One of the fundamental problems in algebraic geometry is the classification of algebraic varieties, say up to birational equivalence. Since every irreducible complex algebraic curve is birational to a unique smooth projective curve, the classification problem for one-dimensional varieties is trivial. For the case of two-dimensional varieties or algebraic surfaces, this classification was the main achievement of the Italian school of algebraic geometry of Castelnuovo, Enriques, Severi and many others, in the early twentieth century. To clarify and make this classification more precise was some of the motivation behind the foundational work of O. Zariski and A. Weil.

For higher dimensional varieties, however, the classification problem remained elusive and difficult even to formulate in a precise way. This required new ideas and techniques that were developed in the 1970s: how to factorize birational transformations and to define what would be called a *minimal model* in each birational equivalence class of algebraic varieties.

The fruits of these intense developments by many algebraic geometers would culminate in the 1980s with S. Mori’s proof of the existence of minimal models for complex threefolds. Now it was possible to formulate a program with the goal of classifying all higher dimensional complex projective varieties up to birational equivalence. This program seeks to construct a birational model of each complex projective variety which is as simple as possible, a so-called minimal model, whose precise definition has evolved as the program developed, for example to take into account the types of singularities that would be allowed.

Complex algebraic varieties, considered as complex manifolds, are also part of the playground of analytic geometers, who study them using analytic or transcendental tools. This two-sided approach has been there from the very beginning in the late 19^{th} century. In the last two decades, the interaction between these two approaches has deepened and at the same time the gulf between the languages of these two approaches, algebra and analysis, has widened.

This brings us to some of the main objectives of the book under review, the proceedings of the Park City Mathematics Institute, whose overall goal is to bring researchers, postdoctoral fellows, graduate and undergraduate students and math educators around some highly focused topic of current and lasting interest in the mathematical sciences. The topics chosen for the 2008 Summer School centered on the new developments in the last few years towards the minimal model program in higher dimension. More specifically, the lectures focused on a fundamental result in this program, the finite generation of the canonical ring of a complex algebraic variety. This theorem has now two proofs. The first one, by Y.-T. Siu, “A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring” (2006), comes from the analytic side. A second proof, by C. Birker, P. Cascini, C. Hacon and J. McKernan, “Existence of Minimal Models for Varieties of log General Type”, *J. Am. Math. Soc*. **23** (2010), 405–468, uses the algebraic approach.

Siu’s proof uses delicate L^{2}-estimates for the ̅∂-equation that allow him to apply a multiplier ideal version of Skoda’s ideal generation theorem. Background and details for this approach are given in the first four set of lectures that make about one half of the book, culminating with a chapter summarizing the proof of the finite generation theorem using the analytic approach. The next four sets of lectures are on the algebraic approach. These lectures range from an introduction to resolution of singularities and a short course on multiplier ideals to the higher dimensional minimal program for varieties of log general type showing that flips exist in all dimensions and that the canonical ring of a smooth complex projective variety is finitely generated.

The book under review succeeds making explicit the bridges between the algebraic and analytic approaches, closing the language differences and at the same time introducing graduate students and researchers to a major development in complex algebraic geometry.

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