Test Configurations, Stabilities and Canonical Kähler Metrics: Complex Geometry by the Energy Method (SpringerBriefs in Mathematics)

✍ Scribed by Toshiki Mabuchi

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 134
• Category: Library

✦ Synopsis

The Yau-Tian-Donaldson conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, this conjecture is still open for general polarizations or more generally in extremal Kähler cases. In this book, the unsolved cases of the conjecture will be discussed.

It will be shown that the problem is closely related to the geometry of moduli spaces of test configurations for polarized algebraic manifolds.

Another important tool in our approach is the Chow norm introduced by Zhang. This is closely related to Ding’s functional, and plays a crucial role in our differential geometric study of stability. By discussing the Chow norm from various points of view, we shall make a systematic study of the existence problem of extremal Kähler metrics.

✦ Table of Contents

Preface
Contents
1 Introduction
1.1 Preliminaries
1.2 The Deligne Pairings with Metrics
1.3 Definition of the Chow Norm
1.4 The First and Second Variation Formulas for the Chow Norm
Problems
2 The Donaldson–Futaki Invariant
2.1 Test Configurations
2.2 Test Configurations Associated to One-Parameter Groups
2.3 Definition of the Donaldson–Futaki Invariant
2.4 Expression of DF1 as an Intersection Number
2.5 The Relationship Between the Chow Norm and DFi
Problems
3 Canonical Kähler Metrics
3.1 Canonical Kähler Metrics on Compact Complex Manifolds
3.2 Conformal Changes of Metrics by Hamiltonian Functions
Problems
4 Norms for Test Configurations
4.1 Norms for Test Configurations of a Fixed Exponent
4.2 The Asymptotic 1-norm of a Test Configuration
4.3 Relative Norms for Test Configurations
4.4 The Twisted Kodaira Embedding
4.5 The Donaldson–Futaki Invariant for Sequences
Problems
5 Stabilities for Polarized Algebraic Manifolds
5.1 The Chow Stability
5.2 The Hilbert Stability
5.3 K-stability
5.4 Relative Stability
Problems
6 The Yau–Tian–Donaldson Conjecture
6.1 The Calabi Conjecture
6.2 The Yau–Tian–Donaldson Conjecture
6.3 The K-Energy
6.4 Extremal Kähler Versions of the Conjecture
Problems
7 Stability Theorem
7.1 Strong K-Semistability of CSC Kähler Manifolds
7.2 Relative Balanced Metrics
7.3 Strong Relative K-Semistability of Extremal Kähler Manifolds
7.4 K-Polystability of Extremal Kähler Manifolds
7.5 A Reformulation of the Definition of the Invariant F ({μj})
Problems
8 Existence Problem
8.1 A Result of He on the Existence of Extremal Kähler Metrics
8.2 Some Observations on the Existence Problem
Problems
9 Canonical Kähler Metrics on Fano Manifolds
9.1 Kähler Metrics in Anticanonical Class
9.2 Extremal Vector Fields
9.3 An Obstruction of Matsushima's Type
9.4 An Invariant as an Obstruction to the Existence
9.5 Examples of Generalized Kähler–Einstein Metrics
9.6 Extremal Metrics on Generalized Kähler–Einstein Manifolds
9.7 The Product Formula for the Invariant γX
9.8 Yao's Result for Toric Fano Manifolds
9.9 Hisamoto's Result on the Existence Problem
Problems
A Geometry of Pseudo-Normed Graded Algebras
A.1 Differential Geometric Viewpoints
A.2 Lp-Spaces
A.3 An Orthogonal Direct Sum of Lp-Spaces
A.4 A Boundedness Theorem for Lp-Spaces
A.5 The Moduli Space of Lp-Spaces
A.6 A Multiplicative System of Lp-Spaces
A.7 Degeneration Phenomena
Solutions
Problems of Chap.1
Problems of Chap.2
Problems of Chap.3
Problems of Chap.4
Problems of Chap.5
Problems of Chap.6
Problems of Chap.7
Problems of Chap.8
Problems of Chap.9
Bibliography