Li Yi

Li Yi

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

Blog posts

2025

Metric method in Birational Geometry

less than 1 minute read

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In this series of notes we will summarize some standard metric methods which will be useful in birational geometry.

Fibration and Foliation in Algebraic Geometry

1 minute read

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The aim of this series of note is to introduce the fibration in Algebraic geometry (classification theory of complex algebraic/analytic varieties). Fibration is the most powerful tool for the classification of varieties and fibration structure is very suitable for the induction process. In this notes we will major focus on the possible applications of the fibrations in the classification results. Fibration is central in Algebraic Geometry.

In algebraic geometry (birational geometry), we understand variety using fibration.

Birational Geometry Note: Kahler minimal model program

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The aim of this note is to introduce the minimal model program for Kahler varieties. Compared with the well-known minimal model program for projective varieties, the theory of Kahler minimal model program will have the following three major difficulties: (1) Mori bend-and-break technique: Mori bend and break is not known for Kahler varieties, (2) The base point free theorem and cone theorem: Since Kahler manifold with a big line bundle is automatic projective, thus a big line bundle does not make sense on (non-projective) Kahler 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.

Birational Geometry Reading Notes: BCHM

2 minute read

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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.

Birational Geometry Notes: Around Moduli theory

2 minute read

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In this part of the notes, we will focus on the properties of moduli spaces. The guiding philosophy behind moduli theory is that

The moduli of a geometric object completely determine the object itself. In birational geometry, one of the central theme is that the geometry of a variety is largely governed by the curves and divisors it contains.

Birational Geometry and Deformation Theory

1 minute read

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The aim of this series of notes is to give a brief introduction to the basics of deformation theory, and then to focus on its applications to certain problems in birational geometry.

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 Note: Adjunction Theory

less than 1 minute read

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The aim of this note is to introduce adjunction theory in birational geometry. Among the various techniques in birational geometry, adjunction theory is one of the most powerful tools. It allows us to relate the geometry — particularly the singularities — of an ambient variety to those of suitable subvarieties.