Birational Geometry Notes: Rationality Questions
Published:
The aim of this series notes is to give a brief introduction to rationality questions in birational geometry.
You May Also Enjoy
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.
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.
My PhD Thesis Project: Kähler minimal model program
2 minute read
Published:
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
2 minute read
Published:
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.