The book under review gives a panoramic survey of Nevanlinna’s theory of valued distribution and Diophantine approximation in several complex variables. The exposition is systematic and self-contained, assuming from the reader some previous background on analytic geometry.
The book starts with the one-dimensional classical case in chapter one and the first subsection of chapter two. Next, the higher-dimensional case is developed from the very beginning, starting with the notion of plurisubharmonic maps of several variables in the second subsection of chapter two and, picking up the pace, to divisors and line bundles, metrics and curvature of line bundles and the first main theorem for coherent ideal sheaves. As we all know, complex variables are better dealt with using the modern tools of analytic geometry.
The book includes complete proofs of some of the fundamental results in this area: Nevanlinna’s main theorems and Cartan’s theorem. All the requisites are developed in the intervening chapters, including semi-abelian varieties and Kobayashi hyperbolicity.
Diophantine approximation is treated in the last two chapters, including Roth’s theorem, Schmidt’s subspace theorem and a discussion of the abc conjecture. The last chapter includes formulations of Faltings’ theorem on the finiteness of the set of rational points on smooth algebraic curve of genus at least two over a number field, and its generalized conjectural higher dimensional version by Serge Lang.
The book is addressed to researchers and graduate students, and an interested reader will be rewarded with a clear exposition of classical and recent progress in this active area of mathematics.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.