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This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. Paula Tretkoff emphasizes those finite covers that are free "ients of the complex two-dimensional ball. Tretkoff also includes background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function.The material in this book began as a set of lecture notes, taken by Tretkoff, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded by Hirzebruch and Tretkoff over a number of years. In this book, Tretkoff has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers.Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry.

Curves, Elliptic. --- Geometry, Algebraic. --- Projective planes. --- Unit ball. --- Riemann surfaces. --- Surfaces, Riemann --- Functions --- Ball, Unit --- Holomorphic functions --- Planes, Projective --- Geometry, Projective --- Algebraic geometry --- Geometry --- Elliptic curves --- Curves, Algebraic --- Appell hypergeometric function. --- Chern numbers. --- Euler number. --- Friedrich Hirzebruch. --- Gauss hypergeometric function. --- Gaussian curvature. --- Hermitian metric. --- Kodaira dimension. --- Mbius transformation. --- Miyaoka-Yau inequality. --- Riemann surface. --- Riemannian metric. --- algebraic geometry. --- algebraic surface. --- arithmetic monodromy group. --- b-space. --- ball "ient. --- canonical divisor class. --- complete quadrilateral. --- complex 2-ball. --- complex manifold. --- complex surface. --- covering group. --- covering space. --- differential geometry. --- divisor class group. --- divisor. --- elliptic curve. --- finite covering. --- first Chern class. --- fractional linear transformation. --- free 2-ball "ient. --- fundamental group. --- geometry. --- intersection point. --- line arrangement. --- line bundle. --- linear arrangement. --- log-canonical divisor. --- minimal surface. --- monodromy group. --- orbifold structure. --- orbifold. --- partial differential equation. --- plurigenus. --- projective plane. --- proportionality deviation. --- ramification indices. --- rational curve. --- regular point. --- signature. --- solution space. --- topological invariant. --- transversely intersecting divisor. --- triangle groups. --- weight.