Basic two-dimensional versions of Hurwitz genus formula (Q1086643)

Let Y be a complex compact normal algebraic surface and assume that all singularities are of Hirzebruch-Jung type. Upto surface singularities, a regular arrangement on Y is a cycle \(\sum v_ iD_ i,\) where \(v_ i\) is a natural number, the \(D_ i's\) are smooth (compact) curves on Y and \(\sum D_ i\) has nice (''regular'') crossings. A pair \(Y=(Y,(reg.) arrangement)\) is called a (regularly) arranged surface. For \(c=c_ 2\) (Euler number symbol) and \(c=\tau\) (signature symbol) the author defines in an explicit manner rational numbers c(Y). The definitions need the Chern numbers of Y, the Euler numbers of the curves \(D_ i\), the selfintersection numbers of the \(D_ i's\) (taken on the minimal resolution of singularities of Y), the weights \(v_ i\), Dedekind sums and the classification data of the singularities of Y. The main result of the paper is the proportional relation \(c(X)=card(G)\cdot c(Y)\) for \(Y=X/G\), \(\sum v_ iD_ i\) the branch divisor of the quotient map \(X\to Y\) weighted with ramification indices. The action of the finite group G on X is assumed to be regular, X smooth, that means that the branch arrangement is regular. c(Y) appeared in earlier papers of the author as non-euclidean volumes of fundamental domains of arithmetic groups acting on the unit ball. The proof uses the formulas of equivariant K-theory, some Dedekind sum calculus due to Hirzebruch/Zagier, and a selfintersection formula for quotient curves on quotient surfaces. The latter has been proved by the author in an earlier paper. The proportional formula is applied to prove that each regular (and smooth) Galois covering of a ruled surface or of \({\mathbb{P}}^ 2\) has non-positive signature. For a compact ball quotient Y with regular branch arrangement one knows that \(c_ 2(Y)=3\tau (Y)\) by proportionality. Using this identity the following finiteness property is proved for a given pair \((Y,\sum D_ i)\): There are at most finitely many ball quotient arrangement \(\sum v_ iD_ i\) supported on \(\sum D_ i.\) The constructive proof is applied to two examples, which are closely connected with differential equations of classical hypergeometric functions of two variables.