 → H 0(M) −→ H 1(M) −→···−→ H n(M) → 0.

This is exactly the Aomoto complex corresponding to the 1-form ω, namely

ω∧ ω∧ ω∧ 0 → A0(M) −→ A1(M) −→···−→ An(M) → 0. 132 7 Logarithmic Connections and Mixed Hodge Structures

To prove (a version of) the Tangent Cone Theorem 6.1, note that the exponential map 1 1 ∗ takes ωα ∈ H (M, C) to Lα = ker ∇ω ∈ H (M, C ).Ifωα is very close to 0, then ∗ all the residues β of the pull-back connection π (∇α) on (X, D) satisfy β/∈ Z≥1,as they are also very close to zero. Indeed, any such residue β is a ﬁnite sum of residues α j of the form ωα. The terms involved in this sum depend on the resolution map π but not on the 1-form ωα, exactly as in our Example 7.6. This completes the proof of the following. Theorem 7.3 The exponential map

exp : H 1(M(A ), C) → T (A ) = H 1(M(A ), C∗) induces an isomorphism of germs

( j (A ), ) =∼ ( j (A ), ), Rm 0 Vm 1 for any integers j, m and any arrangement A in Pn. This result is in fact a reformulation of results in [112, 204], where higher rank local systems are also considered. Notice that for a rank one local system L ∈ T (A ), given by its monodromies (λ ,...,λ ) λ = α 1 d with j j 1, there are inﬁnitely many choices for the residues j in order to have L = Lα, the only condition being given by the formula (7.1)(in α = addition to j j 0). However, some choices may be better than others, in the following sense.

Deﬁnition 7.4 A rank one local system L ∈ T (A ) is admissible if there exists α, ∗ a choice of residues for L , such that any residue β of π (∇α) satisﬁes β/∈ Z≥1.

If L is admissible, it follows from the above discussion that one has, for any positive integer m and any choice of residue α for L as in Deﬁnition 7.4,anisomor- phism m m ∗ H (M, L ) = H (A (A ,ωα)).

We have also seen in our proof of Theorem 7.3 above that there is a neighborhood U of 1 in T (A ) formed entirely of admissible local systems. For more on admissible rank one local systems, see [103, 176, 223]. m One may wonder what can be said about the dimensions dim H (M(A ), Lα) m ∗ and dim H (A (A ,ωα)) in the case when the local system Lα is not admissible. We have the following corollary of Theorem 7.3.

Proposition 7.3 For any rank one local system Lα = exp(ωα) on M(A ) and any integer m, we have an inequality

m m ∗ dim H (M(A ), Lα) ≥ dim H (A (A ,ωα)). 7.1 Two Theorems of Pierre Deligne 133

m ∗ Proof Let dim H (A (A ,ωα)) = k.Ifk = 0 there is nothing to prove, so we > ω ∈ m (A ) may assume k 0. This means that α Rk and we have to show that ( m(A )) ⊂ m (A ) m (A ) exp Rk Vk . Since the subset Rk is algebraic, it is defined in 1 d−1 H (M(A ), C) = C by some polynomial equations Q1(y) =···= Q M (y) = 0. m(A ) 1( (A ), C∗) = (C∗)d−1 In a similar way, Vk is defined in H M by the vanishing of some Laurent polynomials P1(z) =···= PN (z) = 0. It remains to show that Pj (exp(y)) = 0ifQ1(y) =···= Q M (y) = 0. Consider the analytic function g j (y) = Pj (exp(y)) and let E be an irreducible component of the algebraic set m (A ) ( , ) m(A ) Rk . Then the germ E 0 is sent by the exponential map into the set Vk , by Theorem 7.3. Therefore, the function g j vanishes in a neighborhood of 0 in E and hence the function g j vanishes everywhere on E, since an irreducible algebraic variety is also irreducible when regarded as an analytic variety. Remark 7.2 Similar results to Theorem 7.3 and Proposition 7.3 hold with practically the same proof, for a quasi projective smooth variety M for any j ≤ q, if we assume that the cohomology groups H j (M, C) are pure Hodge structures of type ( j, j) for j ≤ q,see[88, Proposition 4.5]. The next example shows that the inequality in Proposition 7.3 may be strict. Example 7.7 Consider the Ceva line arrangement A = A (3, 3, 3) in P2 defined by

(x3 − y3)(x3 − z3)(y3 − z3) = 0, and the corresponding regular map

f : M(A ) → S = P1 \{(1 : 0), (0 : 1), (1 : 1)},(x : y : z) → (x3 − y3 : x3 − z3).

Let u, v be the coordinates on P1 and consider the 1-form α ∈ H 1(S, C) given by

du dv du − dv α = + − 2 . u v u − v

= ∗( 1( , C)) 1(A ) ω = Then E f H S is a 2-dimensional irreducible component of R1 , f ∗(α) is a nonzero form in E and hence by Exercise 6.1 we have

dim H 1(A∗(A ), ω∧) = dim E − 1 = 1.

1(A ) On the other hand, if W denotes the irreducible component of V1 associated with E under the Tangent Cone Theorem, then we have that L ∈ W, where L is the rank one local system on M(A ) whose monodromy about each line is λ = exp(−2πi/3). 1 1 By Proposition 5.4 (ii), it follows that dim H (M(A ), L ) = dim H (F)λ, and it is 1 known that dim H (F)λ = 2, see [32], Remark 3.2 and [82], or Theorem 8.16 further on in this book for a more general result. In some cases, one can obtain the opposite inequality, using the following deep result of Papadima and Suciu. For a more general version see [187]. 134 7 Logarithmic Connections and Mixed Hodge Structures

Theorem 7.4 Let A be an affine arrangement and fix a prime number p and an integer s > 0. Consider a local system L on M(A ) whose monodromy about any hyper- 2πi plane H ∈ A is given by λ = exp( ). Consider the Aomoto complex AF (A ,ω) H ps p F ω = with p-coefficients and H∈A aH . Then

k ( (A ), L ) ≤ β (A ) := k ( (A ), ω), dimC H M p dimFp H AFp for any k. Let A be a line arrangement in P2 with only double and triple points as singularities, and let A denote the corresponding central arrangement in C3. Then the only eigenvalues of h∗ on H 1(F), with F the Milnor ﬁber of A , are the cubic roots of unity, recall Corollary 5.5. By applying the previous theorem for p = 3 and s = 1, as well as adding a lot of new ideas, Papadima and Suciu proved the following key result in [184]. Theorem 7.5 Let A be a line arrangement in P2 with only double and triple points as singularities. Then the multiplicity of a cubic root of unity η = 1 as an eigenvalue ∗ 1 1 of the monodromy h : H (F, C) → H (F, C) is given by β3(A ). In particular, all the Betti numbers bi (F) of the Milnor ﬁber of such a line arrangement are combinatorially determined. Moreover, the following are equivalent.

(i) β3(A ) = 0; (ii) A admits a 3-net; (iii) A admits a 3-multinet.

More precisely, it is shown in [184] that β3(A ) = 0 if and only if d =|A |=3e is a multiple of 3 and A is a (3, e)-net as in Deﬁnition 6.3. The fact that the equality 1 1 H (F) = H (F)1 holds exactly when A is not a (3, e)-net was previously shown by Libgober in [160]. Remark 7.3 Let A be an arrangement of d lines in P2, such that A has only double and triple points and d = 3m for some integer m.LetT ⊂ P2 be the set of triple points in A .IfS = C[x, y, z] denotes the graded ring of polynomials in x, y, z, consider the evaluation map T ρ : S2m−3 → C (7.5)

3 obtained by picking up a representative st in C for each point t ∈ T and sending a homogeneous polynomial h ∈ S2m−3 to the family (h(st ))t∈T . Then Theorem 2 in [32] implies the key formula

β3(A ) = dim(Cokerρ), (7.6) and the last integer is by definition the superabundance or the defect S2m−3(T ) of the finite set of points T with respect to the polynomials in S2m−3. Since by the work of S. Papadima and A. Suciu we know that β3(A ) ∈{0, 1, 2}, this gives a very strong restriction on the position of the triple points in such a line arrangement. For other relations to algebraic geometry of a similar flavor we refer to [12, 82, 110, 162, 174]. 7.2 Mixed Hodge Structure and Spectrum 135

7.2 Mixed Hodge Structure and Spectrum

Let A be a central arrangement of d hyperplanes in Cn+1, with d ≥ 2 and n ≥ 1, given by a reduced equation Q(x) = 0. Consider the corresponding global Mil- nor fiber F defined by Q(x) − 1 = 0inCn+1 with monodromy action h : F → F, h(x) = exp(2πi/d) · x. In studying the cohomology H ∗(F, Q) of the Milnor fiber or the monodromy action h∗ : H ∗(F, Q) → H ∗(F, Q), we can, without any loss of generality, suppose that the arrangement A is essential, recall Exercise 5.1, and we do this in the sequel. For basic facts on mixed Hodge structures we refer to our discussion in the previous section and, for general definitions, results and complete proofs, to the excellent book [192]. The interplay between the monodromy h∗ : H ∗(F, Q) → H ∗(F, Q) and the mixed Hodge structure (MHS) on H ∗(F, Q) is reflected by the spectrum of A defined as Sp(A ) = Sp(Q), where α Sp(Q) = mαt , (7.7) α∈Q with = (− ) j−n p ˜ j ( , C) , mα 1 dim GrF H F β j where p =[n + 1 − α], β = exp(−2πiα)and H˜ j (F, C) is the reduced cohomology of the Milnor fiber F.Here[x] denotes the integral part of the rational number x. To explain this definition, note that H k (F, Q) is an MHS for any k ≥ 0by[62], since F is a smooth affine variety, the monodromy h : F → F is a regular map and hence it induces a morphism of MHSs hk : H k (F, Q) → H k (F, Q) whichinturn k : p k ( , C) → p k ( , C). p k ( , C) induces a linear map h GrF H F GrF H F Then GrF H F β in the above formula is by definition the β-eigenspace of the linear map T k = (hk )−1, the so-called local system monodromy, see [96] for an explanation of this twist. Note that one obviously has T = h∗, the complex conjugate of h∗. Moreover, we use the same letter F to denote both the Milnor fiber and the Hodge filtration, in order to respect the traditional notation and to keep it simple, being sure that no confusion can arise from this fact. The rational number α is called a spectral number for the homogeneous polynomial Q, or for the hyperplane arrangement A ,ifmα = 0inSp(Q). The corresponding integer mα is called the multiplicity of α. N. Budur and M. Saito have obtained the following fundamental result, see [33]. Theorem 7.6 The spectrum Sp(A ) of a central hyperplane arrangement A is determined by the intersection lattice L(A ). Since T d = Id,itfollowsthatT : H ∗(F, Q) → H ∗(F, Q) is semisimple, and hence we get a direct sum decomposition as MHSs over Q

∗ ∗ ∗ H (F, Q) = H (F, Q)1 ⊕ H (F, Q)=1, (7.8) 136 7 Logarithmic Connections and Mixed Hodge Structures

∗ ∗ d−1 where H (F, Q)1 = ker(T − Id) and H (F, Q)=1 = ker(T +···+T + Id). One has the following result, see [86, 90, 141, 149].

Theorem 7.7 Let A be a central hyperplane arrangement in Cn+1 and let F denote its Milnor ﬁber. Then the following hold. (i) For any afﬁne hyperplane arrangement B in Cn, the cohomology groups of the complement H m (M(B), Q) are pure Hodge structures of type (m, m), for any m m m = 0,...,n. In particular, H (F, Q)1 = H (M(A ), Q) is a pure Hodge structure of type (m, m) for any integer m ∈[0, n], where A is the projective arrangement associated to A . W m ( , Q) = < ≤ . (ii) Gr2m H F =1 0, for any m with 0 m n 1 (iii) For any n ≥ 1, the group H (F, Q)=1 is a pure Hodge structure of weight 1.

Proof We prove here only the ﬁrst and the last claims. We prove the claim (i) by induction on d =|B|. When d = 0 there is nothing to prove, so assume from now on that d > 0 and let H ∈ B be a hyperplane. Let (B, B, B) be the triple of arrangements associated to B and the distinguished hyperplane H as in Deﬁnition 2.13. Denote by M, M and respectively M the corresponding complements. Then the proof of Theorem 3.5 shows that we have a short exact sequence

R 0 → H p(M, Q) → H p(M, Q) −→ H p−1(M, Q) → 0, for any integer p. The Leray residue morphism R here is known to be an MHS morphism of type (−1, −1),see[74, Appendix C]. Since both arrangements B and B have at most d − 1 hyperplanes, by induction the claim (i) holds for them. Then the above exact sequence gives the result for the arrangement B. Now we prove the claim (iii). When n = 1, F is the Milnor fiber of an isolated homogeneous curve singularity, and the result follows from [215]. When n > 2, we can consider a generic 3-dimensional linear space E ⊂ Cn+1 and recall that the inclusion E ∩ F → F induces an isomorphism on H 1 compatible with the respective monodromy actions, see for instance [74, Theorem 4.1.24], which is essentially the Lefschetz Theorem for affine varieties. So it is enough to consider the case n = 2. We do this now, but in the more general case of the Milnor fiber F : f = 1 associated to any reduced plane curve C : f = 0ofdegreed. Consider the surface S : f (x, y, z) + wd = 0, which is a singular compactification of the Milnor fiber F, and the exact sequence of cohomology groups with compact supports

→ 2( , Q) → 2( , Q) → 3( , Q) → 3( , Q) → 3( , Q) = . H S H C Hc F H S H C 0

Let μd be the multiplicative group of d-th roots of unity. Note that the monodromy action on F ⊂ S can be extended to a μd -action on S given by

λ · (x : y : z : w) = (x : y : z : λ−1w). 7.2 Mixed Hodge Structure and Spectrum 137

2 2 μ The quotient space S/μd is clearly isomorphic to P , which implies dim H (S) d = 1 μ and dim H 3(S) d = 0. If r denotes the number of irreducible components of C, then clearly b2(C) = r, and the μd -action on C = S \ F is trivial. The ﬁrst morphism H 2(S) → H 2(C) is clearly non-trivial, and hence the kernel of the next mor- ( − ) 3( , Q) 1( , Q) = phism is r 1 -dimensional. On the other hand Hc F is dual to H F 1( , Q) ⊕ 1( , Q) , 1( , Q) = − 3( , Q) H F 1 H F =1 and dim H F 1 r 1. It follows that Hc F 1 ( − ) ( , ) 3( , Q) is r 1 -dimensional and pure of type 1 1 , while Hc F =1 is isomorphic to H 3(S), and hence it is a pure Hodge structure of weight 3. Indeed, the surface S has only isolated singularities and the claim follows from [192, Theorem 6.33]. The 1 claim for H (F, Q)=1 follows by duality, see [192].

Remark 7.4 Let S˜ be a resolution of singularities for the surface S introduced in the proof above. As noticed in [74], p. 218, one has the following equality of Betti ˜ ˜ numbers b3(S) = b3(S). Applying the Poincaré duality to the smooth manifold S gives, in view of the proof above, the following useful equality

1 ˜ ˜ dim H (F, Q)=1 = b1(S) = 2q(S), (7.9) where q(S˜) denote the irregularity of the surface S˜,see[174]. This equality can also be seen in the following way. Let j˜ : F → S˜ be the inclusion. Since F is a ˜ Zariski open subset, it follows that the inclusion j induces an epimorphism π1(F) → ˜ ˜∗ 1 ˜ 1 ˜ π1(S), and hence a monomorphism j : H (S, Q) → H (F, Q). Since S is a smooth compactiﬁcation of F, it follows that

˜∗ 1 ˜ 1 1 j (H (S, Q)) = W1 H (F, Q) = H (F, Q)=1.

Here the ﬁrst equality follows from the general properties of MHSs, see for instance [74, Theorem C24, (iii)], where one has to add in the statement that the compact- iﬁcation X is smooth. The second equality follows from Theorem 7.7.Thesame argument implies the following inequality

h p,q (H 2(F)) ≤ h p,q (S˜) (7.10) for all pairs (p, q) with p + q = 2. With the notation in the above proof of this theorem, one can see that the inclusion j : F → S gives rise to a singular compactiﬁcation of F such that H 1(S, Q) = 0, and hence the equality

∗ 1 1 j (H (S, Q)) = W1 H (F, Q) fails in general for non-smooth compactiﬁcations.

Coming back to the central hyperplane arrangements, the cohomology groups m m m H (F, Q)1 and H (F, Q)−1 are mixed Hodge substructures in H (F, Q) as (T − Id) and (T + Id) are MHS morphisms. Moreover, for β in the group of d- th roots of unity μd , with β =±1, the subspace 138 7 Logarithmic Connections and Mixed Hodge Structures

m m n m 2 m H (F, C)β,β = H (F, C)β ⊕ H (F, C)β = ker[(T ) − 2Re(β)T + Id] (7.11) is again a mixed Hodge substructure which, this time, is deﬁned over R as the last m m equality shows. For β =−1, we set H (F, C)β,β = H (F, C)−1 for uniformity of notation. −1 Let D = Q (0) =∪H∈A H. For a point x ∈ D, x = 0, let L x =∩H∈A ,x∈H H be the minimal ﬂat containing x, and denote by Ax the central hyperplane arrangement induced by A on a linear subspace Tx , passing through x and transversal to L x .We may choose dim Tx = codimL x and identify x with the origin in the linear space Tx . ∗ : ∗( , C) → ∗( , C) Let hx H Fx H Fx be the corresponding monodromy operator at x, i.e. the monodromy operator associated to the central hyperplane arrangement Ax . With this notation, we have the following result, see [80].

Proposition 7.4 Let β ∈ μd , β = 1, be a root of unity which is not an eigenvalue ∗ ∈ = for any monodromy operator hx for x D, x 0. Then the following hold. m (i) H (F, C)β,β = 0, for any m with m < n; n (ii) The eigenspace H (F, C)β,β is a pure Hodge structure of weight n. In particular, if β = exp(−2πiα) for some α ∈ Q, then the coefﬁcients in the corresponding spectrum Sp(A ) have the following symmetry property:

mα = mn+1−α. (7.12)

In the case n = 2, the paper [33] gives the following very simple formulas for the coefﬁcients mα.Letν j be the number of points in the projective line arrangement A , of multiplicity j, j ≥ 3.

Theorem 7.8 When n = 2, then mα = 0 if either α/∈ (0, 3) or αd ∈/ Z, where d = |A | α = i ∈] , ] ∈[ , ]∩Z . Fora rational number d 0 1 with i 1 d , one has the following. i − 1 ij/d−1 mα = − ν , 2 j 2 j mα+1 = (i − 1)(d − i − 1) − ν j (ij/d−1)( j −ij/d), j d − i − 1 j −ij/d mα+ = − ν − δ , , 2 2 j 2 i d j

where β:=min{k ∈ Z | k ≥ β}, and δi,d = 1 if i = d and 0 otherwise.

Similar formulas in the case n = 3aregivenin[235].

Example 7.8 Consider the monomial arrangement A = A (2, 2, 3) in C3 given by

A : (x2 − y2)(y2 − z2)(z2 − x2) = 0, 7.2 Mixed Hodge Structure and Spectrum 139

, Table 7.1 The mixed Hodge numbers h p q (H 2(F)λ for the arrangement A (2, 2, 3) H 1(F, C) H 2(F, C) 1,1 1 2,2 2 λ0 = 1 h (H (F)1) = 5 h (H (F)1) = 6 2,0 2 λ3 =−1 h (H (F)−1) = 0,2 2 h (H (F)−1) = 1 λ 0,1( 1( ) ) = 1,2( 2( ) ) = 2 h H F λ2 1 h H F λ2 3 λ = λ 1,0( 1( ) ) = 2,1( 2( ) ) = 4 2 h H F λ4 1 h H F λ4 3 λ 0,2( 2( ) ) = 1 h H F λ1 2 λ = λ 2,0( 2( ) ) = 5 1 h H F λ5 2

already discussed in Exercise 2.12 in relation to the braid arrangement Br4, see also Example 2.11. Then d = 6 and the only nonzero number ν j is ν3 = 4. Theorem 7.8 implies the following formula for the spectrum of the plane arrangement A

Sp(A ) = t5/2 − t7/3 + 2t13/6 − 5t2 − t5/3 + 3t4/3 + 6t + 2t5/6 + 3t2/3 + t1/2.

1 1 Set λ = exp(−2πik/6). It is known that H (F, C)= = H (F, C) has dimen- k 1 λ2,λ2 sion 2, see for instance [82, 164, 208]. Using this, we can determine from the above formula for the spectrum Sp(A ) all the nonzero equivariant mixed Hodge numbers, namely the dimensions

p,q m p,q m h (H (F)λ) = dim H (H (F)λ)

p,q m where H (H (F)λ) is the λ-eigenspace of the monodromy operator T acting on

p,q ( m( )) = p W m( , C). H H F GrF Gr p+q H F

The corresponding results are given in Table7.1. Clearly one has the following useful formula for any central hyperplane arrangement p j ( , C) = p,q ( j ( )) . dim GrF H F λ h H F λ (7.13) q≥ j−p

This follows from the general properties of MHSs, the Milnor ﬁber F being smooth. 1,0( 1( ) ) = Note that in [82] it is shown that h H F λ4 1 using a different approach. The 2 p,q 2 above results show that H (F, Q)=1 has elements of weight 2, i.e. h (H (F)λ = 0, p,q 2 for p + q = 2 and λ = 1, as well as elements of weight 3, i.e. h (H (F)λ = 0, for 2 p + q = 3 and λ = 1. Hence in general H (F, Q)=1 is not a pure Hodge structure. A different approach to computing such equivariant mixed Hodge numbers is described in [86]. 140 7 Logarithmic Connections and Mixed Hodge Structures

7.3 Polynomial Count Varieties

A very interesting connection between Arithmetic and Geometry is provided by the ‘polynomial count’ varieties, see [80, 131, 142]. Definition 7.5 Let Y be an algebraic variety defined over Q. The variety Y has polynomial count with counting polynomial PY (t) if there exists a polynomial PY (t) ∈ Z[t] such that for all but finitely many prime p and for any finite field s Fq , with q = p , s ≥ 1, the number of points of Y over Fq is given by PY (q). Proposition 7.5 Assume that the affine hyperplane arrangement A in Cn is defined over Q, in the sense that each hyperplane H ∈ A can be defined by a linear equation H = 0 with integer coefficients. Then the complement M(A ) has polynomial count with polynomial PM(A )(t) = χ(A , t).

Proof Notice that for such an arrangement A and for any prime number p, we can A ⊂ Fn consider the arrangement p p obtained by taking the image of the coefficients of H under the obvious composition morphism Z → Fp → Fq . The key fact is that the intersection lattice L(Ap) is the same for all but finitely many p. Indeed, the rank of each element in L(A ) corresponds to the rank of a matrix, which is determined by the vanishing of finitely many minors in the ring of integers. Choose p such that p is bigger than the absolute values of all those minors arising for A (of which there are finitely many for a given arrangement). Note also that for any q and any triple of arrangements (A , A , A ) obtained from a non-empty arrangement A , we obviously have

|M(A )(Fq )|=|M(A )(Fq )|−|M(A )(Fq )|.

By induction on |A | and using Theorem 2.7 we have |M(A )(Fq )|=χ(A , q) for the prime powers q associated with the good prime numbers p, since we need the characteristic polynomial χ(Ap, t) to be independent of p. One can ask the same question for the Milnor ﬁber of F of a central arrangement A in Cn. Here are some answers given in [80].

Proposition 7.6 If h∗ = Id on H ∗(F, C) and A is deﬁned over Q as above, then the Milnor ﬁber F has polynomial count with same count polynomial as M(A ), where A is the associated projective arrangement.

To use this result, we need the following characterization of hyperplane arrangements satisfying the condition h∗ = Id on H ∗(F, C),see[80].

Theorem 7.9 For an essential central hyperplane arrangement A in Cn, the following conditions are equivalent. (i) The monodromy action h∗ is trivial on all the cohomology groups H ∗(F, C). 7.3 Polynomial Count Varieties 141

(ii) The arrangement A is reducible and satisﬁes the following: if

A = A1 ×···×Aq

is the decomposition of A as a product of irreducible arrangements, then G.C.D.(d1,...,dq ) = 1, where d j =|A j | denotes the number of hyperplanes in A j ,forj= 1,...,q.

Moreover, if A is deﬁned over Q, then all the irreducible arrangements A j are also deﬁned over Q.

Example 7.9 An essential central plane arrangement A in C3 which is not irreducible has, up to a linear change of coordinates in C3, a defining equation of the form Q(x, y, z) = R(x, y)z = 0, where R(x, y) is a homogeneous polynomial in x, y of degree d − 1, with d =|A |. Such an arrangement clearly has the property that h∗ = Id on H ∗(F, C). Indeed, the Milnor fiber F is defined in C3 by R(x, y)z = 1 and the family of maps

ht : F → F,(x, y, z) → (exp(2πit)x, exp(2πit)y, exp(−2πi(d − 1)t)z),

for t ∈[0, 1/d], gives a homotopy between the identity map Id : F → F and the monodromy homeomorphism h : F → F. Moreover, it is clear that one has an isomorphism of algebraic varieties F ={(x, y) ∈ C2 : R(x, y) = 0}, the complement of a central line arrangement in the plane.

There are deep relations between the varieties having polynomial count and the Hodge theoretic properties of algebraic varieties. We say that a complex variety Y is cohomologically Tate if for any cohomology group H m(Y, C), one has the following vanishing of mixed Hodge numbers: h p,q (H m (F, C)) = 0forp = q.The fact that the hyperplane arrangement complements M(A ) are cohomologically Tate has been known for a long time: any cohomology group H m(M(A ), Q) is a pure Hodge structure of type (m, m),see[141, 148]. This fact is also stated in Theorem 7.7 (i) above. When the monodromy action h∗ is trivial on all the cohomology groups H ∗(F, C), it follows that we have an equality H m (F, Q) = H m (M(A ), Q) for any 0 ≤ m ≤ n, and hence in this case F is cohomologically Tate. One may ask whether this is the only possibility for a hyperplane arrangement Milnor ﬁber F to be cohomologically Tate. The claim that F cohomologically Tate implies h∗ trivial is true in the case n = 2, recall Theorem 7.7 (iii). In the case n = 3, we have the following very similar result, see [80, Theorem 1.1].

Theorem 7.10 Let A be an essential central arrangement of planes in C3 and let F denote its Milnor ﬁber. The following conditions are equivalent. (i) The mixed Hodge numbers h p,q (H 2(F, C)) vanish for p = q. (ii) The arrangement A is reducible. (iii) The monodromy action h∗ is trivial on all the cohomology groups H ∗(F, C). 142 7 Logarithmic Connections and Mixed Hodge Structures

(iv) The multiplicities mα in Sp(A ) vanish for all α ∈ (0, 1).

Proof Here we give the idea of a proof of this result, different from the proof in [80] in the main step. It follows from Example 7.9 that, assuming (ii), the Milnor ﬁber F is isomorphic to the complement of the central line arrangement given by R = 0 in C2. Hence the implication (ii) ⇒ (i) is obviously true. By Theorem 7.8, it follows that for α ∈ (0, 1], the corresponding multiplicity is

2,0 2 2,1 2 2,2 2 mα = h (H (F, C)β ) + h (H (F, C)β ) + h (H (F, C)β ) (7.14)

2,2 2 with β = exp(−2πiα). Moreover, h (H (F, C)β ) = 0forα ∈ (0, 1) as follows from Theorem 7.7 (ii). 2 Since H (F, C)1 is known to be of type (2, 2), the equivalence of the claims (i) and (iv) in Theorem 7.10 follows. Moreover, the equivalence of the claims (ii) and (iii) follows from Theorem 7.9. So the main step in the proof is to assume that (i) and (iv) hold and we prove (ii). The fact that the arrangement is essential implies that d =|A |≥3 and that the lines in A do not all pass through the same point, i.e. there is no point s of multiplicity ms = d. To prove (ii) we have to show the existence of a point of multiplicity d − 1. The following result establishes a more precise version of this implication, which completes the proof of Theorem 7.10.

Proposition 7.7 Let A : f = 0 be an essential line arrangement in P2 with deg f = ≥ = α = d−1 A d 4 and assume that mα 0 for d . Then has a point of multiplicity − = α = d−1 = α ∈ ( , ) d 1. In particular, mα 0 for d implies mα 0 for all 0 1 . <α= j < Proof The following formula follows from Theorem 7.8:if0 d 1, then j − 1 jms /d−1 mα = − , (7.15) 2 2 s;ms ≥3

where the sum is over all multiple points s in A with multiplicity ms ≥ 3. By a = < = − convention b 0ifa b. We apply the formula (7.15)for j d 1. Since one clearly has (d − 1)m m − 1 < < m d

for m < d, it follows that (d − 1)m/d=m and hence the vanishing mα = 0in this case is equivalent to the equality d − 2 m − 1 = s . (7.16) 2 2 s;ms ≥3

On the other hand, Exercise 2.8 implies that 7.3 Polynomial Count Varieties 143 d m = s . (7.17) 2 2 s;ms ≥2

Subtracting Eq. (7.16) from (7.17) we get 2d − 3 = (ms − 1). (7.18)

s;ms ≥2

Subtracting Eq. (7.18) from (7.17) multiplied by 2 yields 2 2 τ(A ) = (ms − 1) = d − 3d + 3, (7.19)

s;ms ≥2 where τ(A ) is the total Tjurina number of the arrangement A , i.e. the sum of the Tjurina numbers of all its multiple points s. The claim now follows from Corollary 8.2. Note that although this result is further on in this book, its proof has nothing to do with spectral numbers or Hodge theory, being essentially elementary.

Remark 7.5 One has the following easy analog of Proposition 7.7.LetA : f = 0 2 be a line arrangement in P with deg f = d ≥ 4 and assume that m1 = 0. Then A has a point of multiplicity d, i.e. A is not essential. Indeed, note that the formula (7.14) implies in this case that

2,2 2 b2(M(A )) = h (H (F, C)1 = 0.

We conclude using Remark 4.6 and Corollary 4.5. In particular, m1 = 0 implies mα = 0 for all α ∈ (0, 1].

On the other hand, the claim that F cohomologically Tate implies h∗ trivial is false in general, as shown in [80], where the following example was discovered. Example 7.10 Consider the central arrangement A in C8 deﬁned by

Q(x) = x1 · x2 · x3 · (x1 + x2 + x3) · x4 · x5 · x6 · x7 · x8 · (x4 + x5 + x6 + x7 + x8) = 0.

Then the corresponding Milnor ﬁber F is cohomologically Tate and does not have polynomial count. Moreover, the monodromy operator satisﬁes h∗ = Id. For a central arrangement A in Cn with 3 < n < 8, it is an open question whether F cohomologically Tate implies h∗ trivial.

Recall that the (compactly supported) Hodge–Deligne polynomial (also called the E-polynomial) associated to a complex variety Y is given by ⎛ ⎞ ( , ) = ⎝ (− ) j u,v( j ( , C))⎠ u v. HDY x y 1 h Hc Y x y (7.20) u,v j 144 7 Logarithmic Connections and Mixed Hodge Structures

The main property of this polynomial, and the reason for using compact supports in the deﬁnition, is its additivity, namely if X is a complex algebraic variety and Y ⊂ X is a closed algebraic subvariety, then one has

HDX (x, y) = HDX\Y (x, y) + HDY (x, y). (7.21)

This property can be used to compute the Hodge–Deligne polynomial HDX (x, y) when the variety X has a decomposition into very simple pieces, as the following example shows. To do the computation, the following formula coming from the Poincaré duality is very useful, see [192]. If X is a smooth connected n-dimensional complex algebraic variety, then

p,q ( j ( , C))) = n−q,n−p( 2n− j ( , C))), h Hc X h H X (7.22) for any integers p, q, j. Example 7.11 Note that the complex projective space Pn has a natural constructible partition n k P =∪k=0,nC .

The only non-zero cohomology group of an afﬁne space Ck is H 0(Ck , C) = C, which has Hodge type (0, 0). By duality (7.22), it follows that the only non-zero compactly supported group is H 2k (Ck , C) = C, which has Hodge type (k, k).Using the additivity (7.21) repeatedly we get k k HDPn (x, y) = x y . k=0,n

n Since P is compact and smooth, the formula for HDPn (x, y) given above implies n n in particular the well-known fact about the Betti numbers of P , namely b j (P ) = 1 n for j even and satisfying 0 ≤ j ≤ 2n, and b j (P ) = 0 otherwise. The same approach yields the Betti numbers of any toric variety which is either smooth and proper, or cohomologically Tate. Hyperplane arrangement complements can also be treated in this way, see the next example, or Exercise 7.3. Remark 7.6 Another similar approach consists in replacing the Hodge–Deligne polynomial with a universal additive invariant, namely the class of a variety in the Grothendieck group of varieties over a fixed field K ,see[6]. This leads to a description of the class of the complement M(A ) of a projective arrangement A in terms of the corresponding characteristic polynomial χ(A , t), evaluated at the class of the affine line K . When applied to finite fields, this result gives a new proof of Proposition 7.5. See also Corollary 7.3 for a related result. Example 7.12 Let A be a line arrangement in the projective plane P2 and d =|A |. Assume that there are nk points of multiplicity k in A ,fork ≥ 2. Then note that the complement M(A ) = P2 \ C, where C is the union of the lines in A , satisfies 7.3 Polynomial Count Varieties 145

HDM(A )(x, y) = HDP2 (x, y) − HDC (x, y).

On the other hand, let P be the set of singular points of C and set C = C \ P. Then one has HDC (x, y) = HDC (x, y) + HDP (x, y). ( , ) =| |= ( , ) = + + 2 2 One obviously has HDP x y P k≥2 nk and HDP2 x y 1 xy x y by Example 7.11. Note that C is a disjoint union of d projective lines L1,...,Ld each with a number m j of deleted points. Since m j = knk , j=1,d k≥2 ( , ) = ( + ) − it follows that HDC x y d 1 xy k≥2 knk . This yields 2 2 HDM(A )(x, y) = (k − 1)nk − d + 1 − (d − 1)xy + x y . k≥2

In view of (7.22), this equality is equivalent to b0(M(A )) = 1, b1(M(A )) = d − 1, b2(M(A )) = (k − 1)nk − d + 1. k≥2

Note that obviously the number of points of a variety over a finite field also behaves in an additive way with respect to constructible partitions. We have the following result, relating arithmetic and Hodge theory. Theorem 7.11 If Y is an algebraic variety defined over Q, such that Y has polynomial count with count polynomial PY , then the compactly-supported Hodge–Deligne polynomial of Y is given by

HDY (x, y) = PY (xy).

For a proof, see [131, Theorem 2.1.8], as well as the remark at the bottom of page 563 and Example 2.1.10. Another proof of this result is given in [86, Appendix A.5]. Combining Theorem 7.11 with Proposition 7.5 yields the following. Corollary 7.3 Let A be a hyperplane arrangement in Cn, deﬁned over Q. Then the compactly-supported Hodge–Deligne polynomial of M(A ) is given by

HDM(A )(x, y) = χ(A , xy).

Theorem 7.11 implies that a variety Y which has polynomial count is not too far from being cohomologically Tate, i.e. the non-Tate part of the cohomology should cancel out in the compactly-supported Hodge–Deligne polynomial HDY (x, y).For surfaces we have the following more precise result, see [80]. 146 7 Logarithmic Connections and Mixed Hodge Structures

Proposition 7.8 Let Y be a smooth surface defined over Q and consider the following conditions. (i) the surface Y has polynomial count; (ii) the surface Y is cohomologically Tate. If the surface Y is affine, then (i) implies (ii). Moreover, if Y = F is the Milnor fiber of a central plane arrangement A in C3, then (i) ⇐⇒ (ii) and both conditions are equivalent to h∗ = Id on H ∗(F, C).

Hodge–Deligne polynomials allow us to reformulate Theorem 2.4 in the following way.

Theorem 7.12 Let A be a ﬁber type arrangement in Cn. With the notation from Theorem 4.17, one has

( , ) = ( , ) · ...· ( , ). HDM(A ) x y HDF1 x y HDFn x y

Proof To prove this result we apply [85, Theorem 6.1]. Here is the main step. n n−1 Consider the first linear projection p¯n : C → C , inducing the first fiber bundle pn : M(A = An) → M(An−1), with fiber Fn. To apply [85, Theorem 6.1], we 1 have to show that the local system R pn∗C is a trivial local system on M(An−1). ¯ =¯−1( ) = −1( ) ∈ (A ) Set Lt pn t and Lt pn t for t M n−1 . With this notation, it is enough 1 1 to show that in each fiber H (Lt , C) of the local system R pn∗C we can choose a canonically defined C-vector space basis, for any t ∈ M(An−1). Note that there is a unique subarrangement B in A , such that, for t ∈ M(An−1), the fiber Lt is an affine ¯ ¯ line C = Lt , from which we delete the intersection points pH = Lt ∩ H for H ∈ B. If H : H = 0 are the corresponding equations of the hyperplanes H ∈ B, then the 1 restrictions of the 1-forms dH /H to Lt give the canonical basis in H (Lt , C).

Example 7.13 For the Ceva arrangement A = C(3) in P2, the corresponding pencil gives rise to a fiber bundle p : M(A ) → S = P1 \{3 points}, as explained in Remark 6.15. The fiber F of this fibration is a smooth cubic curve with the 9 points in the base locus of the pencil deleted. It follows that

HDF (x, y) = xy − x − y − 8, HDS(x, y) = xy − 2.

Since a direct computation shows that

2 HDM(A )(x, y) = χ(A , xy) = (xy − 4) ,

it follows that HDM(A )(x, y) = HDF (x, y)DHS(x, y). In particular, the local sys- 1 tem R p∗C is not trivial in this case. Hence the ﬁbrations coming from pencils of plane curves of degree >1 behave differently from the ﬁbrations induced by linear projections, which correspond to pencils of plane curves of degree one, as explained in Example 4.9. 7.3 Polynomial Count Varieties 147

If a ﬁnite group Γ acts regularly on a complex algebraic variety X, then each of the graded pieces

p,q ( j ( , C)) := p W j ( , C) H H X GrF Gr p+q H X (7.23) becomes a Γ -module in the usual functorial way, and hence defines an element in the representation ring R(Γ ). These modules are the building blocks of the equivariant Hodge–Deligne polynomial HDΓ (X; u, v) ∈ R(Γ )[u, v], defined by HDΓ (X; u, v) = EΓ ;p,q (X)u pvq , (7.24) p,q Γ ;p,q ( ) = (− ) j p,q ( j ( , C)) ∈ (Γ ) where E X j 1 H H X R . One may consider an even finer invariant, namely the equivariant Poincaré–Deligne polynomial Γ p,q j p q j PD (X; u, v, t) = H (H (X, C))u v t ∈ R+(Γ )[u, v, t]. (7.25) p,q, j

Clearly one has PDΓ (X; u, v, −1) = HDΓ (X; u, v). When Γ is the trivial group, we set HDΓ (X; u, v) = HD(X; u, v) and PDΓ (X; u, v, t) = PD(X; u, v, t), and get the non-equivariant versions of these polynomials, similar to the (compactly supported) Hodge–Deligne polynomial introduced in (7.20). The case of interest to us is when Γ = μd and the action on the Milnor ﬁber F of a central hyperplane arrangement is determined by

exp(2πi/d) · x = h−1(x).

The following result is proved in [83]. Theorem 7.13 Let A be an arrangement of d lines in P2, such that A has only double and triple points. Then the equivariant Poincaré–Deligne polynomial μ PD d (F; u, v, t) of F coming from the monodromy action is determined by the number d of lines in A , the number n3(A ) of triple points in A and the Papadima–Suciu invariant β3(A ).

7.4 Exercises

Exercise 7.1 Assume that the rank one local system L is situated on a translated component W of a characteristic variety, but not on a component W passing through the origin. Show that the rank one local system L is not admissible. Exercise 7.2 Let A : Q(x, y, z) = 0 be a line arrangement in P2 having only double points. Consider the surface S : Q(x, y, z) + wd = 0inP3, where d is the degree of Q. 148 7 Logarithmic Connections and Mixed Hodge Structures

(i) Show that the surface S has only simple surface singularities of type Ad−1, i.e. locally deﬁned by an equation u2 + v2 + td = 0. Show that the link of such a singularity is a Q-homology sphere, refer to [74, Theorem 3.4.10] if necessary. (ii) Show that the surface S is a Q-manifold and deduce that

b3(S) = b1(S) = 0.

(iii) Show that the corresponding monodromy operator

h1 : H 1(F, C) → H 1(F, C)

is the identity. In particular, show that b1(F) = d − 1. This gives a new proof of a special case of Corollary 5.4. (iv) Extend the above properties to the case of a line arrangement A having only points of multiplicity relatively prime to the degree d. Compare this result to Proposition 7.4.

Exercise 7.3 Use the additivity property (7.21) of the Hodge–Deligne polynomials to obtain a very simple proof of Proposition 1.1. To do this, note that M(A ) is obtained from the plane C2 by removing d lines isomorphic to C, and that each intersection point of multiplicity k has to be put back (k − 1)-times in order to get the right answer.

Exercise 7.4 Let A be a line arrangement in the projective plane P2 and d =|A |. Assume that there are nk points of multiplicity k in A ,fork ≥ 2. (i) Show that if A has only double and triple points, then the second Betti number of the complement M(A ) determines the numbers n2 and n3 and hence the spectrum Sp(A ). (ii) Consider the following two line arrangements

A : xyz(x − y)(x + z)(y + z) = 0 and A : xyz(x − y)(x + y)(x + z) = 0.

Show that b2(M(A )) = b2(M(A )) and A has only double and triple points, namely n2 = 3 and n3 = 4. On the other hand, show that A has points of multiplicity k for k = 2, 3, 4, namely n2 = 6, n3 = 1, n4 = 1. Compute the two spectra Sp(A ) and Sp(A ).

Exercise 7.5 Consider again the two distinct realizations (93)1 and (93)2 of the conﬁguration (93) from Example 2.15, namely

A : ( − )( − )( − − )( + + )( + − )( − + ) = 1 xyz x y y z x y z 2x y z 2x y z 2x 5y z 0 and

A : ( + )( + )( + )( + + )( + + )( + + ) = . 2 xyz x y x 3z y z x 2y z x 2y 3z 2x 3y 3z 0 7.4 Exercises 149

β (A ) = β (A ) = Z/ Z (i) Show that 3 1 1 and 3 2 0, by a direct computation using 3 - coefﬁcients. β (A ) = β (A ) = (ii) Show that 3 1 1 and 3 2 0, by a direct computation based on the formula (7.6) in Remark 7.3 and Theorem 7.5. (iii) Conclude that the Hodge–Deligne polynomials HD(F(A1); u, v) and respectively HD(F(A2); u, v) of the corresponding Milnor ﬁbers are distinct. Chapter 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

Abstract We consider free projective hypersurfaces, stressing the relation with the Jacobian syzygies of the defining equation and giving a proof of K. Saito’s Criterion in this setting. The factorization property for π(A , t) when A is a free arrangement, the fact that any supersolvable arrangement is free, and the freeness of reflection arrangements are all stated in the first section. Then we restrict to the case of curves (and in particular, line arrangements) in P2, and state several characterizations of such free curves. Some applications to H. Terao’s conjecture are also given. Next we introduce a spectral sequence approach to the computation of the Alexander polynomial of a plane curve. Two algorithms are described, one for free plane curves, the other for curves in P2 having only weighted homogeneous singularities.

8.1 Free Divisors and Logarithmic Differential Forms

Let S =⊕k Sk = C[x0,...,xn] be the graded polynomial ring in n + 1 indeter- minates with complex coefﬁcients, where Sk denotes the vector space of degree k homogeneous polynomials. Consider for a degree d polynomial f ∈ Sd the corresponding Jacobian ideal J f generated by the partial derivatives f j of f with respect to x j for j = 0,...,n and the graded Milnor algebra M( f ) =⊕k M( f )k = S/J f . The graded modules of all Jacobian syzygies is deﬁned by

n+1 AR( f ) ={r = (a0, a1,...,an) ∈ S : a0 f0 + a1 f1 +···+an fn = 0} (8.1) and tells us a lot about the geometry of the hypersurface V : f = 0, assumed in the sequel to be reduced. To each Jacobian relation r ∈ AR( f ), one can associate a derivation ( ) = ∂ + ∂ +···+ ∂ D r a0 x0 a1 x1 an xn (8.2) of the polynomial ring S such that D(r) kills f , that is D(r)( f ) = 0. The set of all the derivations killing f is denoted by D0( f ), a graded S-module isomorphic to the module AR( f ). One can consider the Euler derivation

= ∂ + ∂ +···+ ∂ DE x0 x0 x1 x1 xn xn (8.3) and then the graded S-module

D( f ) = S · DE ⊕ D0( f ) (8.4) consists of all the derivations δ of the polynomial ring S preserving the ideal ( f ). Instead of derivations of the polynomial ring S, one can associate to a syzygy r ∈ AR( f ) a differential form on Cn+1 with polynomial coefﬁcients. There are two ways to do this. The ﬁrst approach is to consider the n-form j ω(r) = (−1) a j dx0 ∧···∧dx j ∧···∧dxn, (8.5) j=0,n

∈ ( ) ∧ ω( ) = Ωn( ) and note that r AR f is equivalent to d f r 0. We denote by 0 f the set of all such forms for r ∈ AR( f ). A different approach is to consider the gradient vector ﬁeld of f , namely the vector ﬁeld on Cn+1 given by

= ∂ + ∂ +···+ ∂ X f f0 x0 f1 x1 fn xn (8.6)

∇ and the associated contraction X f . Then, a 1-form of the type j η(r) = (−1) a j dx j (8.7) j=0,n

∇ (η) = ∈ ( ) Ω1( ) satisfies X f 0 if and only if r AR f . We denote by 0 f the set of all such forms for r ∈ AR( f ). Note that the case n = 1 is special, since we have one notation for two slightly different objects, but in our discussion below n ≥ 2. Definition 8.1 The projective hypersurface (or the divisor) V : f = 0inPn,orthe affine cone CV over V given by CV : f = 0inCn+1, is said to be free if one of the following equivalent conditions holds. (i) the module AR( f ) of all Jacobian relations of f is a free graded S-module; (ii) the module D0( f ) of all derivations killing the polynomial f is a free graded S-module; (iii) the module D( f ) of all derivations preserving the ideal ( f ) is a free graded S-module; Ωn( ) (iv) the module 0 f of all n-forms killed by wedge product with df is a free graded S-module; Ω1( ) ∇ (v) the module 0 f of all 1-forms killed by the contraction X f is a free graded S-module. In such a case, the rank of the S-module AR( f ) is n and if

n+1 ri = (ri0,...,rin) ∈ AR( f ) ⊂ S 8.1 Free Divisors and Logarithmic Differential Forms 153 for i = 1,...,n is a homogeneous basis of AR( f ) with deg ri = di , then the integers di are called the exponents of V (or of f ). Note that deg ri = di means deg rij = di for any j = 0,...,n. Consider also the vector

r0 = (r00,...,r0n) = (x0, x1,...,xn), (8.8) which is not in the module AR( f ), but it corresponds to the Euler derivation. This vector enters into the following key result.

Theorem 8.1 (Saito’s Criterion) The homogeneous Jacobian syzygies ri ∈ AR( f ) for i = 1,...,n form a basis of this S-module if and only

φ(f ) = c · f, (8.9) where φ(f ) is the determinant of the (n + 1) × (n + 1) matrix Φ( f ) = (rij)0≤i, j≤n and c is a nonzero constant.

Proof This is proved in detail for a hyperplane arrangement in [180, Theorem 4.19]. However, the general case, though a folklore result, does not seem to have a proof in print. Here is the proof we suggest, following the basic ideas from the proof of [180, Theorem 4.19]. Assume ﬁrst that φ(f ) = c · f , with c ∈ C∗. Since c = 0, it follows that the elements ri are linearly independent over S, hence it is enough to show that they span the S-module AR( f ).Letr ∈ AR( f ) be a new syzygy and note that fr is in the n+1 submodule spanned by r0, r1,...,rn in S , simply by using Cramer’s rule. Hence

fr = a1r1 +···anrn, (8.10) since the coefficient of r0 should clearly vanish due to the relation DE ( f ) = d · f . Let φ j ,for j = 2,...,n + 1, be the determinant of the matrix Φ j obtained by Φ( ) ( ,..., ) replacing in the matrix f the j-th line by the coordinates r0 rn of the syzygy r . Consider the reduced affine hypersurface X defined by f = 0inCn+1 and let p ∈ X be a regular point. Then all the lines in the matrix Φ j correspond to vectors in the tangent space Tp X, which is n-dimensional. It follows that the polynomial φ j vanishes on X (since the regular points are dense in a reduced variety) and hence φ j is divisible by f in S, say we have φ j = h j f for some h j ∈ S. Note also that φ Φ f j can be regarded as the determinant of the matrix j obtained by replacing in Φ( ) ( ,..., ) the matrix f the j-th line by the coordinates fr0 frn of the syzygy fr . Using (8.10), we get f φ j = ca j f , and hence h j f = ca j . It follows that all the coefficients a j are divisible by f and hence (8.10) implies that r is in the submodule spanned by r1,...,rn. Assume now conversely that r1,...,rn form a basis of the module AR( f ).It follows as above that φ = hf, and it remains to show that h should be a constant. Indeed, h = 0 since the elements r j for r = 0,...,n are linearly independent over S, in fact they form a basis of D( f ) when regarded as derivations. Consider 154 8 Free Arrangements and de Rham Cohomology of Milnor Fibers the matrix Ψ0 constructed in the following way: the first line is ( f, 0,...,0).For j = 2,...,n + 1, the j-th line corresponds to the Koszul syzygy

f j−1∂0 − f0∂ j−1.

It follows that there is an (n + 1) × (n + 1)-matrix M such that

Ψ0 = M · Φ( f ).

n n In particular, fh = det(Φ) divides det(Ψ0) =±f ( f0) . Hence h divides ( f0) . Using the other Koszul relations obtained by ﬁxing ∂1,...,∂n instead of ∂0, we get n in a similar way that h divides ( f j ) for j = 1,...,n as well. Since D : f = 0isa reduced hypersurface in Pn, its singular locus is given by

f0 = f1 =··· fn = 0 and has codimension at least 2 in Pn. This implies that h is a constant and completes our proof.

In particular, if the condition (8.9) holds, then the syzygies (ri )i=1,n form a basis for the free S-module AR( f ) and hence one gets

d1 + d2 +···+dn = d − 1.

For more on this see [180, 197, 234].

Remark 8.1 A central arrangement A : f = 0, or the corresponding projective arrangement A , is said to be free if the hypersurface obtained as a union of all the hyperplanes in A is free, i.e. the module D( f ), or equivalently the module D0( f ), is free. To get the exponents of the central arrangement A from the exponents of the projective arrangement A , we just have to add d0 = 1 to the set of exponents d1,...,dn. Because of this remark, in the sequel we usually give only the exponents of A .

Example 8.1 The Boolean arrangement A : f = x0x1 ···xn = 0 is free with exponents d1 = d2 = ··· = dn = 1. To see this, it is enough to apply Saito’s Criterion above with

r j = (x0, 0,...,0, −x j , 0,...,0). B : = ( − ) = Example 8.2 The braid arrangement r n f 0≤i< j≤n xi x j 0isfree with exponents d1 = 0 and dk = k for k = 2,...,n. To see this, it is enough to = ( , ,..., ), = ( k , k ,..., k ) apply Saito’s Criterion above with r1 1 1 1 and rk x0 x1 xn for k = 2,...,n. To complete the computation, one should recall the Vandermonde formula, giving the equation of Br n as the determinant, which turns out to be just Saito’s Criterion in this case. Also note that D(r1)(xi − x j ) = 0 and 8.1 Free Divisors and Logarithmic Differential Forms 155

( )( − ) = k − k , D rk xi x j xi x j which easily implies that D(r j ) ∈ D( f ) for any j ≥ 1. By a modified version of Saito’s Criterion, it follows that D( f ) is a free graded S-module. In fact we have the following very general result about the freeness of the reflection arrangements, see [180, Theorem 6.60]. Theorem 8.2 If G is a finite reflection group, then its reflection arrangement A (G) is free. When G is an irreducible group of rank k, then the exponents of the free arrangement A (G) are exactly the coexponents n2 ≤···≤nk , with the first one n1 = 1 omitted. Indeed, the projective arrangement in this case is in Pk−1, hence we need k − 1 exponents according to our definition. Example 8.3 We refer to [180, Table B.1] for the following useful results. One can find there the coexponents for the exceptional reflection groups as well. n (i) For the braid arrangement Br n, regarded in C , the coexponents are

(1, 2,...,n − 1).

(ii) For the full monomial arrangement A (r, 1, n) the coexponents are

(1, r + 1, 2r + 1,...,(n − 1)r + 1).

(iii) For the monomial arrangement A (r, r, n) the coexponents are

(1, r + 1, 2r + 1,...,(n − 2)r + 1,(n − 1)(r − 1)).

We have seen that the Poincaré polynomial π(A , t) for the Boolean arrangement and for the braid arrangement factors as a product of linear factors with integer coefﬁcients, recall Example 2.14. A beautiful result generalizing this fact, and due to Terao [222], is the following. For the case n = 2, see Exercise 8.3. Theorem 8.3 Let A be a free hyperplane arrangement in Pn, with exponents d1,...,dn. Then the Poincaré polynomial π(A , t), and hence the Betti polynomial B(M(A ), t) of the projective complement M(A ) are given by

π(A , t) = B(M(A ), t) = (1 + d1t) ···(1 + dnt).

In particular, the exponents of a free hyperplane arrangement A are determined by the intersection lattice L(A ). Since the supersolvable arrangements also have Poincaré polynomials π(A , t) which split into linear factors, recall Theorems 2.4 and 3.3, one may wonder if there is a relation between these two important classes of hyperplane arrangements. One indeed has the following result. 156 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

Theorem 8.4 Any supersolvable arrangement is free. As a matter of fact, it is proved in [180, Theorem 4.58] that any supersolvable arrangement is inductively free, and an inductively free arrangement is free by definition. Since there are free arrangements which are not inductively free, see [180, Example 4.59], it follows that there are free arrangements which are not supersolvable. A simple example is given below in Example 8.7. In this direction, see also [138, Theorem 4.2].

Example 8.4 The deleted B3-line arrangement is known to be supersolvable, recall Example 2.9. Combining this fact with Example 2.13 and the above theorem, it follows that the deleted B3-line arrangement is free with exponents (3, 4).

Remark 8.2 Note that some free hyperplane arrangements, such as the Boolean arrangements discussed in Example 8.1 (resp. the full monomial line arrangements A (m, 1, 3) for m ≡ 0orm ≡ 2 mod 3 discussed in Examples 5.5 and 8.6), have trivial monodromy on H ∗(F, C) (resp. on H 1(F, C)). Other free line arrangements, such as the monomial arrangement discussed in Examples 5.4 and 8.6, have a non- trivial monodromy on H 1(F, C). So it seems that there is no direct relation between these two properties.

Remark 8.3 K. Saito has conjectured that the complement of a free hyperplane arrangement is a K (π, 1)-space, see Conjecture 5.18 in [180]. This conjecture was perhaps motivated by the fact that a supersolvable arrangement is both free by Theo- rem 8.4 and a K (π, 1)-space by Theorems 4.15 and 4.16. Many free line arrangement complements are K (π, 1)-spaces, see [114, Example 3.13]. There the arrangements 2 2 2 2 2 2 2 2 Ap : (x − y )z (k x − z )(k y − z ) = 0 1≤k≤p are considered, which are free with exponents (2p + 1, 2p + 1), but note that there is a misprint in their exponents in that paper. The ﬁrst counterexamples to the above conjecture were constructed in [108].

The graded S-module AR( f ) gives rise to a coherent sheaf, usually denoted by T D or Der(− log D), and called the sheaf of logarithmic vector ﬁelds along D.Thisis a reﬂexive sheaf on Pn,see[197], in particular a locally free sheaf when n = 2. The dual sheaf of Der(− log D) is denoted by Ω1(log D) and is called the sheaf of logarithmic 1-forms on Pn with poles along the divisor D. Note that when D is a free divisor with exponents d1 ≤ ··· ≤ dn, it follows that AR( f ) = S(−d1) ⊕···⊕S(−dn), and hence in this case one has

Der(− log D) = O(−d1) ⊕···⊕O(−dn) and 1 Ω (log D) = O(d1) ⊕···⊕O(dn). 8.1 Free Divisors and Logarithmic Differential Forms 157

2 n If ct (E) = 1 + c1(E)t + c2(E)t +···+cn(E)t denotes the total Chern class of a vector bundle E on Pn, regarded as an element in the truncated polynomial ring

Z[t]/(tn+1) = H ∗(Pn, Z), i.e. t denotes the standard generator of H 2(Pn, Z), then Theorem 8.3 can be restated as the equality 1 π(A , t) = ct (Ω (log A )) (8.11) for any free hyperplane arrangement A in Pn. Since both polynomials in this equality have degree at most n, this equality can be regarded as an equality in the polynomial ring C[t] as well. If we go beyond the class of free hyperplane arrangements, one may ask when the sheaf Der(− log A ) or, equivalently, its dual Ω1(log A ), is locally free. The answer is given by the following result, see [173].

Theorem 8.5 Let A be a central, essential hyperplane arrangement in Cn+1 and denote by A the corresponding arrangement in Pn. The sheaf Ω1(log A ) is locally free (of rank n) if and only if for all ﬂats X ∈ L(A ),X ={0}, the central hyperplane arrangement AX is free. A central hyperplane arrangement in Cn+1 satisfying the equivalent conditions in the above theorem is said to be locally free, see [236]. Example 8.5 It follows from the deﬁnition that a free hyperplane arrangement is also locally free. The converse fails, as the following example shows, see for more details [107, 173]. Consider in C4 the arrangement A : f = (a0x0 + a1x1 + a2x2 + a3x3) = 0

a=(a0,a1,a2,a3)

4 where a = (a0, a1, a2, a3) ∈{0, 1} , a = (0, 0, 0, 0). Then A is locally free but not free. For locally free arrangements we have the following analog of the formula (8.11), see [173]. Theorem 8.6 Let A be a central, essential hyperplane arrangement in Cn+1 and denote by A the corresponding arrangement in Pn.IfA is locally free, then

1 π(A , t) = ct (Ω (log A )) in the polynomial ring C[t].

Remark 8.4 For other interesting relations between hyperplane arrangements and Chern classes we refer the reader to [6]. This includes a description of the Chern– Schwartz–MacPherson class of a hyperplane arrangement complement M(A ) in terms of the characteristic polynomial χ(A , t). 158 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

For a central arrangement A : f = 0inCn+1, consider the module Ω p(∗A ) of global regular p-forms on M(A ). Using Grothendieck’s Theorem 1.8, it follows that the cohomology of the corresponding de Rham complex (Ω∗(∗A ), d) coincides with the cohomology groups H ∗(M(A ), C) of the complement M(A ). We introduce the set Ω p(A ) of logarithmic p-forms along A consisting of the forms ω ∈ Ω p(∗A ) such that f ω and f dω both admit regular extensions to Cn+1. Then it is easy to see that Ω p(A ) is a graded S-module of ﬁnite type. For instance, Ω0(A ) = S and Ω1(A ) has as associated coherent sheaf on Pn, exactly the sheaf of logarithmic 1-forms Ω1(log A ) considered above. Moreover, the arrangement A is free if and only if the S-module Ω1(A ) is free, and then all the other S-modules Ω p(A ) are free, since then they satisfy

Ω p(A ) =∧pΩ1(A ), see [197]. We get in this way a sub-complex

(Ω∗(A ), d) ⊂ (Ω∗(∗A ), d) and it is natural to ask about the corresponding cohomology groups H p(Ω∗(A ), d)). Recall that the projective dimension of a ﬁnite type graded S-module M, denoted by pd(M), is by deﬁnition the length k of a minimal free graded resolution

0 → Fk →···→ F1 → F0 → M → 0, see for instance [109]. Then clearly such a module M is free exactly when pd(M) = 0 and we set by convention pd(0) =−∞. Deﬁnition 8.2 A central arrangement A is called tame if one has pd(Ω p(A )) ≤ p for any positive integer p. By the above discussion it follows that any free arrangement is tame, and it is shown in [182] that A is tame if the associated projective arrangement A is a general position arrangement. Is is also shown in [182] that any central arrangement in Cn+1 is tame when n ≤ 2 and that the central arrangement in C4 considered in Example 8.5 above is not tame. The following result is proved in [229].

Theorem 8.7 For any tame central hyperplane arrangement A , the logarithmic comparison theorem holds, i.e. the inclusion (Ω∗(A ), d) ⊂ (Ω∗(∗A ), d) is a quasi- isomorphism. In particular, for any positive integer p one has

H p(M(A ), C) = H p(Ω∗(A ), d) = H p(Ω∗(∗A ), d).

In fact, the above isomorphisms also hold for any central arrangement A : f = 0 in Cn+1 if either n ≤ 3orn arbitrary, but p = n + 1, see [229]. It is in fact conjectured that these isomorphisms always hold, see [221, 229]. For other versions of the logarithmic comparison theorem, see [41, 124]. Tame arrangements also play 8.1 Free Divisors and Logarithmic Differential Forms 159 a key role in the paper [51]. A more general discussion of tame divisors can be found in [227]. A free divisor can also be an irreducible hypersurface. This is usually the case with discriminants, objects that played a key role in the introduction of K. Saito’s notion of freedivisorsin[197]. Here is one example of this situation. Let Δn ∈ C[a0,...,an] be the discriminant of the general degree n binary form

n n−1 n−1 n a0x + a1x y +···+an−1xy + an y .

One has the following result, see [31, 53].

n Proposition 8.1 For n ≥ 3, the hypersurface Dn : Δn = 0 in P has degree d = 2n −2, is irreducible and free with exponents d1 = d2 = d3 = 1,d4 =···=dn = 2.

8.2 Free Curves, Free Line Arrangements, and Terao’s Conjecture

From now on we discuss in detail the case of plane curves. To simplify the notation, let S = C[x, y, z] be the graded polynomial ring in three variables x, y, z and let C : f = 0 be a reduced curve of degree d in the complex projective plane P2.The minimal degree of a Jacobian relation for f is the integer mdr( f ) deﬁned to be the smallest integer m ≥ 0 such that there is a nontrivial relation

afx + bfy + cfz = 0 (8.12) among the partial derivatives fx , fy and fz of f with coefﬁcients a, b, c in Sm ,the vector space of homogeneous polynomials of degree m. When mdr( f ) = 0, then C is a union of lines passing through one point, a situation easy to analyse. We assume from now on that mdr( f ) ≥ 1.

Denote by τ(C) the global Tjurina number of the curve C, which is the sum of the Tjurina numbers of the singular points of C, recall Deﬁnition 2.17. We denote by J f the Jacobian ideal of f , i.e. the homogeneous ideal in S spanned by fx , fy, fz, and by M( f ) = S/J f the corresponding graded ring, called the Jacobian (or Milnor) algebra of f .

Deﬁnition 8.3 For a homogeneous reduced polynomial f ∈ Sd one deﬁnes (i) the coincidence threshold

ct( f ) = max{q : dim M( f )k = dim M( fs )k for all k ≤ q}, 160 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

with fs a homogeneous polynomial in S of the same degree d as f and such that 2 Cs : fs = 0 is a smooth curve in P . (ii) the stability threshold st( f ) = min{q : dim M( f )k = τ(C) for all k ≥ q}. It is clear that one has

ct( f ) ≥ mdr( f ) + d − 2, (8.13)

with equality for mdr( f )

if and only if afx + bfy + cfz = 0 and a, b, c are in Sm . The following result is well known, see for instance [209], except for the last claim which is proved in [69].

Proposition 8.2 The curve C : f = 0 is a free divisor if the following equivalent conditions hold. (i) N( f ) = 0, i.e. the Jacobian ideal is saturated. (ii) The minimal resolution of the Milnor algebra M( f ) has the following form

( , , ) 3 fx fy fz 0 → S(−d1 − d + 1) ⊕ S(−d2 − d + 1) → S (−d + 1) −−−−−→ S

for some positive integers d1, d2. (iii) The graded S-module AR( f ) is free of rank 2, i.e. there is an isomorphism

AR( f ) = S(−d1) ⊕ S(−d2)

for some positive integers d1, d2. (iv) The equality ct( f ) + st( f ) = T holds, where T = 3(d − 2).

When C is a free divisor, the integers d1 ≤ d2 are called the exponents of C.They satisfy the relations

2 d1 + d2 = d − 1 and τ(C) = (d − 1) − d1d2, (8.14)

where τ(C) is the total Tjurina number of C, see for instance [97] or Theorem 8.8 below. The following two more geometric characterizations of freeness also play a key role in the theory. Consider ﬁrst the rank two vector bundle T C=Der(− log C) of logarithmic vector ﬁelds along C, which is the coherent sheaf associated to the graded S-module AR( f )(1). Then the curve C is free if and only if this vector bundle splits as the sum of two line bundles on P2. 8.2 Free Curves, Free Line Arrangements, and Terao’s Conjecture 161

Secondly, we have the following result of A. du Plessis and C.T.C. Wall, see Theorem 3.2 in [106]. Theorem 8.8 The global Tjurina number τ(C) of a degree d reduced plane curve C : f = 0 satisﬁes 2r + 2 − d τ(C) ≤ (d − 1)2 − r(d − 1 − r) − , 2 where r = mdr( f ) is the minimal degree of a Jacobian relation for f . Moreover, equality holds if and only if C : f = 0 is a free curve and in this case r = d1 < d/2. This theorem yields in particular the following result.

Corollary 8.1 For a reduced curve C : f = 0 in P2 of degree d, the following conditions are equivalent. (i) τ(C) = (d − 1)2; (ii) mdr( f ) = 0; (iii) C is a union of d lines passing through a common point.

Proof The fact that (i) implies (ii) follows from Theorem 8.8. As already mentioned above, mdr( f ) = 0 implies that C is a union of d lines passing through a common point. And clearly such a curve has a unique singularity, with Tjurina number equal to (d − 1)2.

Remark 8.5 For a line arrangement A : f = 0inP2, it turns out that the invariants mdr( f ),ct( f ) and st( f ) are not determined by the combinatorics. This can be seen in the following example, essentially coming from Ziegler’s paper [240]. We consider two line arrangements A : f = 0 and A : f = 0, both consisting of nine lines, and having only double and triple points. More precisely, they have n2 = 18 double points and n3 = 6 triple points, and hence τ(A ) = τ(A ) = 42. The ﬁrst arrangement is

A : f = xy(x − y − z)(x − y + z)(2x + y − 2z)(x + 3y − 3z)(3x + 2y + 3z)

(x + 5y + 5z)(7x − 4y − z) = 0.

A picture of the afﬁne version (not of its decone) of the line arrangement A is given in Fig. 8.1. In this case the six triple points are on a conic, and a direct computation shows that mdr( f ) = 5, ct( f ) = 12 and st( f ) = 14.

The second arrangement is

A : f = xy(4x − 5y − 5z)(5x + 2y − 10z)(16x + 13y − 20z)(x − 3y + 15z)

(2x − y + 10z)(6x + 5y + 30z)(3x − 4y − 24z) = 0. 162 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

Fig. 8.1 Ziegler’s arrangement A

Fig. 8.2 Ziegler’s arrangement A

A picture of the afﬁne version of the line arrangement (not of its decone) A is given in Fig. 8.2. In this case the six triple points are not on a conic, and a direct computation shows that 8.2 Free Curves, Free Line Arrangements, and Terao’s Conjecture 163

mdr( f ) = 6, ct( f ) = 13 and st( f ) = 13.

Note also that both arrangements A and A are far from being free, for instance

ct( f ) + st( f ) − T = ct( f ) + st( f ) − T = 26 − 21 = 5.

Moreover, A and A are lattice-isotopic arrangements, see Exercise 8.5. See also [205, Example 13] and [227, Example 5.10] for more information on these two line arrangements.

For the line arrangements in P2, there are deep relations between the multiplicity of their intersection points and the minimal degree of a Jacobian syzygy. One example of such relations occurs in the following result, see [70] for a proof.

Theorem 8.9 If A : f = 0 is a line arrangement in P2 and m is the multiplicity of one of its intersection points, then either mdr( f ) = d − mormdr( f ) ≤ d − m − 1, and then one of the following two cases occurs.

(i) mdr( f ) ≤ m − 1. Then equality holds, i.e. mdr( f ) = m − 1, one has the inequality 2m < d + 1 and the line arrangement A is free with exponents d1 = mdr( f ) = m − 1 and d2 = d − m; (ii) m ≤ mdr( f ) ≤ d − m − 1, in particular 2m < d.

This theorem yields in particular the following result.

Corollary 8.2 For a line arrangement A : f = 0 in P2 with deg f = d, the following conditions are equivalent. (i) τ(A ) = (d − 1)2 − (d − 2) = d2 − 3d + 3; (ii) mdr( f ) = 1; (iii) A is a union of d−1 lines passing through a common point P, plus a transversal line not passing through P.

Proof The fact that (i) implies (ii) follows from Theorem 8.8 and Corollary 8.1. Assume now that r = mdr( f ) = 1 and let m be the maximal multiplicity of a point in A . Then m < d by Corollary 8.1 and if m = d − 1 then we are done. Assume now that 2 ≤ m ≤ m − 2. Then Theorem 8.9 implies that either r = d − m = 1, or r = m − 1, or r ≥ m. The ﬁrst and the last case are clearly impossible. If the second case holds, then A has only double points and such a line arrangement has r = mdr( f ) = d − 2, see [102]. Hence d = 3 and our nodal arrangement is of the type described in (iii). To complete the proof, we have to show that a line arrangement as in (iii) has the Tjurina number given by (i). This is obvious, since such a line arrangement has d − 1 nodes and a point P of multiplicity d − 1, and hence τ(A ) = d − 1 + (d − 2)2 = d2 − 3d + 3. 164 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

As another application of Theorem 8.9 we show, following ideas from [98], that there are free line arrangements with exponents (d1, d2) chosen arbitrarily. To do this, for two integers i ≤ j we deﬁne a homogeneous polynomial in C[u, v] of degree j − i + 1 by the formula

gi, j (u, v) = (u − iv)(u − (i + 1)v) ···(u − jv). (8.15)

Theorem 8.10 Consider the line arrangement A : f = 0 of d ≥ 3 lines in P2 given by ( , , ) = ( , ) ( , ) = , f x y z xg1,d1 x y g1,d2 x z 0 for 1 ≤ d1 < d/2 and d2 = d − 1 − d1. Then the following holds.

(i) The line arrangement A is free with exponents (d1, d2). (ii) The line arrangement A has at most two points of multiplicity > 2. (iii) If M(A ) denotes the complement of A in P2, then one has

M(A ) = (C \{d1points}) × (C \{d2points}).

A (π, ) π = F × F F In particular, is a K 1 -arrangement where d1 d2 , with e the free group on e generators.

Proof Note that the line arrangement A has the following two points of multiplicity (possibly) strictly greater than 2: the point A = (0 : 0 : 1) of multiplicity d1 + 1 and the point B = (0 : 1 : 0) of multiplicity d2 + 1. Then we apply Theorem 8.9 by taking m = d2 +1. It follows that either mdr( f ) = d −m = d1,ormdr( f ) ≤ d1 −1 and 2m < d + 1, which is false since

2m = 2d2 + 2 ≥ d1 + d2 + 2 ≥ d + 1.

Hence the only possibility is mdr( f ) = d1. As we have seen above, in the line arrangement A there is one point of multiplicity d1 + 1, one point of multiplicity d2 + 1 and d1d2 additional nodes. Since the Tjurina number of a k-multiple singular point of A is (k − 1)2, it follows that

τ(A ) = 2 + 2 + = ( − )2 − . d1 d2 d1d2 d 1 d1d2

Hence the arrangement A is free with exponents (d1, d2) by Theorem 8.8. In fact, the freeness also follows from the fact that it is supersolvable. Indeed, the point A is a modular flat, which gives rise to an obvious maximal modular chain. It remains to use Theorem 8.4. To prove the last claim, we first delete the line x = 0, and we get the affine plane with coordinates y, z. The trace B of A on this affine plane is given by the equation ( , ) ( , ) = , g1,d1 1 y g1,d2 1 z 0 which implies our claim. 8.2 Free Curves, Free Line Arrangements, and Terao’s Conjecture 165

For the proof of the claims in the next example we refer to [70], where the computer algebra software Singular is used to derive these formulas.

Example 8.6 (i) The monomial line arrangement

A (m, m, 3) : f = (xm − ym )(ym − zm)(xm − zm) = 0,

for m ≥ 2 was introduced in Example 5.4. This arrangement is of pencil type, more precisely it is a (3, m)-net, and this structure can be used to show that A (m, m, 3) is free with exponents (m + 1, 2m − 2). The 2-forms ωi = ω(ρi ) for i = 1, 2 corresponding to a basis ρ1,ρ2 of the S-module AR( f ),asin Eq. (8.5), can be taken to be the forms ω1 and ω2 given by

m+1 m m m+1 m m ω1 = (x − 2xy − 2xz )dy ∧ dz − (y − 2yz − 2yx )dx ∧ dz

+(zm+1 − 2zxm − 2zym )dx ∧ dy,

and

m−1 m−1 m−1 m−1 m−1 m−1 ω2 = y z dy ∧ dz − x z dx ∧ dz + x y dx ∧ dy.

(ii) The full monomial line arrangement

A (m, 1, 3) : f = xyz(xm − ym )(ym − zm )(xm − zm) = 0

for m ≥ 2 was introduced in Example 5.5. This arrangement contains the pencil type arrangement A (m, m, 3), and this structure can be used to show that A (m, 1, 3) is free with exponents (m +1, 2m +1). The 2-forms ωi = ω(ρi ) for i = 1, 2 corresponding to a basis ρ1,ρ2 of the S-module AR( f ),asinEq.(8.5), can be taken to be in this case the forms ω1 and ω2 given by

ω1 = ady ∧ dz − bdx ∧ dz + cdx ∧ dy,

where a = (m + 2)xm+1 − (2m + 1)xym − (2m + 1)xzm ,

b =−(2m + 1)xm y + (m + 2)ym+1 − (2m + 1)yzm ,

c =−(2m + 1)xm z − (2m + 1)ym z + (m + 2)zm+1,

and ω2 = Ady ∧ dz − Bdx ∧ dz + Cdx ∧ dy, 166 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

where

A = (2m + 1)xy2m − (3m2 + 4m + 2)xym zm + (2m + 1)xz2m ,

B = 3(m+1)xm ym+1−(m+2)y2m+1−3(m+1)2xm yzm +(2m+1)yzm (ym +zm),

+ + C =−3(m +1)2xm ym z +(2m +1)ym z(ym +zm)+3(m +1)xm zm 1 −(m +2)z2m 1.

(iii) The Hessian line arrangement

A : f = xyz[(x3 + y3 + z3)3 − 27x3 y3z3]=0

was introduced in Exercise 5.5 and consists of all the four singular members of the pencil P : uc1 + vc2, where

3 3 3 c1 = x + y + z and c2 = xyz.

This arrangement is of pencil type, more precisely it is a (4, 3)-net, and this structure can be used to show that A is free with exponents (4, 7).Asetof generating syzygies of AR( f ) correspond to the forms ω1 and ω2 below.

3 3 3 3 3 3 ω1 =−x(y − z )dy ∧ dz − y(x − z )dx ∧ dz + z(y − x )dx ∧ dy

and ω2 = Ady ∧ dz − Bdx ∧ dz + Cdx ∧ dy,

where A = x7 + 3x4 y3 + 3xy6 + 13x4z3 − 81xy3z3 + 34xz6,

B = y7 + 16y4z3 − 44yz6,

and C =−10x6z − 21x3 y3z − 13y6z − 31x3z4 + 47y3z4 + z7.

In the above formulas, note that when d1 < d2, the syzygy ρ1 of degree d1 is uniquely determined up to a constant factor, while the next syzygy ρ2 is determined modulo a constant factor only modulo multiples of ρ1. In particular, one can try to ﬁnd simpler formulas for the differential forms ω2 above.

Example 8.7 The Ceva line arrangement A = C(3), regarded as a central arrangement in C3, is not supersolvable, recall Remark 5.5. Nevertheless, A = C(3) is a free arrangement by Example 8.6 (i). 8.2 Free Curves, Free Line Arrangements, and Terao’s Conjecture 167

We also mention the following result, stating that any line arrangement is a subarrangement of a supersolvable line arrangement. This is a more precise version of the claim made in Exercise 4.9, which is valid in arbitrary dimensions.

Theorem 8.11 For any line arrangement A in P2 and any point p of P2 not in A , consider the line arrangement B(A , p) obtained from A by adding all the lines determined by the point p and by each of the multiple points in A . Then the central arrangement C in C3 associated to the line arrangement B(A , p) is a supersolvable arrangement. In particular, B(A , p) is a free K (π, 1) line arrangement.

The case when p and several multiple points of A are collinear is allowed, and the line added in such a case is counted just once, hence B(A , p) is a reduced line arrangement.

Proof The point p ∈ P2 gives rise to a modular 1-dimensional ﬂat X in the central arrangement C . The ﬂat X gives rise to an obvious maximal modular chain in C .It remains to use Theorems 4.15, 4.16 and 8.4, and recall that C is free (resp. K (π, 1)) if and only if B(A , p) is free (resp. K (π, 1)).

A similar result holds when the line arrangement A is replaced by an irreducible curve C in P2 having only nodes and cusps as singularities, and p is a generic point in P2,see[70]. We say that Terao’s Conjecture holds for a free hyperplane arrangement A if any other hyperplane arrangement B, of the same dimension and having an isomorphic intersection lattice L(B) = L(A ),isalsofree,see[180, 234]. This conjecture is open even in the case of line arrangements in the complex projective plane P2, in spite of a lot of efforts, see for instance [2, 3]. For line arrangements, since the total Tjurina number τ(A ) is determined by the intersection lattice L(A ), a possible approach to proving Terao’s Conjecture may be to check that A : f = 0 and B : g = 0 satisfy mdr( f ) = mdr(g) and then apply Theorem 8.8. Some partial results in this direction collected from [2, 70, 113] are stated in the next result. Much more can be found in [180, 234].

Proposition 8.3 Let A be a free projective line arrangement with exponents d1 ≤ d2. Let m be the maximal multiplicity of an intersection point in A . Then, if m ≥ d1, Terao’s Conjecture holds for the line arrangement A . In particular, this is the case when one of the following conditions holds.

(i) d1 = d − m; (ii) m ≥ d√/2; (iii) d1 ≤ 2d + 1 − 1. Moreover, Terao’s Conjecture holds for the projective line arrangement A when either d =|A |≤12 or d1 ≤ 5.

Example 8.8 Terao’s Conjecture holds for the line arrangement A from Theorem 8.10. Indeed, in this case m = d2 + 1 ≥ d/2. 168 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

8.3 Spectral Sequences and Alexander Polynomials

Let C : f = 0 be a curve in the complex projective plane P2, having only isolated singularities. Assume C is defined by a homogeneous polynomial f ∈ S = C[x, y, z] of degree d ≥ 3. Let U = P2 \ C, and consider the Milnor fiber F defined by f (x, y, z) = 1inC3 with monodromy action h : F → F,

h(x) = exp(2πi/d) · (x, y, z).

Our aim is to study the characteristic polynomials of the monodromy operators induced by h, namely the Alexander polynomials

Δ j ( ) = ( · − j | j ( , C)), C t det t Id h H F (8.16) for j = 0, 1, 2. These polynomials were already introduced for a hyperplane arrangement in (5.4), where they were denoted by Δ j (h, t). It is clear that, when the curve Δ0 ( ) = − C is reduced, one has C t t 1. Moreover,

Δ0 ( )Δ1 ( )−1Δ2 ( ) = ( d − )χ(U), C t C t C t t 1 (8.17) where χ(U) denotes the Euler characteristic of the complement U, see for instance the zeta-function formula (5.5)or[74, Proposition 4.1.21]. On the other hand, one has χ(U) = χ(P2) − χ(C) = (d − 1)(d − 2) + 1 − μ(C), (8.18) where μ(C) is the global Milnor number of C, that is the sum of the Milnor numbers Δ ( ) = Δ1 ( ) of all the singularities of C. The polynomial C t C t is called the Alexander Δ2 ( ) polynomial of the plane curve C and it determines the remaining polynomial C t via the formula (8.17). To compute the Alexander polynomial ΔC (t) is the same as computing the eigenvalues of the monodromy operator

h1 : H 1(F, C) → H 1(F, C). (8.19)

This problem, going back to O. Zariski, has been considered by many authors, from a multitude of viewpoints and using a variety of techniques, see for instance [74, 110, 154, 178]. Our approach, described below, essentially follows Grifﬁths’ idea of represent- ing cohomology classes by rational differential forms, see [126]. Let Ω j denote the graded S-module of regular differential j-forms on C3,for0≤ j ≤ 3. The complex ∗ = (Ω∗, ∧) K f d f can be regarded as the Koszul complex in S of the partial derivatives fx , fy and fz of the polynomial f . Since C has only isolated singularities, it follows that 0( ∗ ) = 1( ∗ ) = , H K f H K f 0 (8.20) 8.3 Spectral Sequences and Alexander Polynomials 169

2( ∗ ) = and moreover H K f 0 if and only if C is a smooth curve. Similarly to the Koszul ∗,∗ complex, one may consider the following double complex (C , dI , dII), where

Cs,t = Ωs+t+1(td) (8.21)

s,t for t ≥ 0 and C = 0fort < 0, and the anti-commuting differentials (dI , dII) are given by s,t s+1,t s,t s,t+1 dI = d : C → C and dII = d f ∧:C → C . (8.22)

Fix an integer k such that 1 ≤ k ≤ d and set

λ = exp(−2πik/d). (8.23)

( ∗,∗, , ) Note that for any such integer k, we have a subcomplex Ck dI dII of the double ∗,∗ complex (C , dI , dII), where we set

s,t = Ωs+t+1( ) = Ωs+t+1. Ck td k td+k s,t C Now each term Ck is a ﬁnite dimensional -vector space. Moreover, there is a standard construction of an E1-spectral sequence E∗( f )k starting with the double ( ∗,∗, , ) complex Ck dI dII and converging to the cohomology of the associated total complex ( ∗,∗, , ) = (⊕ s,t, + ), Tot Ck dI dII s+t=mCk dI dII refer to [127], pp. 438–443, or to [168, Theorem 2.15], where a more general version is discussed. The E1-page of this spectral sequence is obtained by taking the cohomology of the double complex with respect to the vertical differential dII, and hence is given by

s,t ( ) = s,t ( ∗,∗, , ) = s+t+1( ∗ ) . E1 f k HII Ck dI dII H K f td+k

The following result, see [73], [74, Chap. 6], [95], explains the relation of this spectral sequence to the reduced Milnor ﬁber cohomology H˜ ∗(F, C). Theorem 8.12 With the above notation, for any integer k with 1 ≤ k ≤ d, there is an E1-spectral sequence E∗( f )k such that

s,t ( ) = s+t+1( ∗ ) E1 f k H K f td+k and converging to s,t ( ) = s ˜ s+t ( , C) E∞ f k Gr P H F λ where P∗ is a decreasing filtration on the Milnor fiber cohomology, called the pole order filtration. 170 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

− , , 1 2( ) d1 0 2( ) E1 f k E1 f k

d 0,1( ) 1 1,1( ) E1 f k E1 f k

1,0( ) 2,0( ) E1 f k d1 E1 f k

Fig. 8.3 The E1-term of the spectral sequence E∗( f )k

The differential d1 on the E1-term of the spectral sequence E∗( f )k , represented by the horizontal arrows in Fig.8.3, is induced by dI , and hence it is completely determined by the morphism of graded C-vector spaces

: 2( ∗ ) → 3( ∗ ) d H K f H K f (8.24) induced by the exterior differentiation of forms, i.e. d :[ω] →[d(ω)].

Example 8.9 When C : f = 0 is a smooth curve, it follows from the vanishings in the formula (8.20) that the spectral sequence E∗( f )k degenerates at the E1-term for any k. In this case, the pole order ﬁltration on the cohomology group H 2(F) is exactly the Hodge ﬁltration, as considered by Steenbrink in [215]. Conversely, it is obvious that E1( f )k = E∞( f )k for some k implies that the curve C is smooth.

We have the following key result due to M. Saito [199], telling us when E2( f )k = E∞( f )k for all k.

Theorem 8.13 If the reduced plane curve C : f = 0 has only weighted homogeneous singularities, then the spectral sequence E∗( f )k degenerates at the E2-term for any k. In particular, this applies to any line arrangement A : f = 0 in P2.

n Terao has conjectured that E2( f )k = E∞( f )k for any hypersurface f = 0inP and any k,see[221] where a rather different notation is used. It was already noticed in [73] n that, for a hypersurface f = 0inP with isolated singularities, E2( f )k = E∞( f )k for all k implies that all the singularities should be weighted homogeneous. It seems however that Terao’s conjecture holds for any projective hyperplane arrangement, and 8.3 Spectral Sequences and Alexander Polynomials 171 this has signiﬁcant computational consequences, see [100] for additional information. Moreover, we also have the following result, see [95, Theorem 5.3], which is of fundamental importance in doing explicit computations.

Theorem 8.14 Assume that the plane curve C : f = 0 in P2 has only weighted homogeneous singularities. Then the dimension of

1−t,t ( ) = { : 2( ∗ ) → 3( ∗ ) } E2 f k ker d H K f td+k H K f td+k α is upper bounded by the total number of spectral numbers pi , j of the singularities ( , ) td+k C pi of C equal to d , where pi ranges over all the singularities of C and j = 1,...,μ(C, pi ).

For a hypersurface singularity (X, 0) : g = 0 at the origin of Cn+1, the definition of the spectrum Sp(g) is given by the formula (7.7), where we replace the global Milnor fiber F : Q = 1 by the local Milnor fiber F0 of the holomorphic germ g at the origin of Cn+1, and the monodromy operator h∗ by the semisimple part of the ∗ monodromy operator h0 of g at the origin. For more on the spectrum and spectral numbers (a.k.a. spectrum numbers, or exponents of the singularity), especially in the case of an isolated singularity (X, 0), see for instance [192, Sect. 12.1.3] or [145, Chap. 8], but note a shift by +1 in our book. Namely, the spectral numbers of a plane curve singularities are contained in the interval (−1, 1) in [145, (8.3.4)], but in the interval (0, 2) in [96, Theorem 5.3] and here. We prefer to avoid using the term ‘exponent’ in this context, not to get mixed up with the exponents of free divisors.

Example 8.10 The spectral numbers (including their multiplicities) of a plane curve ( , ) : ( , ) = a + b = , ≥ a1 + b1 singularity X 0 g u v u v 0fora b 2aregivenby a b for all 1 ≤ a1 < a and 1 ≤ b1 < b,see[145]. In particular, for a node a = b = 2 and we get only one spectral number, namely α = 1, with multiplicity 1.

It follows from Theorem 8.14 that, for a reduced plane curve having only weighted = −1,2 homogeneous singularities, the top differential d1 d1 in Fig. 8.3 is injective, as = 1−t,t ≥ well as the differentials d1 d1 for t 3, which are in fact isomorphisms, and for this reason are not represented in Fig. 8.3. In conclusion, we have the following. Corollary 8.3 For a degree d reduced plane curve C : f = 0 having only weighted homogeneous singularities, the term E2( f )k = E∞( f )k is contained in the ﬁrst quadrant for any 1 ≤ k ≤ d.

Example 8.11 (i) For a nodal line arrangement, Theorem 8.14 implies that we have 1−t,t ( ) = = = E2 f k 0 except for k d and t 0. Hence in this case the monodromy operator (8.19) can have only the eigenvalue 1. This should be compared to the stronger result in Corollary 5.4. (ii) For the monomial line arrangement A (m, m, 3)deﬁned by

A (m, m, 3) : f = (xm − ym )(xm − zm )(ym − zm) = 0 172 8 Free Arrangements and de Rham Cohomology of Milnor Fibers for m ≥ 2, there are two types of singularities, the triple points, with local equation 3 3 u +v = 0 and hence with maximal spectral number α3 = 4/3 and the three m-fold intersection points, e.g. [1 : 0 : 0], with local equation um + vm = 0 and maximal α = 2m−2 ≥ spectral number m m . Hence, for m 3, it follows that

: 2( ∗ ) → 3( ∗ ) dl H K f l H K f l is injective for l > 6m − 6 = 2(d − 3).

Remark 8.6 If the curve C : f = 0 has only weighted homogeneous singularities and we can determine the terms E2 in all the spectral sequences E∗( f )k , then we have complete information on the monodromy action and on the pole order ﬁltration P on H 1(F, C). Indeed, one has the following

1 1( , C) = 1,0( ) = 1,0( ) Gr P H F λ E∞ f k E2 f k (8.25) and 0 1( , C) = 0,1( ) = 0,1( ) . Gr P H F λ E∞ f k E2 f k (8.26)

If the curve C : f = 0 has not only weighted homogeneous singularities, then the relation (8.25) still holds, but the relation (8.26) has to be replaced by

0 1( , C) = 0,1( ) = 0,1( ) . Gr P H F λ E∞ f k E3 f k (8.27)

In all the examples computed so far we noticed that the Hodge ﬁltration F and the pole order ﬁltration P coincide on H 1(F, C), see Examples 8.14–8.16 and Theorems 8.17, 8.19.

Even when the spectral sequence E∗( f )k does not degenerate at E2,theE2-page gives us valuable information on the Alexander polynomial ΔC (t), as shown in the next result, see [99, 100].

Theorem 8.15 Let C : f = 0 be a reduced degree d curve in P2, and let

λ = exp(−2πik/d) = 1, with k ∈ (0, d) an integer. Then λ is a root of the Alexander polynomial ΔC (t) if and 1,0 1,0 ( ) = ( ) = = − only if either E2 f k 0 or E2 f k 0, where k d k. The multiplicity m(λ) of the root λ is given by

1,0 1,0 (λ) = ( ) + ( ) . m dim E2 f k dim E2 f k

Remark 8.7 (i) The direct sum

⊕ ( ∗,∗, , ) k=1,...,d Tot Ck dI dII 8.3 Spectral Sequences and Alexander Polynomials 173

∗ of the total complexes discussed above can be identiﬁed with the complex (Ω , D f ), where the differential D f is given by |ω| D (ω) = dω − d f ∧ ω, (8.28) f d with ω ∈ Ω j a homogeneous differential form of degree |ω|,see[74], p. 190. This complex has the advantage that one has explicit isomorphisms

j ∗ j−1 ∗ H (Ω , D f ) = H (F, C), [ω] → ι (Δ(ω)), (8.29) for any j, where ι : F → C3 denotes the inclusion of the Milnor fiber F into C3 and Δ : Ω j → Ω j−1 denotes the contraction with the Euler vector field, see [74], p. 193. In fact, Δ : Ω∗ → Ω∗−1 is the unique derivation of the graded algebra Ω∗ which is S-linear and satisfies

Δ(dx) = x,Δ(dy) = y,Δ(dz) = z.

(ii) For any j deﬁne

j ={ω ∈ Ω j : ω = ∧ ω = } Z f d 0 and d f 0 and j ={ω ∈ Ω j : ω = ∧ (η) η ∈ Ω j−2}. B f d f d for some

j ⊂ j Then it is clear that B f Z f and that there are natural maps from the quotient j = j / j j ( ∗ ) j (Ω∗, ) H f Z f B f to both cohomology groups H C f and H D f . We relate now the Jacobian syzygies AR( f ) considered in the previous section to our spectral sequences. For any polynomial f and any integer j, there is an identiﬁcation

( ) = ( ) := { ∧:Ω2 → Ω3 } AR f j Syz f j+2 ker d f j+2 j+2+d such that the Koszul relations KR( f ) inside AR( f ) correspond to the submodule d f ∧ Ω1 in Syz( f ). Since C : f = 0 has only isolated singularities, it follows from the vanishings in (8.20) that the next sequence is exact for any j

→ Ω0 → Ω1 → ( ∧ Ω1) → . 0 j−2d j−d d f j 0

Here the morphisms are the wedge product by d f . In particular, one has

1 dim(d f ∧ Ω ) j = 0for j ≤ d, (8.30) and 174 8 Free Arrangements and de Rham Cohomology of Milnor Fibers j − d + 1 dim(d f ∧ Ω1) = 3 for d < j < 2d. (8.31) j 2

On the other hand, we get epimorphisms ( ) 2 ∗ Syz f q AR( f ) − = Syz( f ) → H (K ) = , (8.32) q 2 q f q 1 (d f ∧ Ω )q

2 = { : ( ) → Ω3}, for any q. Note that Z f ker d Syz f and there is an obvious morphism

2 → 1−t,t ( ) . H f,td+k E∞ f k (8.33)

Remark 8.8 When C : f = 0 is a free divisor with exponents d1 ≤ d2, then one clearly has dim AR( f )k = 0fork < d1, k − d1 + 2 dim AR( f ) = dim S − = (8.34) k k d1 2 for d1 ≤ k < d2, and k − d1 + 2 k − d2 + 2 dim AR( f ) = dim S − + dim S − = + (8.35) k k d1 k d2 2 2 for k ≥ d2. For non-free divisors, the computation of the dimensions dim AR( f )k (resp. of a basis for AR( f )k ) is much more complicated. These dimensions give an indication of the size of the linear systems to be solved in the algorithm described below.

We describe now an algorithm to compute the Alexander polynomial for a free plane curve having only weighted homogeneous singularities. Assume that C : f = 0 is such a curve, with f a reduced, homogeneous polyno- α = q0 mial of degree d.Let max d be the maximal spectral number of the singularities of C which can be written in this form, i.e. a rational number with denominator d. Note that always q0 < 2d, since all the spectral numbers of plane curve singularities are contained in the interval (0, 2). Assume that d1 ≤ d2 are the exponents of C and let ρ1 and ρ2 be a basis of the S-module AR( f ) with deg ρl = dl for l = 1, 2. Then AR( f ) j = 0for j < d1 and AR( f ) j for j ≥ d1 (resp. j ≥ d2) has a basis as a C-vector space obtained by taking all the products μ · ρ between a monomial μ in x, y, z and ρ = ρ1 (resp. ρ = ρ1 or ρ = ρ2) having the total degree j. The corresponding differential forms μ · ω(ρ) form a basis as a C-vector space for Syz( f )q where q = j + 2. Consider now the composition map δ : ( ) −→d Ω3 → 3( ∗ ) q Syz f q q H K f q (8.36) where the second map is the canonical projection. In down-to-earth terms, we have 8.3 Spectral Sequences and Alexander Polynomials 175

δq (ω(ρ)) = δq (ady∧dz−bdx ∧dz+cdx ∧dy =[ax +by +cz]∈M( f )q−3. (8.37)

Here M( f ) = S/J f is the Milnor algebra of f , with J f = ( fx , fy, fz) the Jacobian ideal of f . Using computer algebra software such as CoCoA [45]orSingular [55], one can compute κq := dim ker δq , (8.38) for d1 + 2 ≤ q ≤ q0 < 2d. Then we compute the difference

1 εq = κq − dim(d f ∧ Ω )q (8.39)

1 using the formulas (8.30) and (8.31). Since (d f ∧ Ω )q is obviously contained in ker δq , it follows that ε = 1−t,t ( ) , q dim E2 f k (8.40) where k ∈[1, d], q − k is divisible by d and t = (q − k)/d. Moreover, Theorem 1−t ,t ( ) 8.14 implies that these are the only terms E2 f k which might be non-zero. Finally, we describe an algorithm computing the Alexander polynomial ΔC (t) for any reduced plane curve having only weighted homogeneous singularities, in particular for an arbitrary line arrangement in P2,see[101]. Since the curve C : f = 0 is no longer assumed to be free, the ﬁrst question we have to address is the size of the syzygy graded module Syz( f ). Let again J f be the Jacobian ideal spanned by fx , fy, fz in S, and denote by M( f ) = S/J f the corresponding Milnor algebra of f .Letm( f ) j = dim M( f ) j for j ≥ 0, invariants which can easily be computed using the Singular software [55]. This is a ﬁnite process, since m( f ) j = τ(C) for j ≥ T = 3(d − 2),see[102]. These numbers enter into the following formula

2( ∗ ) = ( ) − ( ) ≤ ≤ − , dim H K f j m f j+d−3 m fs j+d−3 for 2 j 2d 3 (8.41)

2( ∗ ) = τ( ) ≥ − : = and dim H K f j C for j 2d 2, where Cs fs 0 denotes a smooth curve 2( ∗ ) = ( ) /( ∧ Ω1) of degree d,see[81]. Since H K f j dim Syz f j d f j , the combination of the formulas (8.30), (8.31) and (8.41) above gives us formulas for the dimensions syz( f ) j = dim Syz( f ) j for any 3 ≤ j < 2d. Note that Syz( f ) j = AR( f ) j−2 = 0 for any j < 3 by our assumption d1 = mdr( f )>0, i.e. C is not a union of d lines passing through one point. s,t ( ) + = ≤ ≤ ≤ ≤ Next we compute dim E2 f k for s t 1, 0 t 2. For 3 q q0, where q0 = d · αmax < 2d is exactly as deﬁned above, we set q1 = q − d and consider the linear mapping 3 3 φ : S × S → S − + × S − (8.42) q q−2 q1−2 q 3 d q 3 given by

((a, b, c), (u, v,w))→ (afx + bfy + cfz, ax + by + cz − ufx − vfy − wfz). 176 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

Since S j = 0for j < 0, this map has a simpler form for q < d +2, and we encourage the reader to write down this special case himself. If ω ∈ Ω2 (resp. α ∈ Ω2)isgiven by the formula (8.5), with x0 = x, x1 = y and x2 = z, and starting with the ( , , ) ∈ 3 ( , ,w) ∈ 3 triple a b c Sq−2 (resp. the triple u v Sq −2), one sees that essentially φ (( , , ), ( , ,w)) 1 q a b c u v corresponds to the pair

(d f ∧ ω,dω − d f ∧ α).

It is clear that (( , , ), ( , ,w))∈ := φ a b c u v Kq ker q if and only if d f ∧ ω = 0 and dω = d f ∧ α, i.e. ω gives rise to an element in E2( f ). Note that if ω = d f ∧ η, one can take α =−dη and the corresponding pair (ω, α) gives rise to an element in Kq , which corresponds to the trivial element in ( ) ⊂ 3 E2 f . Consider the projection Bq Sq−2 of Kq on the first component and note /( ∧Ω1) 1,0( ) ≤ ≤ that Bq d f q can be identified with E2 f q ,for3 q d, and respectively 0,1( ) + ≤ ≤ to E2 f q−d ,ford 1 q q0. → The kernel of the projection Kq Bq can be identified with the set of forms α ∈ Ω2 such that d f ∧ α = 0, i.e. α ∈ Syz( f ) .Ifwesetk = dim K , then q1 q1 q q these invariants can be computed using the Singular software [55]. It follows that we have ε = − ( ) − ( ∧ Ω1) , q kq syz f q1 dim d f q

ε = 1,0( ) ≤ ≤ ε = 0,1( ) and also q dim E2 f q for 3 q d, and respectively q dim E2 f q−d + ≤ ≤ ε = < > for d 1 q q0. We set by convention q 0forq 3orq q0. Since E2( f )k = E∞( f )k , by Corollary 8.3 we get that the multiplicity m(λ) of λ = exp(−2πik/d) as a root of the Alexander polynomial ΔC (t) is given by

(λ) = ε + ε , m k k+d (8.43) for any 1 ≤ k ≤ d.

Remark 8.9 If the curve C : f = 0 has only weighted homogeneous singularities and we can determine the terms E2 in all the spectral sequences E∗( f )k , then we have complete information on the monodromy action and on the pole order ﬁltration P on H 2(F, C) as well. Indeed, one has the following

2 2( , C) = 2,0( ) = 2,0( ) , Gr P H F λ E∞ f k E2 f k

1 2( , C) = 1,1( ) = 1,1( ) Gr P H F λ E∞ f k E2 f k and 0 2( , C) = 0,2( ) = 0,2( ) . Gr P H F λ E∞ f k E2 f k 8.3 Spectral Sequences and Alexander Polynomials 177

However, the Hodge filtration F and the pole order filtration P are distinct on H 2(F, C) in general, see Example 8.14. To compare these two filtrations, it is convenient to introduce the following spectra for f . First we consider the pole order spectra of f defined by j ( ) = j α SpP f m P, f,αt (8.44) α>0 for j = 0, 1, where j = p 2− j ( , C) m P, f,α dim Gr P H F λ with p =[3 − α] and λ = exp(−2πiα). Using this, one sets

( ) = 0 ( ) − 1 ( ). SpP f SpP f SpP f

One can deﬁne the Hodge spectra of f similarly, namely j ( ) = j α SpF f m F, f,αt (8.45) α>0 for j = 0, 1, where j = p 2− j ( , C) m F, f,α dim GrF H F λ with p =[3 − α] and λ = exp(−2πiα). It is obvious that these new Hodge spectra relate to the spectrum introduced in (7.7) by the formula

( ) = 0 ( ) − 1 ( ) Sp Q SpF Q SpF Q when n = 2. Moreover, combining Remarks 8.6 and 8.9, we see that our spectral sequences allow us to determine the pole order spectra for any curve C : f = 0 having only weighted homogeneous singularities.

Remark 8.10 If the curve C : f = 0 has not only weighted homogeneous singularities, then as we have seen in Remark 8.6, we have to compute at least the dimension 0,1( ) of the term E3 f k from the E3-page, in order to get the Alexander polynomial Δ ( ) 1 ( ) C t and the spectrum SpP f . Moreover, it is clear that

2,0( ) ≤ 2,0( ) , 1,1( ) ≤ 1,1( ) dim E∞ f k dim E3 f k dim E∞ f k dim E3 f k and 0,2( ) ≤ 0,2( ) . dim E∞ f k dim E3 f k

It is possible to extend the algorithm presented above such that it yields the dimensions of these four terms from the E3-page. Moreover, when the three inequalities above are equalities, a fact which can be tested using the formula (8.17)afterthe 178 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

0,1( ) 0 ( ) computation of dim E3 f k , we get also the spectrum SpP f . The reader interested in these results, which go beyond the line arrangement case, can refer to [101] where all the details of the extended algorithm are explained.

8.4 The de Rham Cohomology of Milnor Fibers

In this section we apply the approach described in the previous section to a number of free and non-free line arrangements, in order to ﬁnd explicit bases (or, at least some non-zero cohomology classes) for some eigenspaces of the monodromy action on the cohomology of their Milnor ﬁbers. To get a feeling for this type of result, we start with the following very simple situation, which can be regarded as a special case of the isolated singularities treated in [215]. Example 8.12 Consider the central line arrangement A : f = xd + yd = 0inC2. It is clear that a C-vector space basis of the corresponding Milnor algebra M( f ) is given by the monomials xa yb where 0 ≤ a < d − 1 and 0 ≤ b < d − 1. Let k be an integer, 1 ≤ k ≤ d, and set λ = exp(−2πik/d). Then one can develop the spectral sequence approach from the previous section in the case of two variables, and the analog of Example 8.9 holds in this new setting. It follows that a C-vector 1 space basis for the eigenspace H (F, C)λ is given by the differential 1-forms

∗ a b ωa,b = ι (Δ(x y dx ∧ dy) for 0 ≤ a < d −1, 0 ≤ b < d −1 and a+b+2 ≡ k mod d.Hereι : F → C2 denotes the inclusion of the corresponding Milnor ﬁber F into C2 and Δ is the contraction with 2 1 1 1 the Euler vector ﬁeld on C . Moreover, the subspace F H (F, C)λ ⊂ H (F, C)λ is spanned by the 1-forms ωa,b satisfying a + b + 2 = k. Example 8.13 Consider the central line arrangement A : f = 0inC3 with Milnor

ﬁber F. Assume d = deg f is a multiple of 3, say d = 3d and f = f1 f2 f3, where f j are homogeneous polynomials of degree d , with f3 a linear combination of f1 and f2. In other words, A is a reduced (3, d )-multinet. Consider the 2-form ω = d f1 ∧ d f2. It is clear that

d f ∧ ω = d(ω) = 0.

Remark 8.7 implies that this form gives rise to a non-zero cohomology class

∗ 1 α = ι (Δ(ω)) ∈ H (F, C)λ, where ι : F → C3 denotes the inclusion of the Milnor ﬁber F into C3, Δ denotes the contraction with the Euler vector ﬁeld on C3 and λ = exp(2πi/3).

A similar situation is when A is a reduced (4, d )-multinet, i.e. f = f1 f2 f3 f4, d = 4d . Note that the six 2-forms d fi ∧d f j for 1 ≤ i < j ≤ 4 span a 1-dimensional 8.4 The de Rham Cohomology of Milnor Fibers 179 space in the cohomology of the Milnor ﬁber, so the above construction does not give 1 a basis for H (F, C)−1, which is at least 2-dimensional by Corollary 6.1. Some of the cohomology classes in Theorems 8.17 and 8.19 are obtained in this simple way.

Now we consider the monomial arrangement A (m, m, 3) and recall ﬁrst the following result, which is proved in [165]. For more on this question refer to [99]. Theorem 8.16 Consider the monomial line arrangement

A (m, m, 3) : f = (xm − ym )(xm − zm )(ym − zm) = 0 for m ≥ 2 and denote by F the corresponding Milnor ﬁber. Then the monodromy operator h∗ : H 1(F, C) → H 1(F, C) (8.46) has a cubic root λ = 1 as an eigenvalue with multiplicity m(λ), where m(λ) = 2 if m is divisible by 3, and m(λ) = 1 otherwise.

Recall the differential 2-forms ω1 and ω2 introduced in Example 8.6, (i). When = ω = m −1 m −1 m −1ω ι : → C3 m 3m , we also define 1 x y z 1.Let F denote the inclusion of the Milnor fiber F into C3 and Δ denote the contraction with the Euler vector field on C3. Theorem 8.17 The eigenspace

1 1 1 F H (F, C)λ = H (F, C)λ,

∗ where λ = exp(2πi/3), is spanned by the differential form α = ι Δ(ω2) when m is α β = ι∗Δ(ω ) not divisible by 3, and by the forms and 1 when m is divisible by 3. For m = 2, 3 this result was proved by Nancy Abdallah in [1, Sect. 5.5].

Proof Consider ﬁrst the case when m is not divisible by 3. Then Theorem 8.16 1 implies that dim H (F, C)λ = 1. The differential form ω2 introduced in Example 8.6 (i) belongs to Syz( f )2m and satisﬁes dω2 = 0. Therefore ω2 gives rise to a 1 non-zero element α in H (F, C)λ, given by Eq. (8.29), since −2πi(2m) exp = exp(2πi/3) = λ. 3m

= ω When m is divisible by 3, say m 3m , then the differential form 1 also belongs ( ) ω = ω to Syz f 2m , satisfies d 1 0 and it is clearly linearly independent of 2 since the arrangement is free. Finally, we explain why the forms α and β have Hodge type (1, 0). Since the Hodge filtration F p is contained in the pole order filtration P p, namely F p ⊂ P p for any integer p,see[96, Eq. (3.1.3)], it follows that

1 0 1 0 1 H (F, C)λ = F H (F, C)λ = P H (F, C)λ, 180 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

1 1 1 1 2 1 2 1 F H (F, C)λ ⊂ P H (F, C)λ and F H (F, C)λ = P H (F, C)λ = 0.

ω = , > 1,0( ) = Since the forms j for j 1 2 have degrees m, it follows that E1 f m 1,0 1 1 E∞ ( f )m = 0. In view of the above, this implies P H (F, C)λ = 0, where λ is 1 1 the complex conjugate of λ. But then F H (F, C)λ = 0, which can be restated as 1,0 1 H (H (F, C)λ) = 0. Since

1,0 1 0,1 1 dim H (H (F, C)λ) = dim H (H (F, C)λ),

1 1 1 it follows that F H (F, C)λ = H (F, C)λ, as claimed. The following result, which explains why we have considered only cubic roots of unity in our discussion of the monomial line arrangement, was proved in [71]. Similar results hold for all irreducible complex reﬂection arrangements. Theorem 8.18 For m ≥ 2, the monodromy operator (8.46) of the arrangement A (m, m, 3) has only cubic roots of unity as eigenvalues. If the arrangement A (m, m, 3) were deformable into a real arrangement, then this claim would follow from [233]. However, this is not the case for m ≥ 3, by the Sylvester–Gallai property of real line arrangements: any such arrangement has at least 3 nodes, see Theorem 1.5. Example 8.14 Consider the monomial arrangement A = A (2, 2, 3) in C3 given by

A : f = (x2 − y2)(y2 − z2)(z2 − x2) = 0, already discussed in Exercise 7.8. This is a free line arrangement, and the algorithms described in the previous section give the following pole order spectra for f :

3 4 5 6 7 8 9 0 ( ) = 6 + 6 + 6 + 6 + 6 + 6 + 6 , SpP f t 3t 2t 6t 2t 3t t (8.47) and 10 12 14 1 ( ) = 6 + 6 + 6 . SpP f t 5t t (8.48)

It follows that

3 4 5 6 7 8 9 10 12 14 ( ) = 6 + 6 + 6 + 6 + 6 + 6 + 6 − 6 − 6 − 6 . SpP f t 3t 2t 6t 2t 3t t t 5t t

Note that the negative terms in this formula coincide with the negative terms in the spectrum Sp(A ) given in Exercise 7.8, which is a confirmation of the fact that the two filtrations F and P coincide on H 1(F, C). However, the positive terms are distinct, and hence the two filtrations F and P are distinct on H 1(F, C).More 9 3 5 6 = 2 ( ) 2 (A ) precisely, the term t t in SpP f is replaced by a term t in Sp . This tells 2( , C) 1 2( , C) us that an element in H F −1 has a nonzero class in Gr P H F −1 and in 0 2( , C) − 2 2( , C) = GrF H F −1. The other eigenvector for 1 can be found in Gr P H F −1 2 2( , C) . GrF H F −1 8.4 The de Rham Cohomology of Milnor Fibers 181

Next we consider the Hessian line arrangement, see also [99]. This arrangement can be given by the equation A : f = xyz (x3 + y3 + z3)3 − 27x3 y3z3 = 0, and consists of all the four singular members of the pencil P : uc1 + vc2, where

3 3 3 c1 = x + y + z and c2 = xyz.

This description of the Hessian line arrangement implies that it is free by [70, 224]. Note that each of these four members is a triangle, which implies that A has 12 double points and 9 points of multiplicity 4. This arrangement plays a key role in the theory of line arrangements, as it is the only known 4-net, see [217, Example 2.15], [187, Example 8.7]. Its monodromy is computed in [32, Remark 3.3 (iii)], [187, Theorem 1.7] Recall then the 2-forms ω1 and ω2 introduced in Example 8.6 (iii).

Theorem 8.19 For the Milnor ﬁber F of the Hessian arrangement, one has the following. 1 1 ∗ (i) The eigenspace F H (F, C)−1 is spanned by α = ι Δ(ω1). 1 1 1 ∗ (ii) The eigenspace F H (F, C)i = H (F, C)i is spanned by β1 = ι Δ(c1ω1) ∗ and β2 = ι Δ(c2 · ω1). (iii) The pole order spectra of the Hessian arrangement are given by the following formulas.

3 4 5 6 7 8 9 10 11 0 ( ) = 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 SpP f t 3t 6t 10t 12t 15t 19t 18t 18t

12 13 14 15 16 17 18 19 20 21 +28t 12 + 18t 12 + 18t 12 + 19t 12 + 15t 12 + 12t 12 + 10t 12 + 6t 12 + 3t 12 + t 12

and 18 21 24 27 30 1 ( ) = 12 + 12 + 12 + 12 + 12 . SpP f t 2t 11t 2t t

Proof The ﬁrst two claims follow exactly as in the proof of Theorem 8.17 once we check that dω1 = d(c1ω1) = d(c2ω2) = 0.

j ( ) The formulas for the spectra SpP f follow from our algorithm using a direct computation done with the software Singular.

Remark 8.11 Note that in the difference

( ) = 0 ( ) − 1 ( ) SpP f SpP f SpP f 182 8 Free Arrangements and de Rham Cohomology of Milnor Fibers in Theorem 8.19 there are some cancelations of terms, unlike in the situation in 0 ( ) 1 ( ) Example 8.14. This shows that the individual spectra SpP f and SpP f carry ( ) 1 ( ) more information than the spectrum SpP f . Moreover, the formula for SpP f conﬁrms that in this case as the two ﬁltrations F and P also coincide on H 1(F, C). We conclude with a discussion of two non-free line arrangements. Example 8.15 Consider the line arrangement A obtained from the Hessian arrangement by deleting one of the 4 triangles. The equation of this new arrangement can be given by A : f = (x3 + y3 + z3)3 − 27x3 y3z3 = 0.

This arrangement is not free, since it is easy to check that

(d − 1)2 − τ(A ) = 64 − 45 = 19 cannot be a product d1d2 with d1 + d2 = 8. The construction from Example 8.13 1 shows that dim H (F, C)λ ≥ 1, where λ = exp(2πi/3). A computation using our algorithm for non-free line arrangements shows that one has equality in this case, i.e. the corresponding Alexander polynomial is

Δ(t) = (t − 1)8(t2 + t + 1).

The pole order spectra of the Hessian arrangement are given by the following formulas:

3 4 5 6 7 8 9 10 11 0 ( ) = 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 SpP f t 3t 6t 10t 12t 12t 19t 12t 12t

12 13 14 15 +12t 9 + 9t 9 + 6t 9 + 3t 9 and 15 18 21 1 ( ) = 9 + 9 + 9 . SpP f t 8t t

It is easy to show that this line arrangement A is yet another version of the Pappus A : = ( ) line arrangement 1 Q1 0 of type 93 1 described in Example 2.15 and in Exercise 4.2, but this time the coefﬁcients of the lines are not real. Example 8.16 Consider the line arrangement A from Example 6.10, which admits a reduced (3, 4)-multinet structure without being a net. This arrangement is not free, as a direct computation yields

(d − 1)2 − τ(A ) = 121 − 89 = 32 and one concludes as above that there are no d1, d2 with d1 + d2 = 11 (so one of the di ’s must be odd) and d1d2 = 32. The construction from Example 8.13 shows that 1 dim H (F, C)λ ≥ 1, where λ = exp(2πi/3). A computation using our algorithm 8.4 The de Rham Cohomology of Milnor Fibers 183 for non-free line arrangements shows that one has equality in this case, i.e. the corresponding Alexander polynomial is

Δ(t) = (t − 1)11(t2 + t + 1).

The pole order spectra of this arrangement are given by the following formulas:

3 4 5 6 7 8 9 10 11 0 ( ) = 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 SpP f t 3t 6t 10t 15t 20t 21t 22t 22t

12 13 14 15 16 17 18 19 20 21 +32t 12 + 22t 12 + 22t 12 + 21t 12 + 20t 12 + 16t 12 + 12t 12 + 7t 12 + 3t 12 + t 12 and 20 24 28 1 ( ) = 12 + 12 + 12 . SpP f t 11t t

Remark 8.12 The two Ziegler line arrangements, discussed in Remark 8.5,are lattice-isotopic, see Exercise 8.5 below. Hence they have smoothly equivalent Milnor ﬁbrations by Theorem 5.1. However, as noticed in [227, Example 5.10] and [198, 0 ( ) Remark 4.14 (iv)], the corresponding spectra SpP f are distinct. This fact can also be checked using the algorithm described above. In this way one obtains, for the arrangement A : f = 0, the following spectrum

3 4 5 6 7 8 9 10 11 0 ( ) = 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 SpP f t 3t 6t 10t 14t 15t 22t 15t 15t

12 13 14 15 16 +14t 9 + 12t 9 + 9t 9 + 5t 9 + t 9 .

For the second arrangement A : f = 0, we get the following result

3 4 5 6 7 8 9 10 11 0 ( ) = 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 SpP f t 3t 6t 10t 15t 15t 22t 15t 15t

12 13 14 15 +14t 9 + 12t 9 + 9t 9 + 5t 9 .

In other words, the P-ﬁltration on the Milnor ﬁber cohomology is not preserved under lattice-isotopy.

Remark 8.13 Computations of Milnor fiber monodromy and pole order spectra for central arrangements of higer rank can be found in [100]. For the reflection arrangements, the corresponding pole order spectra enjoy a surprising symmetry property, as can be seen for instance in formula (8.47) above. This symmetry is very mysterious, as the corresponding Hodge spectra are not at all symmetric, see Example 7.8 above. It is an open problem to find an explanation for this symmetry. 184 8 Free Arrangements and de Rham Cohomology of Milnor Fibers

8.5 Exercises

n Exercise 8.1 Show that a free hyperplane arrangement in P with exponents d1 ≤ d2 ≤···≤dn is essential if and only if d1 > 0.

Exercise 8.2 Show that for a degree d free curve C : f = 0 with exponents d1 ≤ d2, one has ct( f ) = d1 + d − 2 and st(C) = d2 + d − 3. 2 Exercise 8.3 Show that a free line arrangement A in P with exponents (d1, d2) satisﬁes the relation b2(M(A)) = d1d2.

Hint: use the relations (8.14), the formula for the Euler characteristic of C in terms of the total Milnor number μ(C) (8.18), and the equality μ(C) = τ(C), valid for any curve C which is a union of lines in P2. Exercise 8.4 Consider a free line arrangement A : Q = 0inP2 with exponents (d1, d2) and the corresponding gradient mapping

2 2 ψQ = grad(Q) : P P .

Show that deg ψQ = d1d2. Exercise 8.5 Show that the six triple points of Ziegler’s arrangement A (resp. A ) in Remark 8.5 are situated (resp. are not situated) on a conic in P2. Show that one can pass from the arrangement A to the arrangement A just by moving the triple point L2 ∩ L3 ∩ L5 along the line L2 kept ﬁxed, and moving the lines L3 and L5 in such a way to preserve the triple point. In particular, this shows that A and A are lattice-isotopic. Exercise 8.6 Show that Terao’s Conjecture holds for the full monomial line arrangement A (m, 1, 3) : f = xyz(xm − ym )(ym − zm )(xm − zm) = 0 for m ≥ 2 and for the Hessian line arrangement

A : f = xyz[(x3 + y3 + z3)3 − 27x3 y3z3]=0.

Exercise 8.7 Use Example 8.12 and the formula for the spectrum (7.7) to recover the values of the spectral numbers given in Example 8.10 in the case a = b. Exercise 8.8 Let G be the multiplicative cyclic group of m-th roots of unity, acting on C3 by ξ(x, y, z) = (ξ y,ξx,ξz), for any ξ ∈ G. Then the Milnor ﬁber F of the monomial arrangement A (m, m, 3) is clearly G-invariant, and hence there is an induced G-action on F. Compute the cohomology of the quotient space F/G. 8.5 Exercises 185

3 Exercise 8.9 Let G = Σ3, the permutation group acting on C by permuting the coordinates x, y, z. Then the Milnor ﬁber F of the Hessian line arrangement is G- invariant, hence we get an induced action of G on F. Show that the action of G on 1 1 H (F, C) =1 is trivial and describe the G-action on H (F, C)1. References

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A B Acyclic orientations, 37 B3-arrangement, 19 Alexander polynomial, 11, 87, 90, 168, 172, Bestvina–Brady groups, 81, 89 174 Beta invariant, 57 Aomoto complex, 103, 134 Betti number, 3, 56, 69 Arrangement, 2, 15 Betti polynomial, 5 Boolean arrangement, 18, 22, 23, 25, 26, 42, abelian, 16 47, 63, 74, 97, 154, 156 affine, 15 Bouquet central, 15 of circles, 11, 88 decomposable, 18 of spheres, 64 deleted, 30 Braid arrangement, 18, 22, 24, 27, 37, 39, elliptic, 16 42, 76, 78, 79, 154 essential, 20 Braid group, 75 fiber type, 77, 78, 117, 167 free, 154, 156, 166, 167 general position, 36, 95, 108 C Cauchy formula, 10 generic, 36 Center, 15 hyperplane, 2 Chamber, 31 indecomposable, 18 Characteristic polynomial, 26, 37, 144, 157 inductively free, 156 Characteristic variety, 100, 101 irreducible, 18 Chern class, 157 locally free, 157 Chromatic polynomial, 37, 69 projective, 15 Classifying space K (G, 1), 103, 117 reducible, 18 Coexponents of a group, 155 restricted, 30 Coincidence threshold ct( f ), 159 subspace, 58, 68 Combinatorially equivalent arrangements, supersolvable, 23, 28, 38, 51, 78, 96, 117, 21, 29, 66 156, 166, 167 Complexification, 16 Component tame, 158 essential, 114 toral, 16 global, 111 toric, 16 local, 111 Artin groups, 77 non-essential, 114 affine, 77 translated, 114, 120 Atom, 21 Cone, 16, 17 © Springer International Publishing AG 2017 197 A. Dimca, Hyperplane Arrangements, Universitext, DOI 10.1007/978-3-319-56221-6 198 Index

Connection, 91 Fundamental group, 8, 71, 73, 99 integrable, 91 logarithmic, 127 Contraction with the Euler vector field, 173 G Counting polynomial, 140 Good compactification, 93, 94 Cover Gradient mapping, 67 congruence, 106, 121, 122 Gradient vector field, 152 cyclic, 105 Graphic arrangement, 37, 38 finite abelian, 105 Grothendieck group, 144 free abelian, 108 Gysin exact sequence, 53 Coxeter arrangement, 38 Coxeter group, 38 Cyclotomic polynomial, 103 H Hasse diagram, 22 Hodge structure D mixed, 130 De Rham complex pure of type (p, p), 130 logarithmic, 128 pure of weight k, 130 regular, 10 Hodge–Deligne polynomial, 143 smooth, 10 equivariant, 147 Decone, 17 Hurewicz morphism, 63 Defect, 134 Deleted B3-arrangement, 22, 25, 26, 49, 78, 118 I Differential form, 9, 11, 91 Intersection poset, 21 with logarithmic poles, 127 Inversion Formula, 26 Discriminant, 159 Irreducible group, 38 Irregularity, 137 Isotropic subspace, 112 E maximal, 113 Edge, 21 dense, 93 Elementary loop, 72 J E-polynomial, 143 Jacobian algebra, 159 Euler characteristic, 4, 91 Jacobian ideal, 151, 159 equivariant, 87 Jacobian syzygies, 151 Euler derivation, 151 Exceptional groups, 39 Exponential map, 104 K Exponents, 153, 160 K (π, 1)-arrangement, 74–77, 84, 117, 164, 167 Koszul complex, 168 F Face, 31 Face poset, 31 L Filtration Lattice, 21 Hodge, 129, 130 geometric, 21 pole order, 169, 179 intersection, 21 weight, 130 Lattice-isotopic arrangements, 65, 89, 163, Flat, 21 184 Free divisor, 152, 174 Leray spectral sequence, 92 Free hypersurface, 152 Lie algebra, 82 Full monomial arrangement, 39 graded, 82 Full monomial group, 39 holonomy, 82 Index 199

Malcev, 82 equivariant, 147 Line arrangement, 163 Poincaré homology sphere, 83 Ceva, 95, 111, 112, 117, 122, 133, 146, Poincaré–Leray residue, 53 166 Poincaré polynomial, 26, 49 full monomial, 96, 111, 165, 184 Polynomial periodicity, 106 Hessian, 40, 98, 112, 122, 123, 166, 172, Primitive cohomology, 65 181, 184 Product arrangement, 18 MacLane, 20, 30 Projective hypersurface, 61 monomial, 95, 111, 122, 165, 171, 179 non-Pappus (93)2, 28, 122, 148 Pappus (93)1, 28, 83, 122, 123, 148, 182 Q Ziegler’s arrangements, 161 Quasi-projective group, 81 Local system, 90 admissible, 132 Log concave sequence, 69 R Logarithmic forms, 158 Rank, 20 Reducible group, 38 Reflecting hyperplane, 38 M Reflection, 38 Möbius function, 24 Reflection arrangement, 18, 38, 76, 79, 155, Milnor algebra, 151 180 Milnor fiber, 11, 85, 140, 168 Reflection group, 38, 76 Milnor number, 35, 67, 184 Reflexive sheaf, 156 global, 168 Region Minimal degree mdr( f ), 159 bounded, 31 Minimal space, 66 relatively bounded, 31 Mixed Hodge numbers Representation ring, 87, 147 equivariant, 139 Residue, 127 Mixed Hodge structure, 129 Resonance variety, 103 Modular flat, 23 Roberts’ formula, 1, 41 Moduli spaces, 30 Monodromy operator, 11 Monodromy transformation, 11, 86 S Monomial arrangement, 39 Saito’s Criterion, 153 Monomial group, 39 Salvetti complex, 77, 101 Multiarrangement, 109 Semi-lattice, 21 (k, d)-multinet, 109 Singular software, 165, 175, 176 Multiple fiber, 115 Spectral number, 135, 171, 172, 174 Spectrum, 135, 171, 177 Stability threshold st( f ), 160 N Stallings group, 81 (k, d)-net, 109 Superabundance, 134 Normal crossing divisor, 93 Support, 117 singular, 117 Surface O Burniat, 107 Oriented matroid, 31, 77 Hirzebruch covering, 107, 121 Orlik–Solomon algebra, 46, 48, 51 Sylvester–Gallai property, 7, 8, 180

P T Papadima-Suciu β3(A) invariant, 134, 147 Terao’s Conjecture, 167, 184 Pencil, 109 Theorem Poincaré–Deligne polynomial Arnold–Brieskorn, 10, 56 200 Index

Brieskorn decomposition, 50, 57 Twisted cohomology, 91, 128 cross-cut, 32 deletion-restriction, 32 de Rham, 10 V factorization, 58 Variety Grothendieck, 10, 158 cohomologically Tate, 141 Hurewicz, 63 with polynomial count, 140 logarithmic comparison, 158 Orlik–Solomon, 55 Schläﬂi, 2 tangent cone, 104, 132 W Whitney, 32 Weighted homogeneous singularity, 170, Zaslavsky, 5 171, 174 Tjurina number, 35, 159 Wonderful models, 97 global, 159 Torsion, 61, 97 Torsion character, 114 Z Total turn of monodromy, 92 Zariski’s theorem of Lefschetz type, 60, 69, Triple of arrangements, 30 71