Li Yi

Li Yi

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

Birational Geometry Note: Birational Geometry of surfaces and threefolds

less than 1 minute read

Published:

The aim of this series of note is to introduce the birational geometry for surfaces and threefolds, which will be helpful for deepening the understanding of birational geometry.

Part I. Birational Geometry of Surfaces

I.1. Proof of classification theorem for surfaces

Part II. Family of Surfaces, and Moduli of Surfaces

Part III. Log Calabi-Yau Surfaces

Part IV. Fano threefolds

Part V. Geometry of Calabi-Yau threefolds

You May Also Enjoy

Metric method in Birational Geometry

1 minute read

Published:

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

Published:

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

1 minute read

Published:

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

Published:

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.