The first book on the explicit birational geometry of complex algebraic threefolds arising from the minimal model program, this text is sure to become an essential reference in the field of birational geometry. Threefolds remain the interface between low and high-dimensional settings and a good understanding of them is necessary in this actively evolving area. Intended for advanced graduate students as well as researchers working in birational geometry, the book is as self-contained as possible. Detailed proofs are given throughout and more than 100 examples help to deepen understanding of birational geometry. The first part of the book deals with threefold singularities, divisorial contractions and flips. After a thorough explanation of the Sarkisov program, the second part is devoted to the analysis of outputs, specifically minimal models and Mori fibre spaces. The latter are divided into conical fibrations, del Pezzo fibrations and Fano threefolds according to the relative dimension.

‘This book is an excellent introduction to the classification of complex algebraic threefolds. It includes a thorough modern treatment and a glimpse into many of the recent higher dimensional breakthroughs.’

Christopher Hacon - University of Utah

‘A distinctive feature and a great strength of the book is the wealth of simple yet enlightening examples that illustrate even the most exotic aspects of the theory. They are a most valuable resource for testing questions and conjectures. I strongly recommend the book to anyone who wants to delve deeper into the study of 3-folds. The papers describing the steps of Mori’s program are long and difficult… the author has chosen basic results and special cases that can be explained in a chapter, yet give a true introduction to the main difficulties of the general theory. For the steps of Mori’s program and the plurigenera, Kawakita gives the first textbook treatments that go beyond the elementary results. Anyone wanting to read the full proofs should start with this book and ponder the many examples presented here.’

Janos Kollar Source: Bulletin of the American Mathematical Society