Minimizing normalized volumes of valuations

Abstract

For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, we introduce a normalized volume function $\widehat{\rm vol}$ that is defined on the space of real valuations centered at $o$ and consider the problem of minimizing $\widehat{\rm vol}$. We prove that the normalized volume has a uniform positive lower bound by proving an Izumi type estimate for any $\mathbb{Q}$-Gorenstein klt singularity. Furthermore, by proving a properness estimate, we show that the set of real valuations with uniformly bounded normalized volumes is compact, and hence reduce the existence of minimizers for the normalized volume functional $\widehat{\rm vol}$ to a conjectural lower semicontinuity property. We calculate candidate minimizers in several examples to show that this is an interesting and nontrivial problem. In particular, by using an inequality of de-Fernex-Ein-Musta\c{t}\u{a}, we show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of $\widehat{\rm vol}$ for a smooth point. Finally the relation to Fujita's work on divisorial stability is also pointed out.