Degenerating Kahler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric

• 成果类型: Article
• 署名作者: Biquard, Olivier; Guenancia, Henri
• 署名单位: Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite; Universite Paris Cite; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Institut National des Sciences Appliquees de Toulouse; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01138-5
• 发表日期: 2022
• 页码: 1101-1163
• 关键词: equations; SINGULARITIES

摘要

Let X be a complex projectivemanifold and let D subset of X be a smooth divisor. In this article, we are interested in studying limits when ss -> 0 of Kahler-Einstein metrics omega(ss) with a cone singularity of angle 2 pi ss along D. In our first result, we assume that X \ D is a locally symmetric space and we show that omega(ss) converges to the locally symmetric metric and further give asymptotics of omega(ss) when X \ D is a ball quotient. Our second result deals with the case when X is Fano and D is anticanonical. We prove a folklore conjecture asserting that a rescaled limit of omega(ss) is the complete, Ricci flat Tian-Yau metric on X \ D. Furthermore, we prove that (X, omega(ss)) converges to an interval in the Gromov-Hausdorff sense.