A refinement of Izumi's Theorem

TL;DR

This paper refines Izumi's inequality by demonstrating Lipschitz continuity of valuation functions and volume in a toroidal setting, enhancing understanding of valuations centered at a point on algebraic varieties.

Contribution

It provides a stronger, Lipschitz continuity result for valuations and their volumes, using toroidal and positivity techniques, improving previous bounds.

Findings

Valuation functions are uniformly Lipschitz continuous.

Volume functions of valuations are Lipschitz continuous.

The proof employs toroidal methods and nef divisor positivity.

Abstract

We improve Izumi's inequality, which states that any divisorial valuation v centered at a closed point 0 on an 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--> v(f)/\ord_0(f) 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.