Abstract
We prove that the K-moduli space of cubic fourfolds is identical to their GIT moduli space. More precisely, the K-(semi/poly)stability of cubic fourfolds coincide to the corresponding GIT stabilities, which was studied in detail by Laza. In particular, this implies that all smooth cubic fourfolds admit Kähler–Einstein metrics. Key ingredients are local volume estimates in dimension three due to Liu and Xu, and Ambro–Kawamata’s non-vanishing theorem for Fano fourfolds.
Funding source: National Science Foundation
Award Identifier / Grant number: 2148266
Funding statement: The author was partially supported by the NSF Grant No. 2148266 (transferred from NSF Grant No. DMS-2001317).
Acknowledgements
I would like to thank Chenyang Xu and Ziquan Zhuang for fruitful discussions and helpful comments on a draft, including Remark 4.8 and a simplification of the proof of Proposition 4.6. I would like to thank Kento Fujita, Chen Jiang, Zhiyuan Li, Linquan Ma, Yuji Odaka, Giulia Saccà, Cristiano Spotti, and Gang Tian for helpful discussions and comments.
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