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Original Article

Equivariant R-Test Configurations and Semistable Limits of Q-Fano Group Compactifications

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

Yan Li partially supported by NSFC Grant 12101043 and the Beijing Institute of Technology Research Fund Program for Young Scholars.

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Abstract

Let G be a connected, complex reductive group. In this paper, we classify G×G-equivariant normal R-test configurations of a polarized G-compactification. Then, for Q-Fano G-compactifications, we express the H-invariants of their equivariant normal R-test configurations in terms of the combinatory data. Based on Han and Li “Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties”, we compute the semistable limit of a K-unstable Fano G-compactification. As an application, we show that for the two smooth K-unstable Fano SO 4(C)-compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized Kähler–Ricci flow.

Keywords

K-stabilityKähler–Ricci solitonsKähler–Ricci flow$ {\mathbb {Q}} $-Fano compactifications

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Peking Mathematical Journal
Volume 6 Issue 2,
September 2023
Pages 559-607
DOI: 10.1007/s42543-022-00054-0
Cite this article:
Li, Y., Li, Z. Equivariant R-Test Configurations and Semistable Limits of Q-Fano Group Compactifications. Peking Math J 6, 559-607 (2023). https://doi.org/10.1007/s42543-022-00054-0

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Received: 03 November 2021
Revised: 22 May 2022
Accepted: 08 July 2022
Published: 07 November 2022
© Peking University 2022
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