Complex ball quotients and line arrangements in the projective plane. With an appendix by Hans-Christoph Im Hof (Q2790317)

This book has its origin in the notes prepared by F. Hirzebruch. In complex dimension \(1\), every compact Riemann surface of genus \(g\geq 2\) has representations both as a plane algebraic curve, and so as a branched covering of the complex projective line, and as a quotient of the complex \(1\)-ball, or unit disk, by a freely acting cocompact discrete subgroup of the automorphisms of the \(1\)-ball. In complex dimension \(2\), the smooth compact connected complex algebraic surface, representable as quotients of the complex \(2\)-ball by a freely acting cocompact discrete subgroups of the automorphisms of the \(2\)-ball, are precisely the surfaces of general type satisfying the equality \(c_1^2=3c_2\). Here \(c_2\) is the Euler characteristic and \(c_1^2\) is the self-intersection umber of the canonical divisor. This leads to the notion of \textit{proportionality deviation} of a complex surface, which is defined to be the expression \(3c_2-c_1^2\). This book examines the explicit computation of this proportionality deviation for finite covers of the complex projective plane ramified along certain line arrangements. Candidates for ball quotients among these finite covers arise by choosing weights on the line arrangements such that the proportionality deviation vanishes. Then it is shown that these ball quotients actually exist.NEWLINENEWLINEThe book is organized as follows. Chapter 1 collects the main prerequisites from topology and differential geometry needed for the subsequent discussions. Chapter 2 applies some of the materials from Chapter 1 to Riemann surfaces. Also this Chapter discuss the classical hypergeometric functions of one complex variable, paving a way for Appell hypergeomtric functions of two complex variables. Chapter 3 studies complex surfaces and their coverings branched along divisors, i.e., subvarieties of codimension \(1\). Here the Chern numbers \(c_1^2\), \(c_2\) as well as the proportionality deviation are defined, among other things. Chapter 4 gives a rough classification of (smooth complex connected compact algebraic) surfaces. Two approaches are presented for the Miyaoka--Yau inequality: \(c_1^2\leq 3c_2\) for surfaces of general type. One is algebraic geometric approach due to Miyaoka, and the other is Aubin--Yau approach which makes use of analysis and differential geometry. Also discussion is presented why the equality in the Miyaoka--Yau inequality characterizes surfaces of general type that are free quotients of the complex \(2\)-ball. Chapter 5 discusses the main topic of the book, that is, the free \(2\)-ball quotients arising as finite covers of the projective plane branched along line arrangements. Let \(X\) be the surface obtained by blowing up the singular intersection points of a line arrangement in the complex projective plane. Let \(Y\) be a smooth compact complex surface given by a finite cover of \(X\) branched along the divisors on \(X\) defined by the lines of the arrangement. If \(Y\) is of general type with vanishing proportionality deviation, then it is a free \(2\)-ball quotient. Necessary conditions are derived for weighted line arrangements in the complex projective plane to admit finite covers that are all quotients by finding solutions to \(c_1^2=3c_2\) for such finite covers. Chapter 6 discusses the existence question of such finite covers. Chapter 7 focuses on the complete quadrilateral line arrangement, and its relationship with the space of regular points of the system of partial differential equation defining the Appell hypergeometric function. Deligne and Mostow established criteria for the monodromy groups of this Appell function to act discontinuously on the complex \(2\)-ball. This gives rise to examples of complex \(2\)-ball quotients, determined by freely acting subgroups of finite index in the monodromy groups that are branched along the blown-up complete quadrilateral. The book concludes with two appendices. One is by Im Hof and supplies a proof of Frenchel's Conjcture about the existence of torsion-free subgroups of finite index in finitely generated Fuchsian groups. The second concerns with Kummer covers of line arrangements in the projective plane.