Ball and Surface Arithmetics

This monograph is based on the work of the author on surface theory con nected with ball uniformizations and arithmetic ball lattices during several years appearing in a lot of special articles. The first four chapters present the heart of this work in a self-contained manner (up to well-known ba sic facts) increased by the new functorial concept of orbital heights living on orbital surfaces. It is extended in chapter 6 to an explicit HURWITZ theory for CHERN numbers of complex algebraic surfaces with the mildest singularities, which are necessary for general application and proofs. The chapter 5 is dedicated to the application of results in earlier chapters to rough and fine classifications of PICARD modular surfaces. For this part we need additionally the arithmetic work of FEUSTEL whose final results are presented without proofs but with complete references. We had help ful connections with Russian mathematicians around VENKOV, VINBERG, MANIN, SHAFAREVICH and the nice guide line of investigations of HILBERT modular surfaces started by HIRZEBRUCH in Bonn. More recently, we can refer to the independent (until now) study of Zeta functions of PICARD modular surfaces in the book [L-R] edited by LANGLANDS and RAMAKR ISHN AN. The basic idea of introducing arrangements on surfaces comes from the monograph [BHH], (BARTHEL, HOFER, HIRZEBRUCH) where linear ar rangements on the complex projective plane ]p2 play the main role.

Autorentext

Dr. Rolf-Peter Holzapfel ist Mitglied der Arbeitsgruppe "Allgebraische Geometrie und Zahlentheorie" am Max-Planck-Institut zur Förderung der Wissenschaften, Berlin.

Klappentext

Bei höherdimensionalen komplexen Mannigfaltigkeiten stellt die Riemann-Roch-Theorie die grundlegende Verbindung von analytischen bzw. algebraischen zu topologischen Eigenschaften her. Dieses Buch befaßt sich mit Mannigfaltigkeiten der komplexen Dimension 2, d. h. mit komplexen Flächen. Hauptziel der Monographie ist es, neue rationale diskrete Invarianten (Höhen) in die Theorie komplexer Flächen explizit einzuführen und ihre Anwendbarkeit auf konkrete aktuelle Probleme darzustellen.Als erste unmittelbare Anwendung erhält man explizit und ganz allgemein Formeln vom Hurwitz-Typ endlicher Flächenüberlagerungen für die vier klassischen Invarianten, die auf andere Weise bisher nur in Spezialfällen zugänglich waren. Ein weiteres Anwendungsgebiet ist die Theorie der Picardschen Modulflächen: Neue Resultate werden beschrieben. Letztendlich kann im letzten Kapitel eine Ergänzung des bekannten Satzes von Bogomolov-Miyaoka-Yau mit Hilfe der Höhentheorie gezeigt werden. The monograph presents basically an arithmetic theory of orbital surfaces with cusp singularities. As main invariants orbital hights are introduced, not only for the surfaces but also for the components of orbital cycles. These invariants are rational numbers with nice functorial properties allowing precise formulas of Hurwitz type and a fine intersection theory for orbital cycles. For ball quotient surfaces they appear as volumes of fundamental domains. In the special case of Picard modular surfaces they are discovered by special value of Dirichlet L-series or higher Bernoulli numbers. As a central point of the monograph a general Proportionality Theorem in terms of orbital hights is proved. It yields a strong criterion to decide effectively whether a surface with given cycle supports a ball quotient structure being Kaehler-Einstein with negative constant holomorphic sectional curvature outside of this cycle. The theory is applied to the classification of Picard modular surfaces and to surfaces geography.

Inhalt
1 Abelian Points.- 1.1 Cyclic Points.- 1.2 Graphs of Abelian Points.- 1.3 Geometric Interpretation.- 1.4 Derived Representations.- 1.5 The Differential Relation.- 1.6 Stepwise Resolutions of Cyclic Points.- 1.7 Continued Fractions and Selfintersection Numbers.- 1.8 Reciprocity Law for Geometric Sums.- 1.9 Explicit Dedekind Sums.- 1.10 Eisenstein Sums.- 1.11 Hirzebruch's Sum.- 1.12 Geometric Interpretation.- 1.13 Quotients and Coverings of Modifications.- 1.14 Selfintersections of Quotient Curves.- 1.15 The Bridge Algorithm.- 1.16 First Orbital Properties.- 1.17 Local Orbital Euler Numbers.- 1.18 Absorptive Numbers.- 2 Orbital Curves.- 2.1 Point Arrangements on Curves.- 2.2 Euler Heights of Orbital Curves.- 2.3 The Geometric Local-Global Principle.- 2.4 Signature Heights of Orbital Curves.- 3 Orbital Surfaces.- 3.1 Regular Arrangements on Surfaces.- 3.2 Basic Invariants and Fixed Point Theorem.- 3.3 Euler Heights.- 3.4 Signature Heights.- 3.5 Quasi-homogeneous Points, Quotient Points and Cusp Points.- 3.6 Quasi-smooth Orbital Surfaces.- 3.7 Open Orbital Surfaces.- 3.8 Orbital Decompositions.- 4 Ball Quotient Surfaces.- 4.1 Ball Lattices.- 4.2 Neat Ball Cusp Lattices.- 4.3 Invariants of Neat Ball Quotient Surfaces.- 4.4 ?-Rational Discs.- 4.5 Cusp Singularities, Reflections and Elliptic Points.- 4.6 Orbital Ball Quotient Surfaces and Molecular.- 4.7 Invariants of Disc Quotient Curves.- 4.8 Invariants of Ball Quotient Surfaces.- 4.9 Global Proportionality.- 4.10 Orbital Decompositions and the Finiteness Theorem.- 4.11 Leading Examples.- 4.12 Towards the Count of Ball Metrics on Non-Compact Surfaces.- 5 Picard Modular Surfaces.- 5.1 Classification Diagram.- 5.2 Picard Modular Surface of the Field of Eisenstein Numbers.- 5.3 Picard Modular Surface of the Field ofGauss-Numbers.- 5.4 Kodaira Classification of Picard Modular Surfaces.- 5.5 Special Results and Examples.- 5A Volumes of Fundamental Domains of Picard Modular Groups.- 5A.1 The Order of Finite Unitary Groups.- 5A.2 Index of Congruence Subgroups.- 5A.3 Local Volumina.- 5A.4 The Global Volume.- 6 ?-Orbital Surfaces.- 6.1 Introduction.- 6.2 Arrangements with Rational Coefficients.- 6.3 Finite Morphisms of ?-Orbital Surfaces.- 6.4 Functorial Properties for Rational Invariants.- 6.5 Euler and Signature Heights.- 6.6 Reduction of Galois-Finite Morphisms.- 6.7 Local Base Changes.- 6.8 Global Base Changes.- 6.9 Explicit Hurwitz Formulas for Finite Surface Coverings.- 6.10 Finite Coverings of Ruled Surfaces and the Inequality c12 ? 2c2.