Singular Kähler-Einstein metrics on $ \mathbb Q $-Fano compactifications of Lie groups

Yan Li; Gang Tian; Xiaohua Zhu; Yan Li; Gang Tian; Xiaohua Zhu

doi:10.3934/mine.2023028

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-43. doi: 10.3934/mine.2023028

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Singular Kähler-Einstein metrics on $ \mathbb Q $-Fano compactifications of Lie groups

1.
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China
2.
Beijing International Centre of Mathematical Research and School of Mathematical Science, Peking University, Beijing 100871, China

Received: 04 November 2021 Revised: 14 March 2022 Accepted: 21 March 2022 Published: 26 April 2022

In this paper, we prove an existence result for Kähler-Einstein metrics on $ \mathbb Q $-Fano compactifications of Lie groups by the variational method, provided their moment polytopes satisfy a fine condition. As an application, we prove that there is no $ \mathbb Q $-Fano $ {\rm SO}_4(\mathbb C) $-compactification which admits a Kähler-Einstein metric with the same volume as that of a smooth K-unstable Fano $ {\rm SO}_4(\mathbb C) $-compactification.
- Kähler-Einstein metrics,
- $ \mathbb Q $-Fano compactifications of Lie groups,
- variation method,
- reduced Ding functional,
- K-stability
Citation: Yan Li, Gang Tian, Xiaohua Zhu. Singular Kähler-Einstein metrics on $ \mathbb Q $-Fano compactifications of Lie groups[J]. Mathematics in Engineering, 2023, 5(2): 1-43. doi: 10.3934/mine.2023028

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Abstract

In this paper, we prove an existence result for Kähler-Einstein metrics on $ \mathbb Q $-Fano compactifications of Lie groups by the variational method, provided their moment polytopes satisfy a fine condition. As an application, we prove that there is no $ \mathbb Q $-Fano $ {\rm SO}_4(\mathbb C) $-compactification which admits a Kähler-Einstein metric with the same volume as that of a smooth K-unstable Fano $ {\rm SO}_4(\mathbb C) $-compactification.

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