A refinement of Izumi's Theorem

  • Sébastien Boucksom

    Université Paris 6, France
  • Charles Favre

    École Polytechnique, Palaiseau, France
  • Mattias Jonsson

    University of Michigan, Ann Arbor, USA
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We improve Izumi's inequality, which states that any divisorial valuation vv centered at a closed point 00 on a normal algebraic variety YY is controlled by the order of vanishing at 00. More precisely, as vv ranges through valuations that are monomial with respect to coordinates in a fixed birational model XX dominating YY, we show that for any regular function ff on YY at 00, the function vv(f)/ord0(f)v\mapsto v(f) / {\rm ord}_0(f) d0d_0 is uniformly Lipschitz continuous as a function of the weight defining vv. As a consequence, the volume of vv 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.