A refinement of Izumi's Theorem
  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. 
S\'ebastien Boucksom, Charles Favre, and Mattias Jonsson
math.AC math.AG