Chapter I. Riemann Surfaces -- Chapter II. Harmonic Functions on Riemann Surfaces -- Chapter III. Uniformization -- Chapter IV. Compact Riemann Surfaces -- Appendices to Chapter IV -- Chapter V. Analytic Functions of Several Complex Variables -- Chapter V. Analytic Functions of Several Complex Variable -- Chapter VI. Abelian Functions -- Chapter VII. Modular Forms of Several Variables -- Chapter VIII. Appendix: Algebraic Tools -- References -- Index.
The book provides a complete presentation of complex analysis, starting with the theory of Riemann surfaces, including uniformization theory and a detailed treatment of the theory of compact Riemann surfaces, the Riemann-Roch theorem, Abel's theorem and Jacobi's inversion theorem. This motivates a short introduction into the theory of several complex variables, followed by the theory of Abelian functions up to the theta theorem. The last part of the book provides an introduction into the theory of higher modular functions.