Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)
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Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)
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Overview
This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. Paula Tretkoff emphasizes those finite covers that are free quotients of the complex two-dimensional ball. Tretkoff also includes background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function.
The material in this book began as a set of lecture notes, taken by Tretkoff, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded by Hirzebruch and Tretkoff over a number of years. In this book, Tretkoff has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers.
Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry.
Product Details
| ISBN-13: | 9781400881253 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 02/16/2016 |
| Series: | Mathematical Notes , #51 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 232 |
| File size: | 12 MB |
| Note: | This product may take a few minutes to download. |
About the Author
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Complex Ball Quotients and Line Arrangements in the Projective Plane
By Paula Tretkoff
PRINCETON UNIVERSITY PRESS
Copyright © 2016 Princeton University PressAll rights reserved.
ISBN: 978-1-4008-8125-3
CHAPTER 1
Topological Invariants and Differential Geometry
In this chapter, we compile some prerequisites from topology and differential geometry needed in later chapters. For the most part we do not provide proofs since there are many good references for this material, for example. In §1.1, for a topological space X, we define singular homology and cohomology, as well as the Euler number e(X). The Euler number is the topological invariant that we will encounter the most often in the subsequent chapters. For a complex surface X, it coincides with the second Chern number c2 (X), as we shall see in Chapter 3 (we assume there that X is smooth compact connected algebraic). In that chapter, we also introduce for such a surface the first Chern number c2/1(X), which can be defined as the self-intersection number of the canonical divisor. Some generalities on the first Chern class c1 (X) as well as necessary background on the canonical divisor are given in §1.4, although intersection theory for surfaces is only introduced in Chapter 3. The Miyaoka-Yau inequality for minimal smooth compact connected algebraic surfaces of general type, which is of deep importance for the material of this book, is derived in Chapter 4, and is the inequality c21(X)[less than or equal to]3c2(X) relating the Chern numbers and c2/1(X) and c2(X). In Chapter 6, we derive a version of this inequality for surfaces with an orbifold structure that are not necessarily compact (we touch on the non-compact situation also at the end of Chapter 4). When this inequality is an equality, X is a quotient of the complex two-dimensional ball B2 by a discrete subgroup of the automorphisms of B,2, acting without fixed points in Chapter 4 and with fixed points in Chapter 6. For the summaries of the proofs of the Miyaoka-Yau inequalities in these chapters, we use techniques due to Aubin, S.-T. Yau, and R. Kobayashi coming from differential geometry and partial differential equations. Some of the differential geometry can be found in §1.4 and the rest is derived as needed in Chapters 4 and 6.
In this book, we discuss the Miyaoka-Yau inequality only for surfaces, as our interest is in weighted line arrangements in the complex projective plane. A suitably generalized Miyaoka-Yau inequality due to Aubin and Yau holds, for example, for compact Kähler manifolds of dimension n whose first Chern class vanishes or is negative, meaning it is represented by a real closed negative definite (1, 1)-form (see §1.4 for definitions). For a statement, see p. 323.
1.1 TOPOLOGICAL INVARIANTS
Let X be a topological space. We briefly recall the definition, using singular chains, of the singular homology groups Hi(X, Z) with integer coefficients (see, for example). Viewing Rn as the subset of Rn+1 consisting of the vectors with (n + 1)th coordinate equal to 0, we can consider the union R∞ = [union]n[greater than or equal to]1Rn. Let en, n [greater than or equal to] 1 be the vector whose nth coordinate is 1 and whose other coordinates are 0, and let e0 be the vector with all its coordinates 0. For i ≥ 0, the standard simplex Δi of dimension i is given by the set
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
A singular i-dimensional simplex in X is a continuous map from Δi onto X. The singular i-chains Ci(X) in X are the finite linear combinations, with integer coefficients, of the singular i-dimensional simplices. They form an abelian group.
For i ≥ 1 and k = 0, ... i, we define the kth face map [partial derivative]ki: Δi-1 [arrow right]Δi by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The boundary operator
[partial derivative] : Ci(X) [right arrow] Ci-1(X)
is defined as the alternating sum
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It satisfies [partial derivative][partial derivative] = 0, and so we have a differential complex with homology group
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This definition depends only on X. No triangulation or similar structure is needed. If X is triangulated, we can use the chain groups c(X) consisting of finite linear combinations of the simplices of the triangulation; otherwise the definition is as before. It is known that the homology groups obtained in this way are the groups Hi(X, Z) defined above. If the abelian group Hi (X, Z) is finitely generated, then the rank of Hi (X, Z) is called the ith Betti number, bi(X). The subgroup of elements of finite order is the torsion subgroup Ti(X). If the space X admits a finite triangulation, then the Hi (X, Z) are all finitely generated and trivial for i sufficiently large. If Hi (X, Z) is finitely generated, then Hi (X, Z) [cross product] R is a real vector space of dimension bi(X), which is also denoted by Hi (X, R). The homology groups are homotopy invariants.
The dual construction gives singular cohomology. A singular i-cochain on X is a linear functional on the Z-module Ci(X) of singular i-chains. The group of singular i-cochains is therefore Ci(X) = Hom(Ci(X), Z). The coboundary operator is defined by (dω)(s) = ω ( [partial derivative]s), ω [member of] C*(X), and satisfies dd = 0. The graded group of singular cochains C*(X) = [direct sum] iCi( X) is therefore a differential complex whose homology is called the singular cohomology of X with integer coefficients. The ith cohomology group is denoted by Hi(X, Z) (for more details, see).
The Euler-Poincaré characteristic, which we also call the Euler number, is an important invariant of a topological space X, and is denoted by e(X). Assume H(Xi Z) is finitely generated and trivial for i sufficiently large. Then
e(X) = [summation over (i)](- 1)ibi(X) = [summation over (i)] (-1)idim Hi(X,R). (1.1)
The number e(X) is a topological invariant and a homotopy invariant. If the space has a finite triangulation, then
e(X) = [summation over (i)](-1)i rank (ci(X)).
In other words,
e(X) = [summation over (i)](-1)iei,
where ei is the number of i-dimensional simplices of the triangulation. This generalizes to cellular decompositions.
The Euler-Poincaré characteristic generalizes the notion of the cardinality of a finite set. Indeed, if X is a finite set, then e(X) = |X|. If X and Y are finite sets, then we have
|X [union] Y| = |X| + |Y| - |X [intersection] Y|,
which generalizes to the Mayer-Vietoris relation
e(X [union] Y) = e(X) + e(Y) - e(X [intersection] Y), (1.2)
which is certainly true if X [union] Y admits a finite triangulation that induces triangulations on X, Y, and X [intersection] Y. In all our applications of the Mayer-Vietoris relation, this will be the case. Moreover, if X and X' are homotopically equivalent, then e(X) = e(X'). For products of topological spaces (admitting finite triangulations), we have the formula
e(X x Y) = e(X)e(Y),
generalizing
|X x Y| = |X| * |Y|
for the cardinality of finite sets. These simple formulas allow us to compute the Euler-Poincaré characteristic (Euler number) of many of the spaces we shall encounter. We can also calculate the homology groups, but prefer to use the Mayer-Vietoris relation and the product formula. In fact, in many cases occurring in later chapters, it is easy to calculate the Euler number but very difficult to determine the Betti numbers. For more details and proofs of these facts, see.
A point P is a set with one element; hence e(P) = 1. An interval I is homotopic to a point so that
e(I) = 1.
As the circle S1 is the union of two semicircles homeomorphic to the unit interval and intersecting at two points, we have
e(S1 = e(I) + e(I) - 2 * e(P) = 0.
On the other hand, the 2-sphere S2 (also called the Riemann sphere) is the union of two hemispheres, each homotopic to a point and intersecting in a copy of S1, so that
e(S2) = e(P) + e(P) - e(S1) = 2.
More generally, the n-sphere Sn satisfies
e(Sn) = 2, for n even, e(Sn) = 0, for n odd.
Example: In Chapter 2 we discuss Riemann surfaces (real dimension 2). The real manifold underlying a compact Riemann surface is homeomorphic to S2 with g handles attached, for a certain number g depending only on the topology of the Riemann surface. Each handle is homeomorphic to I x S1 where I is the unit interval and S1 the unit circle. The invariant g is called the genus of the Riemann surface. From S2 we remove g pairs of open disks (the closed disks are supposed to be disjoint, each being a disk with respect to a local coordinate system). By Mayer-Vietoris, the resulting space X has Euler number 2 - 2g. The Euler number of the Riemann surface Fg of genus g also equals 2 - 2g, because the handles have Euler number 0 and intersect X in circles. (Attaching a handle means identifying the two boundary circles of the handle with a pair of circles in X). For Fg we have
e(Fg) = 2 - 2g = e - k + f,
where e is the number of vertices, k the number of edges, and f the number of faces of a triangulation, or more generally a cellular decomposition of Fg. For g = 0 (when Fg = S2), this is the traditional formula of Euler (and perhaps Descartes). The classification of compact-oriented surfaces by their genus g is essentially due to A.F. Möbius (1863) and C. Jordan (1866). See the essay by R. Remmert and M. Schneider in.
1.2 FUNDAMENTAL GROUPS AND COVERING SPACES
In this book we encounter quotients of simply-connected complex spaces by groups of automorphisms of the spaces acting discontinuously, and often freely. For example, in Chapter 2 we use the fact that every Riemann surface is the quotient of one of the simply-connected Riemann surfaces, called its universal cover, by a subgroup acting freely discontinuously. This fact is a consequence of the theory of fundamental groups and covering spaces. Here, we give only a rough indication of some of the basics of this theory. More details can be found, for example, in, Chapter IX. Let X be a topological space, and x0 a fixed point of X. We say X is path-connected if there is a path between any two points of X. Suppose X is path-connected. The paths in X with base point x0 are the continuous maps p from the interval [0, 1] to X with p(0) = x0. We can put the compact open topology on this space, with the basic open sets consisting of the paths p in X with base point x0 such that p(K) [subset] U for a fixed compact set K in [0, 1] and a fixed open set U in X. The paths p with p(0) = x0 and p(1) = x0 are called the loops based at x and have a natural composition law. For loops p1p2 the composite p1 ο p2 is obtained by first traversing p1 and then traversing p2. A homotopy between two paths p1 and p2 in X is a continuous map F: [0, 1] x [0, 1] [right arrow] X such that F(t, s) = p1(t) for s = 0 and F(t, s) = p2(t) for s = 1. This induces an equivalence relation between loops (based at x0). The set of homotopy classes of loops (based at x0), with the operation given by the composition of loops, is the fundamental group π1 (X, x0). Then, for any two base points x, y, the groups π1 (X, x0) and π0 (X, y0) are isomorphic. An isomorphism can be constructed by using the path from x0 to y0. Two such isomorphisms are related by inner automorphisms of the two groups. By definition, the path-connected space X is simply connected if and only if π1 (X, x0) is trivial.
Every path-connected space X, which satisfies some additional conditions, has a simply-connected universal covering space Y with a free action of π(X, x0) such that the orbit space is X. The additional conditions (see) are locally path-connected and semi-locally simply connected. These certainly hold for manifolds and for triangulated spaces. For x [member of] X, we denote by ω(X, x0, x) the set of paths from x0 to x. Elements α, β of Ω(X, x0, x) define α ο β-1 in Ω(X, x0, x) and hence an element in π(X, x0). If this element is the identity element the elements α [Beta] are called equivalent. The set of equivalence classes is denoted by Yx The space Y is the disjoint union of all these Yx for x [member of] X. The natural projection π : Y [right arrow] X has the Yx as fibers, that is, π-1(x) = Yx Fixing a path from x0 to x, or rather a point of Yx defines a bijection of Yx to π1(X, x0), with the fixed point of Yx going to the identity. Composing this fixed path with a path from x to an arbitrary point x' in a locally path -connected neighborhood U of x leads to a local trivialization U x π1 (X, x0), where π1(X, x0) carries the discrete topology, and Y carries a global topology which is locally of this form. The covering space Y is simply connected. If π1 (y0) = x0, then a loop α in Y from y0 to y0 projects under π to a loop π (α) from x0 to x0, which by our construction represents the identity element of π1 (X, x0). But a homotopy from π(α) to the constant loop can be lifted to a homotopy in Y deforming α to the constant loop.
(Continues...)
Excerpted from Complex Ball Quotients and Line Arrangements in the Projective Plane by Paula Tretkoff. Copyright © 2016 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- Frontmatter, pg. i
- Contents, pg. vii
- Preface, pg. ix
- Introduction, pg. 1
- Chapter One. Topological Invariants and Differential Geometry, pg. 6
- Chapter Two. Riemann Surfaces, Coverings, and Hypergeometric Functions, pg. 23
- Chapter Three. Complex Surfaces and Coverings, pg. 47
- Chapter Four. Algebraic Surfaces and the Miyaoka-Yau Inequality, pg. 65
- Chapter Five. Line Arrangements in P2(C) and Their Finite Covers, pg. 85
- Chapter Six. Existence of Ball Quotients Covering Line Arrangements, pg. 126
- Chapter Seven. Appell Hypergeometric Functions, pg. 167
- Appendix A. Torsion-Free Subgroups of Finite Index by Hans-Christoph Im Hof, pg. 189
- Appendix B. Kummer Coverings, pg. 197
- Bibliography, pg. 205
- Index, pg. 213