Test Configurations, Stabilities and Canonical Kähler Metrics

Complex Geometry by the Energy Method
1. Auflage 2021
ISBN: 978-981-16-0499-7
Verlag: Springer Nature Singapore

Buch, Englisch, 128 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 224 g

Reihe: 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.

Zielgruppe

Graduate

Autoren/Hrsg.

Mabuchi, Toshiki

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

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