On positivity of the CM line bundle on K-moduli spaces

• 成果类型: Article
• 署名作者: Xu, Chenyang; Zhuang, Zhiquan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.192.3.7
• 发表日期: 2020
• 页码: 1005-1068
• 关键词: kahler-einstein metrics; base loci; STABILITY; VARIETIES; volumes; LIMITS; cone; INVARIANTS; MANIFOLDS; EXISTENCE

摘要

In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties that conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations that can be used to test various K-stability notions of Fano varieties.