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

Chenyang Xu, Ziquan Zhuang

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1005-1068
Number of pages64
JournalAnnals of Mathematics
Issue number3
StatePublished - Nov 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)


  • Cm line bundle
  • Fano varieties
  • K-moduli
  • K-stability
  • Projectivity


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