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. Loci in Birational Geometry

Note-1: Exceptional Locus

Note-2: A brief introduction to Zariski Decomposition

Note-3: Base Locus, Stable Base Locus, Diminished 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

Part II. Classical MMP Theory

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

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

Note-II.3 Cone and contraction theorems

Note-II.4 Mori’s bend and break

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

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 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

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 VI. 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 VII. 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 VIII. 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 X. Non-vanishing and Abundance

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

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My PhD Thesis Project: Kähler minimal model program

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The aim of this note is to introduce the minimal model program for Kähler varieties. Compared with the well-known minimal model program for projective varieties, the theory of Kähler minimal model program will have the following three major difficulties: (1) Mori bend-and-break technique: Mori bend and break is not known for Kähler varieties, (2) The base point free theorem and cone theorem: Since Kähler manifold with a big line bundle is automatic projective, thus a big line bundle does not make sense on (non-projective) Kähler manifold, (3) Contraction theorem: In the classical MMP, we require the base point free theorem to produce some semi-ampleness divisor. This semi-ample divisor will induce the negative contraction morphism, and this approach is not very clear under Kaehler setting.

Fibration and Foliation in Algebraic Geometry

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Moishezon Morphism Seminar Note

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The purpose of this series of notes is to discuss the recent paper Moishezon morphism by Professor Kollár in detail. Let us first briefly answer the question WHY Moishezon?

First, Moishezon spaces have more functorial behavior (compared with projective varieties), as we will see in my notes. Secondly, from almost any projective variety we can construct a Moishezon space via bimeromorphic modification, making Moishezon spaces versatile in birational geometry. Thirdly, by Artin’s fundamental theorem, the category of Moishezon spaces appears naturally in moduli theory. Another compelling reason to consider the Moishezon category is that it allows cutand-paste operations similar to those we can perform in topology.