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This chapter justifies the assumption that ball quotients covering line arrangements exist. It begins with the general case on the existence of finite covers by ball quotients of weighted configurations, focusing on log-canonical divisors and Euler numbers reflecting the weight data on divisors on the blow-up X of P2 at the singular points of a line arrangement. It then uses the Kähler-Einstein property to prove an inequality between Chern forms that, when integrated, gives the appropriate Miyaoka-Yau inequality. It also discusses orbifolds and b-spaces, weighted line arrangements, the problem of the existence of ball quotient finite coverings, log-terminal singularity and log-canonical singularity, and the proof of the main existence theorem for line arrangements. Finally, it considers the isotropy subgroups of the covering group.

This chapter discusses the free 2-ball quotients arising as finite covers of the projective plane branched along line arrangements. It first considers a surface X obtained by blowing up the singular intersection points of a linear arrangement in the complex projective plane, as well as a smooth compact complex surface Y that is a finite covering of X. If Y is of general type with vanishing proportionality deviation, then it is a free 2-ball quotient. The chapter then looks at line arrangements that have equal ramification indices along each of the proper transforms of the original lines, along with cases of blowing down rational curves and removing elliptic curves. It also enumerates all possibilities for the assigned weights of the arrangements, under the assumption that divisors of negative or infinite weight on the blown-up line arrangements do not intersect.

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.

This chapter deals with complex surfaces and their finite coverings branched along divisors, that is, subvarieties of codimension 1. In particular, it considers coverings branched over transversally intersecting divisors. Applying this to linear arrangements in the complex projective plane, the chapter first blows up the projective plane at non-transverse intersection points, that is, at those points of the arrangement where more than two lines intersect. These points are called singular points of the arrangement. This gives rise to a complex surface and transversely intersecting divisors that contain the proper transforms of the original lines. The chapter also introduces the divisor class group, their intersection numbers, and the canonical divisor class. Finally, it describes the Chern numbers of a complex surface in order to define the proportionality deviation of a complex surface and to study its behavior with respect to finite covers.

This chapter explains that the book deals with quotients of the complex 2-ball yielding finite coverings of the projective plane branched along certain line arrangements. It gives a complete list of the known weighted line arrangements that can produce such ball quotients, and then provides a justification for the existence of the quotients. The Miyaoka-Yau inequality for surfaces of general type, and its analogue for surfaces with an orbifold structure, plays a central role. The book also examines the explicit computation of the proportionality deviation of a complex surface 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.

This chapter addresses Drinfeld's lemma for diamonds. It proves a local analogue of Drinfeld's lemma, thereby giving a first nontrivial argument involving diamonds. This lecture is entirely about fundamental groups. A diamond is defined to be connected if it is not the disjoint union of two open subsheaves. For a connected diamond, finite étale covers form a Galois category. As such, for a geometric point, one can define a profinite group, such that finite sets are equivalent to finite étale covers. In this proof, the chapter uses the formalism of diamonds rather heavily to transport finite étale maps between different presentations of a diamond as the diamond of an analytic adic space.