Kähler-Einstein metrics and normalized volumes of valuations

Abstract

In this thesis we study questions relating "global volumes" of Kähler-Einstein Fano varieties and "local volumes" at a singularity. The "local volumes" is interpreted as the normalized volume of real valuations centered at a klt singularity recently introduced by Li. In the first part, we show that the volume of a Kähler-Einstein Fano variety is bounded from above by the normalized volume of any valuation centered at a closed point. This refines a recent result of Fujita. In the second part, we study the minimization problem of normalized volumes over cone singularities. We show that a Fano manifold is K-semistable if and only if the normalized volume function over its affine cone is minimized at the canonical valuation. This confirms a conjecture of Li.