Li Yi

Li Yi

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

Birational Geometry Note: Adjunction Theory

1 minute read

Published:

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.

A useful summary can be found in Anti-pluricanonical Systems on Fano Varieties, Chapter 3. This note is based on that paper, and our goal is to provide a more detailed introduction to adjunction theory.

Part I. Canonical Bundle Formula

For detailed information, see my reading notes:

3. Kawamata’s Canonical Bundle Formula [6.4]

4. Ambro’s Canonical Bundle Formula

5. Fujino–Mori Canonical Bundle Formula

6. Generalized Canonical Bundle Formula of Birkar–Zhang [6.7]

7. Canonical Bundle Formula for Generalized Kähler Pairs [6.4]

8. o-minimality Approach to the b-semiampleness Conjecture

Part II. Subadjunction Theorems

1. Adjunction Formulas

2. The Restriction of Divisors and the Shokurov Difference Divisor

3. Kawamata’s Subadjunction Formula

4. Hacon–McKernan–Xu’s Subadjunction-Type Theorem [6.6]

Part III. Inversion of Adjunction

1. Inversion of Adjunction

Part IV. Applications of Canonical Bundle Formulas and Subadjunction

1. Applications of the Canonical Bundle Formula to Finite Generation Problems

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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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