AI 中文总结
该论文解答了Li–Liu–Xu的问题，证明了n维klt芽满足局部体积与极小对数偏差的严格不等式，还证明了相关比值的离散性，并将其应用于得到对数Fano对极小对数偏差的下界。
AI 中文摘要
我们解答了Li–Liu–Xu提出的问题：每个n维klt芽x∈(X,Δ)（n≥2）满足严格不等式\\(\widehat{\operatorname{vol}}(x,X,\Delta)\le n^{n-1}\operatorname{mld}_x(X,\Delta)\\)，当且仅当在x附近Δ=0且解析上(x∈X)≅\\(\frac{1}{r}(1,\ldots,1)\\)（r≥1）时等号成立。我们还证明，在固定维数且系数取自固定有限集时，\\(\widehat{\operatorname{vol}}/\operatorname{mld}\\)在非零处是离散的。作为该严格不等式的应用，我们得到了对数Fano对的极小对数偏差的下界。
英文摘要
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.
Comments19 pages
