Complex Arrangements

196 Pages·2009·4.05 MB·English

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, andtopology.
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-Brieskornand 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 dimH1(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 B -bundles 91
n
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–Solomonalgebra 109
7.1. Introduction 109
7.2. Resolution of the Orlik-Solomonalgebra 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-Solomonalgebra 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
A
9.1. Logarithmic forms and derivations 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 arrangementcomplements 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
Thisisabookaboutthealgebra,geometry,andtopologyofcomplexhyperplane
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
recentdevelopments,andasaninvitationforfurtherinvestigation. Wetakeafresh
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.
Animportantexampleisthe braidarrangementofdiagonalhyperplanesinCℓ.
In that case, loops in the complement can be viewed as (pure) braids on ℓ strings,
and the fundamental groupcan be identified with the pure braid groupP . For an
ℓ
arbitrary hyperplane arrangement, = H ,...,H , with complement X( ) =
1 n
Cℓ \ ni=1Hi, the identification of tAhe fu{ndamental g}roup, G(A) = π1(X(AA)), is
more complicated, but it can be done algorithmically, using the theory of braids.
S
This 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] proofofthe residualnilpotency ofthe pure braidgroup,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]andmanyothershasprofoundimplicationsinthestudyofKnizhnik-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
motionplanning (finding a collision-freemotionbetweentwoplacements ofagiven
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).