Degenerating Kähler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric - Sorbonne Université
Article Dans Une Revue Inventiones Mathematicae Année : 2022

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

Résumé

Let $X$ be a complex projective manifold and let $D\subset X$ be a smooth divisor. In this article, we are interested in studying limits when $\beta\to 0$ of Kähler-Einstein metrics $\omega_\beta$ with a cone singularity of angle $2\pi \beta$ along $D$. In our first result, we assume that $X\setminus D$ is a locally symmetric space and we show that $\omega_\beta$ converges to the locally symmetric metric and further give asymptotics of $\omega_\beta$ when $X\setminus D$ 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 $\omega_\beta$ is the complete, Ricci flat Tian-Yau metric on $X\setminus D$. Furthermore, we prove that $(X,\omega_\beta)$ converges to an interval in the Gromov-Hausdorff sense.
Fichier principal
Vignette du fichier
cones2zero_final_version.pdf (421.73 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03333520 , version 1 (06-10-2021)
hal-03333520 , version 2 (20-07-2022)

Identifiants

Citer

Olivier Biquard, Henri Guenancia. Degenerating Kähler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric. Inventiones Mathematicae, 2022, 230 (3), pp.1101-1163. ⟨10.1007/s00222-022-01138-5⟩. ⟨hal-03333520v2⟩
93 Consultations
47 Téléchargements

Altmetric

Partager

More