Li Yi

Li Yi

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

Birational Geometry Reading Notes: BCHM

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This aim of this series of notes is to study the classical paper BCHM in details. We will try to finish the proof of the main theorem and summarize some interesting applications of BCHM. We will not strictly follow the structure of the original paper, instead we will divide the paper into several topics discussion, including: Existence of flip, existence of minimal model, termination problem, non-vanishing conjecture and finite generation problem etc.

Part I. Extension theorems and existence of flips

Note-I.1: Nakayama’s extension theorems [update 8.17]

Note-I.2: Hacon Mckernan extension theorem [update 2.21]

Note-I.3: deFernex-Hacon extension theorem [updage 9.12]

Note-I.4: Demailly-Hacon-Paun dlt extension

Part II. Base Point Freeeness and Variants

Note-II.1 Classical Base point free theorem for klt and dlt pairs

Note-II.2 Fujino’s some generalizations of base point free theorems

Part III. Existence of flip, minimal model, good minimal model, canonical model

Note-III.1: Hacon Mckernan’s proof of existence of klt flip

Note-III.2: Hacon-Xu’s and Birkar’s proof on existence of LC flips (with some generalizations)

Note-III.3: Existence of minimal models (BCHM C and beyond)

Note-III.4: Dxistence of good minimal model theorem (DHP and beyond)

Note-III.5: Basic properties of minimal model, good minimal models, canonical model

Note-III.6: Behavior of minimal model, good minimal model, canonical model under birational modification

Note-III.7: Minimal model, good minimal model, canonical model in family

Part IV. Finiteness of minimal model and Termination problems

Note-1: Polyhedron decomposition results

Note-2: Special finiteness

Note-3: Global Termination Problem

Part V. Finite Generation Problems

Finite Generation Note-1: Finite generation of canonical ring, Cox ring

Finite Generation Note-2: Demailly-Hacon-Paun’s analytic proof of finite generation

Finite Generation Note-3: Finite generation and Abundance

Part VI. Partial Modification

Note-1: Log resolution, discrepancy

Note-2: dlt modification, crepant extraction theorem

Note-3: Canonical modification and terminal modification, Q-factorialization

Note-4: Semi log modificaiton

Part VII. Locus in Birational Geometry

The aim of this series of notes is to summarize the base properties about stable base locus, augmented base locus, restricted base locus which will be used over and over again 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 non-klt locus

Note-5: non-nef locus(numerical base locus), non-Kahler Locus

Note-6: Recent developments on the distribution of Hodge Locus

Part VIII. Application of BCHM

Note-1: Deformation of Singularities

Note-2: Deformation of positivities

Part X. Non-vanishing and abundance

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