Ball and surface arithmetics.

*(English)*Zbl 0980.14026
Aspects of Mathematics. E 29. Wiesbaden: Vieweg. xiii, 414 p. (1998).

It is well-known that a Riemann surface occurs as a finite cover of the complex projective plane \(\mathbb P^1\), and it is isomorphic to a compact quotient \(\mathbb D/\Gamma'\) of the complex unit disc \(\mathbb D\) by a suitable discrete subgroup \(\Gamma'\) of the automorphism group of \(\mathbb D\). Similarly, a complex projective surface is a finite cover of \(\mathbb P^2\), and it can be regarded as the quotient \(\mathbb B/ \Gamma\) of the complex unit ball
\[
\mathbb B = \{ (z_1, z_2) \in \mathbb C \mid |z_1|^2 + |z_2|^2 < 1\}
\]
by a ball lattice \(\Gamma\). In this book the author discusses complex algebraic surfaces with the underlying philosophy that, up to birational equivalence and compactifications, all complex algebraic surfaces are ball quotients of the form \(\mathbb B/ \Gamma\).

The main purpose of this book is to introduce special rational invariants, called orbital heights, and to work with them producing interesting results. These heights are invariant with respect to finite coverings up to a well-determined factor, and they are defined for orbital surfaces, which are complex surfaces with an arrangement. An arrangement is a potentially assumed branch locus whose components, consisting of points and irreducible curves, are endowed with natural weights and rational coefficients. Finding such heights of orbital surfaces requires the verification of various functorial properties going inductively through the categories of abelian points, orbital curves, orbital surfaces, and the corresponding categories of related objects.

Chapter 1 is devoted to the category of abelian points, which are embedded cyclic surface singularities together with two weighted embedded curve germs intersecting at each such point. In order to classify abelian points, weighted graphs of abelian points are also discussed. Orbital curves are introduced in chapter 2 as surface germs along a weighted compact curve supporting abelian points. Two kinds of heights can be considered for orbital curves, namely, Euler heights and signature heights. Euler heights generalize Euler numbers, and signature heights are similar to self-intersection numbers for curves on smooth surfaces. These heights can be read off from certain graphs of orbital curves obtained from graphs of abelian points. Chapter 3 is concerned with orbital surfaces. Euler and signature heights are also considered for orbital surfaces, and they can be calculated from certain weighted graphs of orbital surfaces constructed from graphs of orbital curves and points. Chapter 4 discusses ball quotients \(\mathbb B/ \Gamma\), which are considered as open orbital surfaces. The support of the arrangement \(A(\Gamma)\) of a ball quotient \(\mathbb B/ \Gamma\) is the branch locus of the infinite Galois covering \(\mathbb B \to \mathbb B /\Gamma\). Up to universal factors, the heights of the ball quotients occur as volumes of the corresponding ball lattices. Chapter 5 presents Picard modular surfaces, which are quotients \(\mathbb B/ \Gamma\) by certain arithmetic ball lattices \(\Gamma\) associated to imaginary quadratic number fields. Results in earlier chapters are applied to rough and fine classifications of Picard modular surfaces using the arithmetic work of Feustel. The calculation of the heights of Picard modular surfaces is discussed in appendix 5.A. Chapter 6 deals with the extension of the functorial concept of orbital heights to a more general class of orbital surfaces. Such a concept is extended to an explicit Hurwitz theory for Chern numbers of complex algebraic surfaces with mildest singularities.

Over the years, the author has written numerous papers on surface theory connected with ball uniformizations and arithmetic ball lattices. This book makes available in one place various results contained in those papers and should be a valuable addition to the literature.

The main purpose of this book is to introduce special rational invariants, called orbital heights, and to work with them producing interesting results. These heights are invariant with respect to finite coverings up to a well-determined factor, and they are defined for orbital surfaces, which are complex surfaces with an arrangement. An arrangement is a potentially assumed branch locus whose components, consisting of points and irreducible curves, are endowed with natural weights and rational coefficients. Finding such heights of orbital surfaces requires the verification of various functorial properties going inductively through the categories of abelian points, orbital curves, orbital surfaces, and the corresponding categories of related objects.

Chapter 1 is devoted to the category of abelian points, which are embedded cyclic surface singularities together with two weighted embedded curve germs intersecting at each such point. In order to classify abelian points, weighted graphs of abelian points are also discussed. Orbital curves are introduced in chapter 2 as surface germs along a weighted compact curve supporting abelian points. Two kinds of heights can be considered for orbital curves, namely, Euler heights and signature heights. Euler heights generalize Euler numbers, and signature heights are similar to self-intersection numbers for curves on smooth surfaces. These heights can be read off from certain graphs of orbital curves obtained from graphs of abelian points. Chapter 3 is concerned with orbital surfaces. Euler and signature heights are also considered for orbital surfaces, and they can be calculated from certain weighted graphs of orbital surfaces constructed from graphs of orbital curves and points. Chapter 4 discusses ball quotients \(\mathbb B/ \Gamma\), which are considered as open orbital surfaces. The support of the arrangement \(A(\Gamma)\) of a ball quotient \(\mathbb B/ \Gamma\) is the branch locus of the infinite Galois covering \(\mathbb B \to \mathbb B /\Gamma\). Up to universal factors, the heights of the ball quotients occur as volumes of the corresponding ball lattices. Chapter 5 presents Picard modular surfaces, which are quotients \(\mathbb B/ \Gamma\) by certain arithmetic ball lattices \(\Gamma\) associated to imaginary quadratic number fields. Results in earlier chapters are applied to rough and fine classifications of Picard modular surfaces using the arithmetic work of Feustel. The calculation of the heights of Picard modular surfaces is discussed in appendix 5.A. Chapter 6 deals with the extension of the functorial concept of orbital heights to a more general class of orbital surfaces. Such a concept is extended to an explicit Hurwitz theory for Chern numbers of complex algebraic surfaces with mildest singularities.

Over the years, the author has written numerous papers on surface theory connected with ball uniformizations and arithmetic ball lattices. This book makes available in one place various results contained in those papers and should be a valuable addition to the literature.

Reviewer: Min Ho Lee (Cedar Falls)

##### MSC:

14J15 | Moduli, classification: analytic theory; relations with modular forms |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14M17 | Homogeneous spaces and generalizations |

14J25 | Special surfaces |

32N15 | Automorphic functions in symmetric domains |

14G35 | Modular and Shimura varieties |

11F55 | Other groups and their modular and automorphic forms (several variables) |