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  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.
All Science Journal Classification (ASJC) codes
- General Mathematics
- Kähler–Einstein metric
- Normalized volume
- Sasaki–Einstein metric