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Test Configurations, Stabilities and Canonical Kähler Metrics

Complex Geometry by the Energy Method

  • Book
  • © 2021

Overview

Authors:
  1. Toshiki Mabuchi

    1. Department of Mathematics, Osaka University, Graduate School of Science, Toyonaka, Japan

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  • The main theme of our book is the very hot issue known as the Yau-Tian-Donaldson conjecture
  • As to the Yau-Tian-Donaldson conjecture, we focus on the open case on the existence of extremal Kähler metrics
  • In our approach to the existence problem of extremal metrics, we use the Chow norm very effectively

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-x
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  2. Introduction

    • Toshiki Mabuchi
    Pages 1-8

  3. The Donaldson–Futaki Invariant

    • Toshiki Mabuchi
    Pages 9-20

  4. Canonical Kähler Metrics

    • Toshiki Mabuchi
    Pages 21-24

  5. Norms for Test Configurations

    • Toshiki Mabuchi
    Pages 25-33

  6. Stabilities for Polarized Algebraic Manifolds

    • Toshiki Mabuchi
    Pages 35-52

  7. The Yau–Tian–Donaldson Conjecture

    • Toshiki Mabuchi
    Pages 53-57

  8. Stability Theorem

    • Toshiki Mabuchi
    Pages 59-68

  9. Existence Problem

    • Toshiki Mabuchi
    Pages 69-73

  10. Canonical Kähler Metrics on Fano Manifolds

    • Toshiki Mabuchi
    Pages 75-98

  11. Back Matter

    Pages 99-128
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Keywords

About this book

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.

Reviews

“The concise style of exposition likely means that this monograph is best suited for experts with background knowledge in canonical Kähler metrics. … It can be recommended also to those who would like a review of important results concerning the generalised Kähler-Einstein metrics, with various examples, and the moduli space of Lp-spaces.” (Yoshinori Hashimoto, Mathematical Reviews, May, 2023)

Authors and Affiliations

  • Department of Mathematics, Osaka University, Graduate School of Science, Toyonaka, Japan

    Toshiki Mabuchi

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