Test configurations, stabilities and canonical Kähler metrics : complex geometry by the energy method / Toshiki Mabuchi.

Format:

Book

Author/Creator:

Mabuchi, Toshiki, 1950- author.

Series:

SpringerBriefs in mathematics.

SpringerBriefs in Mathematics

Language:

English

Subjects (All):

Geometry, Differential.

Manifolds (Mathematics).

Moduli theory.

Physical Description:

1 online resource (x, 128 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Singapore : Springer, [2021]

System Details:

Mode of access: World Wide Web.

Summary:

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.

Contents:

Intro

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

3 Canonical Kähler Metrics

3.1 Canonical Kähler Metrics on Compact Complex Manifolds

3.2 Conformal Changes of Metrics by Hamiltonian Functions

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

5 Stabilities for Polarized Algebraic Manifolds

5.1 The Chow Stability

5.2 The Hilbert Stability

5.3 K-stability

5.4 Relative Stability

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

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})

8 Existence Problem

8.1 A Result of He on the Existence of Extremal Kähler Metrics

8.2 Some Observations on the Existence Problem

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

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.

Notes:

Includes bibliographical references.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

981-16-0500-9

OCLC:

1243514520