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Test configurations, stabilities and...

Mabuchi, Toshiki.

Test configurations, stabilities and canonical Kahler metricscomplex geometry by the energy method /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Test configurations, stabilities and canonical Kahler metricsby Toshiki Mabuchi.
Reminder of title:	complex geometry by the energy method /
Author:	Mabuchi, Toshiki.
Published:	Singapore :Springer Singapore :2021.
Description:	x, 128 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Geometry, Differential.
Online resource:	https://doi.org/10.1007/978-981-16-0500-0
ISBN:	9789811605000$q(electronic bk.)

Test configurations, stabilities and canonical Kahler metricscomplex geometry by the energy method /
Mabuchi, Toshiki.

Test configurations, stabilities and canonical Kahler metricscomplex geometry by the energy method /[electronic resource] :by Toshiki Mabuchi. - Singapore :Springer Singapore :2021. - x, 128 p. :ill., digital ;24 cm. - SpringerBriefs in mathematics,2191-8198. - SpringerBriefs in mathematics..

Introduction -- The Donaldson-Futaki invariant -- Canonical Kahler metrics -- Norms for test configurations -- Stabilities for polarized algebraic manifolds -- The Yau-Tian-Donaldson conjecture -- Stability theorem -- Existence problem -- Canonical Kahler metrics on Fano manifolds -- Geometry of pseudo-normed graded algebras -- Solutions.

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 Kahler 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 Kahler metrics.

ISBN: 9789811605000$q(electronic bk.)

Standard No.: 10.1007/978-981-16-0500-0doiSubjects--Topical Terms:

182610
Geometry, Differential.

LC Class. No.: QA641

Dewey Class. No.: 516.373

Test configurations, stabilities and canonical Kahler metricscomplex geometry by the energy method /
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505 0 $a Introduction -- The Donaldson-Futaki invariant -- Canonical Kahler metrics -- Norms for test configurations -- Stabilities for polarized algebraic manifolds -- The Yau-Tian-Donaldson conjecture -- Stability theorem -- Existence problem -- Canonical Kahler metrics on Fano manifolds -- Geometry of pseudo-normed graded algebras -- Solutions.
520 $a 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 Kahler 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 Kahler metrics.
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