Algebraic Varieties: Minimal Models and Finite Generation

A self-contained introduction to minimal model theory and the finite generation theorem of the canonical ring of algebraic varieties.

Yujiro Kawamata (Author), Chen Jiang (Translated by)

9781009344678, Cambridge University Press

Hardback, published 27 June 2024

262 pages
22.9 x 15.2 x 1.6 cm, 0.539 kg

The finite generation theorem is a major achievement in modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic 0 is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar‒Cascini‒Hacon‒McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori's bend-and-break method, vanishing theorems, positivity theorems, and Siu's analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum with self-contained explanations of terminology and theorems.

Preface
1. Introduction
2. Algebraic varieties with boundaries
3. The minimal model program
4. The finite generation theorem
Bibliography
Index.

Subject Areas: Geometry [PBM]