Li Yi

Li Yi

I am a 3rd year PhD student majoring in birational geometry

Birational Geometry Reading Notes: BCHM

2 minute read

Published:

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. 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 II. Base Point Freeness and Variants

Note-II.1 Classical Base Point Free Theorem for klt and dlt Pairs

Note-II.2 Fujino’s Generalizations of Base Point Free Theorems

Note-II.3 Positivity in Families and Base Point Freeness

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

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

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

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

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

Note-III.5: Basic Properties of Minimal Models, Good Minimal Models, and Canonical Models

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

Part IV. Minimal Models, Good Minimal Models, Canonical Models, and Abundance in Families

Note-IV.1: A Brief Survey on Minimal Models, Good Minimal Models, Canonical Models, and Abundance in Families [update 10.13]

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

Note-IV.3: Existence of Good Minimal Models in Families

Note-IV.4: Relative MMP, Fiberwise MMP, and Absolute MMP

Note-IV.5: Restriction of the MMP to the Central Fiber

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

Part V. Finiteness of Minimal Models and Termination Problems

Note-V.1: Polyhedral Decomposition Results

Note-V.2: MMP with Scaling

Note-V.3: Special Termination

Note-V.4: Global Termination Problem

Part VI. Finite Generation Problems

Finite Generation Note-VI.1: Finite Generation of the Canonical Ring and Cox Ring

Finite Generation Note-VI.2: Demailly–Hacon–Păun’s Analytic Proof of Finite Generation

Finite Generation Note-VI.3: Finite Generation and Abundance

Part VII. Partial Modifications

Note-1: Log Resolution and Discrepancy

Note-2: Crepant Extraction with Applications

Note-3: dlt Modification with Applications

Note-4: Canonical and Terminal Modifications, Q-factorialization

Note-5: Semi-log Modification

Part VIII. Loci in Birational Geometry

The aim of this series of notes is to summarize the basic properties of the stable base locus, augmented base locus, and restricted base locus, which are used repeatedly in birational geometry.

Note-1: Exceptional Locus

Note-2: Zariski Decomposition

Note-3: Base Locus, Stable Base Locus, Restricted Base Locus, and Augmented Base Locus

Note-4: Geometry of the Non-klt Locus

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

Note-6: Recent Developments on the Distribution of the Hodge Locus

Part X. Non-vanishing and Abundance

Note-X.1: Miyaoka’s Proof of Abundance for Threefolds

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