Kähler–Einstein metrics and volume minimization

Chi Li, Yuchen Liu

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


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 languageEnglish (US)
Pages (from-to)440-492
Number of pages53
JournalAdvances in Mathematics
StatePublished - Jan 7 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • K-semistable
  • Kähler–Einstein metric
  • Normalized volume
  • Sasaki–Einstein metric


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