Birational Geometry Reading Notes: BCHM

The aim of this series of notes is to study the classical paper BCHM in detail. We aim to complete the proof of the main theorem and summarize some important applications of BCHM. We do not strictly follow the structure of the original paper; instead, we divide the material into several thematic topics, including the existence of flips, existence of minimal models, termination problems, the non-vanishing conjecture, and the finite generation problem.

Part I. Loci in Birational Geometry

Note-1: Exceptional Locus and Indeterminacy Locus (in preparation)

Note-2: A Brief Introduction to Zariski Decomposition (in preparation)

Note-3: Base Locus, Stable Base Locus, Diminished Base Locus, and Augmented Base Locus (in preparation)

Note-4: KLT, LC singularities and Geometry of the Non-klt Locus (in preparation)

Note-5: Non-nef Locus (Numerical Base Locus), Non-Kähler Locus (in preparation)

Part II. Classical MMP Theory

Note-II.1 Classical Base Point Free Theorem for klt and dlt Pairs (in preparation)

Note-II.2 Positivity in Families and Base Point Freeness (in preparation)

Note-II.3 Cone and contraction theorems (in preparation)

Note-II.4 Mori’s bend and break (in preparation)

Part III. Extension Theorems

Note-I.1: Nakayama’s Extension Theorems [update 8.17]

Note-I.2: Hacon–McKernan Extension Theorem [update 2.21]

Note-I.3: de Fernex–Hacon Extension Theorem [update 11.12]

Note-I.4: Demailly–Hacon–Păun dlt Extension

Part IV. Existence of Flips, Minimal Models, Good Minimal Models, and Canonical Models

Note-III.1: Hacon–McKernan’s Proof of Existence of klt Flips (in preparation)

Note-III.2: Hacon–Xu and Birkar’s Proof of the Existence of lc Flips (with Generalizations) (in preparation)

Note-III.3: Existence of Minimal Models (BCHM C and Related Results) (in preparation)

Note-III.4: Existence of Good Minimal Models (DHP and Related Results) (in preparation)

Note-III.5: Basic Properties of Minimal Models, Good Minimal Models, and Canonical Models (in preparation)

Note-III.6: Behavior of Minimal Models, Good Minimal Models, and Canonical Models under Birational Modifications (in preparation)

Note-III.7: Behavior of Minimal Models, Good Minimal Models, and Canonical Models under Perturbation (in preparation)

Part V. Minimal Models, Good Minimal Models in Families

Note-IV.1: Good Minimal Models in Families [update 10.13]

Note-IV.2: Existence of Good Minimal Models on the Closure

Note-IV.4: Relative MMP, Fiberwise MMP, and Absolute MMP (in preparation)

Note-IV.5: Restriction of the MMP to the Central Fiber (in preparation)

Note-IV.6: Extension of the MMP from the Central Fiber [update 12.3]

Part VI. Finiteness of Minimal Models and Termination Problems

Note-V.1: Polyhedral Decomposition Results (in preparation)

Note-V.2: MMP with Scaling (in preparation)

Note-V.3: Special Termination

Note-V.4: Global Termination Problem (in preparation)

Part VII. Finite Generation Problems

Note-VII.1: Finite Generation of the Canonical Ring and Cox Ring (in preparation)

Note-VII.2: Demailly–Hacon–Păun’s Analytic Proof of Finite Generation (in preparation)

Note-VII.3: Finite Generation and Abundance (in preparation)

Part VIII. Partial Modifications

Note-1: Log Resolution and Discrepancy (in preparation)

Note-2: Crepant Extraction with Applications (in preparation)

Note-3: dlt Modification with Applications (in preparation)

Note-4: Canonical and Terminal Modifications, Q-factorialization (in preparation)

Note-5: Semi-log Modification (in preparation)

Part IX. Analytic BCHM

Note-1: Cone theorem for analytic varieties (in preparation)

Note-2: From algebraic bchm to analytic bchm (in preparation)

Part X. Non-vanishing and Abundance

Note-X.1: Miyaoka’s Proof of Abundance for Threefolds (in preparation)