Li Yi

Li Yi

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

Metric method in Birational Geometry

1 minute read

Published:

In this series of notes, we summarize several standard metric methods that are useful in birational geometry.

I. Ohsawa–Takegoshi Extension Theorem

I.1. The Ohsawa–Takegoshi Extension Theorem and Why It Is Useful in Birational Geometry [upd 10.8]

II. Invariance of Plurigenera and extension problem

II.1. Păun’s Analytic Proof of Invariance of Plurigenera

II.2. Demailly-Hacon-Paun’s Proof of DLT Extension Theorem

III. Positivity of Direct Images with applications

III.1. Păun–Takayama’s Construction of Singular Hermitian Metrics on Direct Images of Relative (Pluri)canonical Sheaves

III.2. Hacon–Popa–Schnell’s Construction of Singular Hermitian Metrics on Direct Images of Relative (Pluri)canonical Sheaves [upd 10.24]

III.3. Kollár’s Approach to Projectivity of Moduli

III.4. Kawamata’s Proof of the Iitaka Conjecture When the Base Is of General Type

IV. Metric methods in the canonical bundle formulas

IV.1. Hacon–Păun’s Metric Method for the Canonical Bundle Formula

V. Metric Methods in Boundedness Problems

V.1. The Theorem of Angehrn and Siu

V.2. Birational Boundedness for Varieties of General Type

VI. Metric Methods in the Abundance Conjecture

VI.1. Supercanonical Metrics and the Abundance Conjecture

VI.2. Metrics with Minimal Singularities

VI.3. Siu and Păun’s Analytic Proof of Finite Generation of the Canonical Ring and Shokurov’s Non-vanishing

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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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The aim of this series of notes is to introduce fibrations in algebraic geometry (the classification theory of complex algebraic/analytic varieties). Fibrations are among the most powerful tools for the classification of varieties, and fibration structures are particularly well suited to inductive arguments. In these notes, we focus mainly on applications of fibrations to classification results. Fibrations are central in algebraic geometry.

In algebraic geometry (birational geometry), we understand varieties using fibrations.

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.

Birational Geometry Reading Notes: BCHM

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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.