A Sharp inequality between local volumes and minimal log discrepancies

Abstract

We answer a question of Li--Liu--Xu: every $n$-dimensional klt germ $x\in (X,\Delta )$, where $n\ge 2$, 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 $\Delta =0$ near $x$, and analytically, $(x\in X)\cong \frac{1}{r}(1,\dots ,1)$ for some $r\ge 1$. We also prove that, in fixed dimension and with coefficients in a fixed finite set, $\widehat{\mathrm{vol}}/\mathrm{mld}$ is discrete away from zero. As applications of the sharp inequality, we obtain lower bounds for minimal log discrepancies of log Fano pairs.

19 pages