Complex Ball Quotients and Line Arrangements in the Projective Plane

Paula Tretkoff

• Publisher: Princeton University Press
• Year: 2016
• Print ISBN: 9780691144771
• eBook ISBN: 9781400881253
• Topic: Mathematics
• Subtopic: Algebraic Geometry
• Index: Mathematics

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.

Chapter One

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, [19], [49], [55], [112]. 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

, 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

relating the Chern numbers

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 B₂, 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 [20], [84], 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 H_i(X, ℤ) with integer coefficients (see, for example, [19], [112]). Viewing ℝⁿ as the subset of ℝⁿ⁺¹ consisting ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• Introduction
• 1: Topological Invariants and Differential Geometry
• 2: Riemann Surfaces, Coverings, and Hypergeometric Functions
• 3: Complex Surfaces and Coverings
• 4: Algebraic Surfaces and the Miyaoka-Yau Inequality
• 5: Line Arrangements in ℙ₂(ℂ) and Their Finite Covers
• 6: Existence of Ball Quotients Covering Line Arrangements
• 7: Appell Hypergeometric Functions
• A: Torsion-Free Subgroups of Finite Index
• B: Kummer Coverings
• Bibliography
• Index