Complex analysis, 2, Riemann surfaces, several complex variables, Abelian functions, higher modular functions / Eberhard Freitag

The book is the second volume of the textbook Complex analysis, consisting of 8 chapters. It provides an approach to the theory of Riemann surfaces from complex analysis. In other words, most of methods and tools are purely complex analytic while the book contains standard theorems on Riemann surfaces, e.g. the uniformization theorem, the Riemann-Roch theorem, Abel's theorem. For those theorems given in the first three chapters, a fundamental tool is the existence theorem for harmonic differentials. The author uses the alternating method of Schwarz to show the theorem. It expresses the point of view of this book. The book also contains a concise, but sufficiently substantial description of analytic functions of several complex variables in Chapter V. It will help the reader to understand a theory of modular functions given in Chapter VII. The book is self-contained and, moreover, some notions which might be unfamiliar for the reader are explained in appendices of chapters. I think this book is an excellent textbook on Riemann surfaces, especially for graduate students who have taken the first course of complex analysis. (MathScinet)