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

1st ed. 2021

by Toshiki Mabuchi

Book Details

Format Paperback / Softback

Book Series SpringerBriefs in Mathematics

ISBN-10 9811604991

ISBN-13 9789811604997

Edition 1st ed. 2021

Publisher Springer Verlag, Singapore

Imprint Springer Verlag, Singapore

Country of Manufacture GB

Country of Publication GB

Publication Date Mar 26th, 2021

Print length 128 Pages

Weight 238 grams

Dimensions 23.30 x 15.50 x 1.20 cms

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The Yau-Tian-Donaldson conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. 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.

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.

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