Kahler-Einstein metrics and birational geometry

2016 Fiscal Year Final Research Report

• Project/Area Number: 25800050
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Fukuoka University (2016); Kumamoto University (2013-2015)
• Principal Investigator: Sano Yuji 福岡大学, 理学部, 教授 (00399792)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: ケーラー・アインシュタイン計量 / K 安定性 / トーリックファノ多様体 / 端的ケーラー計量 / balanced 計量

Outline of Final Research Achievements

In this research, we achieved the following two results.
(1) For a given toric Fano manifold, we introduced an invariant of a Fano polytope which is likely to be a polar dual of the barycenter of the image under the moment map on a given toric Fano manifold. Under some assumption, it induces a simpler criterion for the existence of Kahler-Einstein metrics (or K stability) on a toric Fano manifold.
(2) With a differential geometrical approach, we extended the moment map interpretation for balanced metrics to relative balanced metrics. Then, it implies the quantization of the extremal metrics and the extremal vector fields on a polarized manifold.

Free Research Field

ケーラー幾何学