Abstract
We prove that if a polarized Q-Fano variety admits a Kähler–Einstein metric, then the normalized volume functional over the associated affine cone is globally minimized at the canonical divisorial valuation obtained by blowing-up the vertex. This is also generalized to the logarithmic and the orbifold settings. A refinement of this result together with the result in [41] gives an equivalent characterization of K-semistability for any smooth Fano manifold. We also prove that the valuation associated to the Reeb vector field of a smooth Sasaki–Einstein metric minimizes the normalized volume over the corresponding Kähler cone. These results strengthen and generalize the minimization result of Martelli–Sparks–Yau.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 440-492 |
| Number of pages | 53 |
| Journal | Advances in Mathematics |
| Volume | 341 |
| DOIs | |
| State | Published - Jan 7 2019 |
Funding
We would like to thank Kento Fujita, Gang Tian and Chenyang Xu for helpful comments. The first author is partially supported by NSF DMS-1405936 and an Alfred P. Sloan research fellowship (Grant Number: FG-2017-9258 ). The first author would like to thank Laszlo Lempert and Sai-Kee Yeung for their interest in this work. Part of this paper is written while the first author visited MSRI at Berkeley in 2016, and he would like to thank the institute for its hospitality. The second author is partially supported by NSF DMS-0968337 . The second author would like to thank his advisor János Kollár for his constant support, encouragement and numerous inspiring conversations. The second author also wishes to thank Charles Stibitz, Yury Ustinovskiy, Xiaowei Wang and Ziquan Zhuang for many useful discussions. The authors would also like to thank an anonymous referee for correcting some references and helpful suggestions. We would like to thank Kento Fujita, Gang Tian and Chenyang Xu for helpful comments. The first author is partially supported by NSF DMS-1405936 and an Alfred P. Sloan research fellowship (Grant Number: FG-2017-9258). The first author would like to thank Laszlo Lempert and Sai-Kee Yeung for their interest in this work. Part of this paper is written while the first author visited MSRI at Berkeley in 2016, and he would like to thank the institute for its hospitality. The second author is partially supported by NSF DMS-0968337. The second author would like to thank his advisor János Kollár for his constant support, encouragement and numerous inspiring conversations. The second author also wishes to thank Charles Stibitz, Yury Ustinovskiy, Xiaowei Wang and Ziquan Zhuang for many useful discussions. The authors would also like to thank an anonymous referee for correcting some references and helpful suggestions.
Keywords
- K-semistable
- Kähler–Einstein metric
- Normalized volume
- Sasaki–Einstein metric
ASJC Scopus subject areas
- General Mathematics