News Source & Photographer: School of Mathematics and Statistics

Editor: Wang Huan

Reviewer: Zhang Xicheng

Translator: Wang Yuhan

Recently, Li Yan, a teacher from the School of Mathematics and Statistics of Beijing Institute of Technology (BIT), and Li Zhenye, a teacher from the College of Mathematics and Physics of Beijing University of Chemical Technology, jointly published a research paper entitled “Finiteness of Q-Fano Compactifications of Semisimple Groups with Kähler–Einstein Metrics” on International Mathematics Research Notices. This study proves that for a connected, semisimple complex Lie group G, there are only at most a finite number of Q-Fano G-compactifications with Kähler–Einstein metrics.

The compact space of reductive complex Lie groups is a algebraic variety with high symmetry, and its geometric structure often has an explicit combinatorial expression. In recent years, people have constructed a series of examples in this type of space, such as constructing the Fano variety with specific canonical metrics (Kähler-Einstein metrics, various soliton metrics etc.), the second type of singularity of Kähler-Ricci flows, and so on. These examples answer many hot topics in geometric analysis, indicating that studying Lie group compact spaces is very meaningful for understanding geometric topics and testing important geometric conjectures. T. Delcroix [1] first provided a combination criterion for the existence of Kähler-Einstein metrics on the smooth Fano compact space of the connected and reductive complex Lie groups; Li Yan, Tian Gang and Zhu Xiaohua [2] proved using the variational method that the above criterion also holds for Q-Fano compact space that satisfies the klt condition. On the other hand, Li Yan, Tian Gang and Zhu Xiaohua [3] proved that the Kähler-Ricci flow on a K-unstable smooth SO4-Fano compact space produced the second type of singularity. If its limit is a Q-Fano SO4 compact space, it must have a (singular) Kähler-Einstein metrics and have the same volume as the initial manifold. [2] It has been proven that a limit space that satisfies both conditions does not exist for the semisimple group SO4, so the corresponding Kähler-Ricci flow limit space is no longer an SO4-compact space.

The paper further exhausts all SO4-compact spaces with (singular) Kähler-Einstein metrics. In fact, the paper proves that for a general connected, semisimple complex Lie group G, there are only at most a finite number of Q-FanoG-compact spaces with (singular) Kähler-Einstein metrics, and provides an algorithm to exhaust these compact spaces.

Paper link address: https://academic.oup.com/imrn/article/2022/15/11776/6231716

A brief introduction to the research group and its leader:

The geometry team of the School of Mathematics and Statistics of BIT has actively carried out research on frontier issues and has made a series of important research achievements in recent years.

Li Yan graduated with a bachelor's degree from the School of Mathematical Sciences at Zhejiang University and a doctoral degree from the School of Mathematical Sciences at Peking University. She has worked as a postdoctoral researcher at the Beijing International Center for Mathematical Research. His main research area is the geometric analysis of Homogeneous space and its compact space. He has hosted projects of the National Natural Science Youth Fund, and previously published SCI papers in Journal of Functional Analysis, Mathematische Zeitschrift, Математичский Сборник and various journals.