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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.
Introduction.
Algebraic Varieties with Boundaries.
Q-divisors and R-divisors.
Rational Maps and Birational Maps.
Canonical Divisors.
Intersection Numbers and Numerical Geometry.
Cones of Curves and Cones of Divisors.
Pseudo-Effective Cones and Nef Cones.
Kleiman’s Criterion and Kodaira’s Lemma.
The Hironaka Desingularization Theorem.
The Kodaira Vanishing Theorem.
The Covering Trick.
Generalizations of the Kodaira Vanishing Theorem.
KLT Singularities for Pairs.
LC, DLT, and PLT Singularities for Pairs.
Various Singularities.
The Subadjunction Formula.
Terminal and Canonical Singularities.
Minimality and Log Minimality.
The 1-Dimensional and 2-Dimensional Cases.
The 1-Dimensional Case.
Minimal Models in Dimension 2.
The Classification of Algebraic Surfaces.
Rational Singularities.
The Classification of DLT Surface Singularities I.
The Classification of DLT Surface Singularities II.
The Zariski Decomposition.
The 3-Dimensional Case.
The Minimal Model Program.
The Basepoint-Free Theorem.
Proof of the Basepoint-Free Theorem.
Paraphrasings and Generalizations.
An Effective Version of the Basepoint-Free Theorem.
The Rationality Theorem.
The Cone Theorem.
The Contraction Theorem.
The Cone Theorem.
Contraction Morphisms in Dimensions 2 and 3.
The Cone Theorem for the Space of Divisors.
Types of Contraction Morphisms and the MinimalModel Program.
Classification of Contraction Morphisms.
Flips.
Decrease of Canonical Divisors.
The Existence and the Termination of Flips.
Minimal Models and Canonical Models.
The Minimal Model Program.
Minimal Model Program with Scaling.
The Existence of Rational Curves.
Deformation of Morphisms.
The Bend-and-Break Method.
The Lengths of Extremal Rays.
The Divisorial Zariski Decomposition.
Polyhedral Decompositions of a Cone of Divisors.
Rationality of Sections of Nef Cones.
Polyhedral Decomposition according to Canonical Models.
Polyhedral Decomposition according to Minimal Models.
Applications of Polyhedral Decompositions.
Multiplier Ideal Sheaves.
Multiplier Ideal Sheaves.
Adjoint Ideal Sheaves.
Extension Theorems.
Extension Theorems I.
Extension Theorems II.
The Finite Generation Theorem.
Setting of the Inductive Proof.
PL Flips.
Restriction of Canonical Rings to Divisors.
The Existence of PL Flips.
The Special Termination.
The Existence and Finiteness of Minimal Models.
The Nonvanishing Theorem.
Summary.
Algebraic Fiber Spaces.
Algebraic Fiber Spaces and Toroidal Geometry.
TheWeak Semistable Reduction Theorem and theSemipositivity Theorem.
The Finite Generation Theorem.
Generalizations of the Minimal Model Theory.
The Case with a Group Action.
The Case when the Base Field Is Not Algebraically Closed.
Remaining Problems.
The Abundance Conjecture.
The Case of Numerical Kodaira Dimension 0.
Generalization to Positive Characteristics.
Related Topics.
Boundedness Results.
Minimal Log Discrepancies.
The Sarkisov Program.
Rationally Connected Varieties.
The Category of Smooth Algebraic Varieties.
References.
Index