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Complex Arrangements: Algebra, Geometry, Topology Dan Cohen Complex Arrangements: Algebra, Geometry, Topology Draft of September 4, 2009 Dan Cohen Graham Denham Michael Falk Hal Schenck Alex Suciu Hiro Terao Sergey Yuzvinsky 2000 Mathematics Subject Classification. Primary 32S22, 52C35; Secondary 14F35, 20E07, 20F14, 20J05, 55R80, 57M05 Key words and phrases. hyperplane arrangement, fundamental group, cohomology ring, characteristic variety, resonance variety Abstract. This is a book about complex hyperplane arrangements: their algebra, geometry, and topology. Contents Preface vii Introduction ix Chapter 1. Aspects of complex arrangements 1 1.1. Arrangements and their complements 1 1.2. Combinatorics 4 1.3. Topology 14 1.4. Algebra 18 1.5. Geometry 26 1.6. Compactifications 30 Chapter 2. Cohomology ring 35 2.1. Arnold-Brieskorn and Orlik-Solomon Theorems 35 2.2. Topological consequences 38 2.3. Geometric consequences 38 2.4. Homology and Varchenko’s bilinear form 41 2.5. Quadratic OS algebra 47 Chapter 3. Special classes of arrangements 49 3.1. Generic arrangements 49 3.2. Reflection arrangements 52 3.3. Simplicial arrangements 55 3.4. Supersolvable arrangements 55 3.5. Hypersolvable arrangements 61 3.6. Graphic arrangements 62 Chapter 4. Resonance varieties 65 4.1. The cochain complex determined by a one-form 65 4.2. Degree-one resonance varieties 69 4.3. Resonance over a field of zero characteristic 75 4.4. Nets and multinets 78 4.5. Bounds on dim H1(A, a) 82 4.6. Higher-degree resonance 87 Chapter 5. Fundamental Group 89 5.1. Fundamental group and covering spaces 89 5.2. The braid groups 90 5.3. Polynomial covers and Bn-bundles 91 5.4. Braid monodromy and fundamental group 94 5.5. Fox calculus and Alexander invariants 96 iii iv CONTENTS 5.6. The K(π, 1) problem and torsion-freeness 97 5.7. Residual properties 98 Chapter 6. Lie Algebras attached to arrangements 101 6.1. Lie algebras 101 6.2. Quadratic algebras, Koszul algebras, duality 103 6.3. Lie algebras attached to a group 105 6.4. The associated graded Lie algebra of an arrangement 105 6.5. The Chen Lie algebra of an arrangement 106 6.6. The homotopy Lie algebra of an arrangement 107 6.7. Examples 108 Chapter 7. Free Resolutions and the Orlik–Solomon algebra 109 7.1. Introduction 109 7.2. Resolution of the Orlik-Solomon algebra over the exterior algebra 110 7.3. The resolution of A∗ 116 7.4. The BGG correspondence 118 7.5. Resolution of k over the Orlik-Solomon algebra 123 7.6. Connection to DGAs and the 1-minimal model 130 Chapter 8. Local systems on complements of arrangements 135 8.1. Three views of local systems 135 8.2. General position arrangements 137 8.3. Aspherical arrangements 139 8.4. Representations ? 143 8.5. Minimality ? 143 8.6. Flat connections 143 8.7. Nonresonance theorems 145 Chapter 9. Logarithmic forms, -derivations, and free arrangements 149 9.1. Logarithmic forms and derivationsA along arrangements 149 9.2. Resolution of arrangements and logarithmic forms 150 9.3. Free arrangements 151 9.4. Multiarrangements and logarithmic derivations 152 9.5. Criteria for freeness 154 9.6. The contact-order filtration and the multi-Coxeter arrangements 155 9.7. Shi arrangements and Catalan arrangements 157 Chapter 10. Characteristic varieties 159 10.1. Computing characteristic varieties 159 10.2. The tangent cone theorem 160 10.3. Betti numbers of finite covers 160 10.4. Characteristic varieties over finite fields 161 Chapter 11. Milnor fibration 163 11.1. Definitions 163 11.2. (Co)homology 164 11.3. Examples 166 Chapter 12. Compactifications of arrangement complements 167 12.1. Introduction 167 CONTENTS v 12.2. Definition of M 167 12.3. Nested sets 168 12.4. Main theorems 171 Bibliography 173 Preface This is a book about the algebra, geometry, and topology of complex hyperplane arrangements. Dan Cohen Graham Denham Michael Falk Hal Schenck Alex Suciu Hiro Terao Sergey Yuzvinsky vii Introduction In the introduction to [171], Hirzebruch wrote: “The topology of the com- plement of an arrangement of lines in the projective plane is very interesting, the investigation of the fundamental group of the complement very difficult.” Much progress has occurred since that assessment was made in 1983. The fundamental groups of complements of line arrangements are still difficult to study, but enough light has been shed on their structure, that once seemingly intractable problems can now be attacked in earnest. This book is meant as an introduction to some recent developments, and as an invitation for further investigation. We take a fresh look at several topics studied in the past two decades, from the point of view of a unified framework. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. In its simplest manifestation, an arrangement is merely a finite collection of lines in the real plane. These lines cut the plane into pieces, and understanding the topology of the complement amounts to counting those pieces. In the case of lines in the complex plane (or, for that matter, hyperplanes in complex ℓ-space), the complement is connected, and its topology (as reflected, for example, in its fundamental group) is much more interesting. An important example is the braid arrangement of diagonal hyperplanes in Cℓ. In that case, loops in the complement can be viewed as (pure) braids on ℓ strings, and the fundamental group can be identified with the pure braid group Pℓ. For an arbitrary hyperplane arrangement, = H1,...,Hn , with complement X( ) = ℓ n A { } A C i=1 Hi, the identification of the fundamental group, G( ) = π1(X( )), is more\ complicated, but it can be done algorithmically, using theA theory ofA braids. ThisS theory, in turn, is intimately connected with the theory of knots and links in 3-space, with its wealth of algebraic and combinatorial invariants, and its varied applications to biology, chemistry, and physics. A revealing example where devel- opments in arrangement theory have influenced knot theory is Falk and Randell’s [132] proof of the residual nilpotency of the pure braid group, a fact that has been put to good use in the study of Vassiliev invariants. A more direct link to physics is provided by the deep connections between ar- rangement theory and hypergeometric functions. Work by Schechtman-Varchenko [285] and many others has profound implications in the study of Knizhnik-Zamolod- chikov equations in conformal field theory. We refer to the recent monograph by Orlik and Terao [244] for a comprehensive account of this fascinating subject. Hyperplane arrangements, and the closely related configuration spaces, are used in numerous areas, including robotics, graphics, molecular biology, computer vision, and databases for representing the space of all possible states of a system characterized by many degrees of freedom. Understanding the topology of com- plements of subspace arrangements and configuration spaces is important in robot ix x INTRODUCTION motion planning (finding a collision-free motion between two placements of a given robot among a set of obstacles), and in multi-dimensional billiards (describing pe- riodic trajectories of a mass-point in a domain in Euclidean space with reflecting boundary). CHAPTER 1 Aspects of complex arrangements In this chapter we introduce the basic tools used in the study of complex hyper- plane arrangements. Along the way, we preview some of the more advanced topics to be treated in detail later in the text: the algebraic, topological, and geometric aspects of arrangements and their interplay. We begin with an elementary discussion of the main objects of interest: cen- tral and projective arrangements and their complements. In 1.2 we introduce the fundamental combinatorial invariants of arrangements: the intersection§ lattice, M¨obius function, and Poincar´epolynomial, the underlying matroid and dual point configuration. We also prove a useful result for inductive arguments: the deletion- restriction formula, which relates the Poincar´epolynomials of an arrangement , H A the deletion ′ = H, and the restriction , for any H . In 1.3 weA discussA\ in more detail the topologyA of the complement.∈ A We describe the braid§ monodromy presentation for the fundamental group of the complement, which is treated in depth in Chapter 5. This section closes with a first glimpse of fiber-type arrangements, where the fundamental group is an iterated semidirect product. In 1.4 we introduce the Orlik-Solomon algebra, which is a combinatorial al- gebra associated§ to any matroid, and define the nbc basis for the algebra. Using this basis, we prove that the Poincar´eseries of the OS-algebra is equal to Poincar´e polynomial of . We then introduceA in 1.5 some of the finer geometric invariants of an arrange- ment : the sheaf of logarithmic§ one-forms with poles along , and the sheaf of derivationsA tangent to . For a normal crossing divisor, Grothendieck’sA algebraic de Rham theorem showsA that the cohomology of the complement may be computed from the complex of logarithmic forms. In the case of arrangements, Solomon-Terao showed that the logarithmic forms determine the Poincar´epolynomial of , even though is not a normal crossing divisor in general. A Finally,A in 1.6 we sketch the construction of wonderful compactifications, com- pact algebraic§ manifolds in which the complement of is realized as the comple- ment of a normal-crossing divisor. This topic is treatedA in detail in Chapter 12. 1.1. Arrangements and their complements 1.1.1. Central and projective arrangements. The primary object of in- terest in this book is a central arrangement of hyperplanes in the complex vector A space Cℓ. A hyperplane is a linear subspace of (complex) codimension one; an ar- rangement is a finite set of hyperplanes.
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