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

• Record Nr.: UNISA996466550003316
• Autore: Mabuchi Toshiki <1950->
• Pubbl/distr/stampa: Singapore : , : Springer, , [2021]; ©2021
• ISBN: 981-16-0500-9
• Descrizione fisica: 1 online resource (x, 128 pages) : illustrations
• Collana: SpringerBriefs in Mathematics
• Disciplina: 516.36
• Soggetti: Geometry, Differential; Manifolds (Mathematics); Moduli theory; Geometria diferencial; Varietats (Matemàtica); Teoria de mòduls; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: 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 -- Problems -- 3 Canonical Kähler Metrics -- 3.1 Canonical Kähler Metrics on Compact Complex Manifolds -- 3.2 Conformal Changes of Metrics by Hamiltonian Functions -- Problems -- 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 -- Problems -- 5 Stabilities for Polarized Algebraic Manifolds -- 5.1 The Chow Stability -- 5.2 The ; Hilbert Stability -- 5.3 K-stability -- 5.4 Relative Stability -- Problems -- 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 -- Problems -- 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}) -- Problems -- 8 Existence Problem -- 8.1 A Result of He on the Existence of Extremal Kähler Metrics -- 8.2 Some Observations on the Existence Problem -- Problems -- 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 -- Problems -- 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.