Test Configurations, Stabilities and Canonical Kähler Metrics: Complex Geometry by the Energy Method

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Toshiki Mabuchi

Test Configurations, Stabilities and Canonical Kähler Metrics

Complex Geometry by the Energy Method

Toshiki Mabuchi

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English

Springer Verlag, Singapore
26 March 2021

Calculus & mathematical analysis; Complex analysis, complex variables; Differential & Riemannian geometry; Algebraic geometry

Series: SpringerBriefs in Mathematics

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.

By: Toshiki Mabuchi
Imprint: Springer Verlag, Singapore
Country of Publication: Singapore
Edition: 1st ed. 2021
Dimensions: Height: 235mm, Width: 155mm,
Weight: 454g
ISBN: 9789811604997
ISBN 10: 9811604991
Series: SpringerBriefs in Mathematics
Pages: 128
Publication Date: 26 March 2021
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Reviews for Test Configurations, Stabilities and Canonical Kähler Metrics: Complex Geometry by the Energy Method

“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)