The theory of Riemann surfaces, and complex analysis in general, is a privileged part of mathematics: It is analysis and geometry, of course, but it is also algebra and differential equations, and a source of problems for many other branches of the mathematical tree.

There are many wonderful books on Riemann surfaces, with various approaches and emphases that reflect the multifaceted nature of these objects. Some classic texts emphasize the algebraic geometry approach, focusing on *compact* Riemann surfaces. A few books are devoted to open Riemann surfaces, perhaps with a view towards applications to complex dynamical systems. There are even fewer books that study compact and open Riemann surfaces from the point of view of complex analytic geometry, many of their methods having been developed in the last five decades. The classical text that immediately comes to mind is *Principles of Algebraic Geometry*, by P. Griffiths and J. Harris (Wiley, 1978), whose approach is similar to the book under review, but requires from the reader a more profound knowledge of several complex variables to be able to grasp the techniques and results on complex manifolds that are the basic objects to be studied.

This brings us to the book we are reviewing. It is an introductory text on the theory of Riemann surfaces that treats both open and compact surfaces. Its approach, as could be guessed from the above remarks, is from the point of view of complex analytic geometry. However, the mathematical requirements are a bit more elementary than those for Griffiths-Harris: complex analysis on one variable. These requirements are reviewed in the first chapter: from the Cauchy-Green formula and the Cauchy-Riemann equations, holomorphic, meromorphic, harmonic and subharmonic functions are introduced, and their main properties are formulated (power series representation, singularities, residues, the maximum principle, Montel and Koebe’s theorems, regularity of harmonic functions, local integrability of subharmonic functions).

The second chapter introduces the basic objects: Riemann surfaces as one-dimensional complex analytic manifolds, proving, to begin with, that every Riemann surface is an oriented (real) two dimensional manifold and, conversely, every smooth oriented (real) two-manifold has a complex structure. Actually, a key ingredient in the proof of the last claim is a result on partial differential equations, the Korn-Lichtenstein theorem, which is not proved until Chapter 11, using Hilbert space methods to obtain estimates for the ∂-bar operator. That the corresponding solution is smooth is also needed, a result whose proof uses the Fourier transform, Plancherel’s theorem and the Sobolev embedding theorem: a wonderful excursion into complex analysis!

Functions on Riemann surfaces are introduced in chapter three, extending the corresponding local notions (holomorphic, meromorphic, harmonic and subharmonic functions). At the end of this chapter, enough machinery has been developed to prove, for example, the Riemann-Hurwitz theorem and the Harnack Principle. Chapter four introduces the main characters for the novel approach taken in this book: complex and holomorphic line bundles. The main examples are the canonical line bundle, the tangent bundle and the line bundle associated to a divisor on a Riemann surface. The text establishes the bijection between the set of divisors modulo linear equivalence and the set of holomorphic line bundles, again assuming some hard facts proven until chapter eleven. The chapter ends with a proof that on a compact Riemann surface the vector space of holomorphic sections of a given holomorphic line bundle is a finite dimensional complex vector space, its dimension being the *arithmetic genus* of the Riemann surface.

Chapter six introduces the differential forms that will play an important part in the development, ending with proofs of the Poincaré lemma and the definition of Dolbeaut cohomology. Chapter six introduces the complex differential methods to be used: connections on line bundles, Hermitian metrics and the curvature of a connection, proving among other things, a theorem of Chern that for each holomorphic line bundle with an Hermitian metric there is a unique connection that is compatible with these structures.

Chapter seven includes a discussion of potential theory, Dirichlet’s problem and Green’s functions on a Riemann surface. This is put to use to prove the existence of solutions for ∂-bar on topologically non-trivial relatively compact domains with smooth boundary in a Riemann surface. This is used in chapter eight to obtain global solutions of the d-bar equation on an open Riemann surface, and is then put to work to prove some classical results such as a solution to the Mittag-Leffler problem on an open Riemann surface and a solution to the Poisson equation.

Chapter nine is central: after introducing the Hodge star-operator and the Laplace-Beltrami associated to a Riemann surface and proving some of their main properties, the author uses de Rham cohomology groups to relate the harmonic forms on a compact Riemann surface to the topology of that surface, proving, for example, that the geometric genus of a compact Riemann surface agrees with its arithmetic genus. The main result of this chapter is the Hodge theorem and, as a consequence, the important fact that every Riemann surface admits a line bundle with a metric of positive curvature, a result that is used in chapter twelve to prove that every compact Riemann surface can be embedded in a projective three dimensional space and every open Riemann surface can be embedded in a three dimensional affine space.

Chapter ten is devoted to the Riemann-Koebe uniformization theorem, obtaining as a by-product another proof of the existence of a line bundle with a metric of positive curvature on any Riemann surface. As has already been previously mentioned, chapter eleven gives the Hilbert space results needed in various previous situations, the main result being a theorem of Hörmander. Chapter twelve addresses the embedding theorems. The last two chapters give a classical approach to Riemann-Roch and Abel’s theorems ending with a discussion of the inversion problem.

Throughout the book, examples are given to illustrate the theory being developed, and at the end of every chapter, with the exception of the last two, there are exercises, few in number but enough to complement and test the reader’s grasp on the subject. The book is a welcome addition to the vast literature on Riemann surfaces, the chosen point of view shows the tight connection with complex analysis, it is mostly self-contained, and provides a down-to-earth introduction to the theory of complex manifolds.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is [email protected].