A Sharp inequality between local volumes and minimal log discrepancies

Abstract: We answer a question of Li--Liu--Xu: every $n$-dimensional klt germ $x\in(X,Δ)$, where $n\ge2$, satisfies the sharp inequality [ \widehat{\operatorname{vol}}(x,X,Δ)\le n^{n-1}\operatorname{mld}_x(X,Δ), ] with equality if and only if $Δ=0$ near $x$, and analytically, $(x\in X)\cong\frac{1}{r}(1,\ldots,1)$ for some $r\ge1$. We also prove that, in fixed dimension and with coefficients in a fixed finite set, $\widehat{\operatorname{vol}}/\operatorname{mld}$ is discrete away from zero. As applications of the sharp inequality, we obtain lower bounds for minimal log discrepancies of log Fano pairs.