A refinement of Izumi's Theorem

Abstract

We improve Izumi's inequality, which states that any divisorial valuation $v$ centered at a closed point $0$ on a normal algebraic variety $Y$ is controlled by the order of vanishing at $0$. More precisely, as $v$ ranges through valuations that are monomial with respect to coordinates in a fixed birational model $X$ dominating $Y$, we show that for any regular function $f$ on $Y$ at $0$, the function $v\mapsto v(f) / {\rm ord}_0(f)$ $d_0$ is uniformly Lipschitz continuous as a function of the weight defining $v$. As a consequence, the volume of $v$ is also a Lipschitz continuous function. Our proof uses toroidal techniques as well as positivity properties of the images of suitable nef divisors under birational morphisms.