[Paper Review] A refinement of Izumi's Theorem

Sébastien Boucksom, Charles Favre+1 more|arXiv (Cornell University)|Sep 18, 2012

Algebraic Geometry and Number Theory33 references16 citations

TL;DR

This paper refines Izumi's Theorem by establishing uniform Lipschitz continuity of the ratio $ v(f)/\operatorname{ord}_0(f) $ across monomial valuations centered at a point on a variety, using toroidal geometry and positivity of nef divisors. The key result shows that both the valuation and its volume are Lipschitz continuous in the weight parameters defining the valuation, with explicit bounds depending on the geometry of the dual complex.

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.

Motivation & Objective

• To strengthen Izumi’s Theorem by quantifying the comparability between divisorial valuations and order of vanishing at a point.
• To establish uniform Lipschitz continuity of the map $ v \mapsto v(f)/\operatorname{ord}_0(f) $ for monomial valuations in a fixed birational model.
• To show that the volume of a valuation is also Lipschitz continuous in the weight parameters defining the valuation.
• To connect the Lipschitz constant to geometric invariants via Newton polyhedra and intersection theory on the dual complex.

Proposed method

• Use of toroidal geometry to model the space of quasimonomial valuations via the dual complex $ \Delta(X,Z) $ of a simple normal crossing divisor.
• Application of convex analysis to study the concavity and Lipschitz continuity of the valuation function $ v \mapsto v(f) $ on each face of $ \Delta $.
• Employment of positivity properties of images of nef divisors under birational morphisms to bound the Lipschitz constant.
• Use of Cohen’s structure theorem to express functions as power series in local coordinates, enabling the definition of Newton polyhedra $ \operatorname{Nw}(f,J) $.
• Reduction of the problem to estimating the norm of extremal points of Newton polyhedra via the Lipschitz constant of the logarithmic function $ \varphi = \log|f| $.
• Derivation of bounds on the Lipschitz constant using intersection numbers $ (L_d \cdot M^{n-|J|-1} \cdot E_J) $ in the proof of Theorem B.

Experimental results

Research questions

• RQ1 Can the constant in Izumi’s inequality be made uniform across a family of monomial valuations?
• RQ2 Is the ratio $ v(f)/\operatorname{ord}_0(f) $ Lipschitz continuous in the weight parameters defining the valuation?
• RQ3 How does the Lipschitz constant of the valuation function relate to geometric invariants of the ambient variety and divisor configuration?
• RQ4 Can the volume of a valuation be shown to vary continuously with respect to the weight parameters?

Key findings

• The function $ v \mapsto v(f) $ on the dual complex $ \Delta $ is concave and Lipschitz continuous with constant at most $ A \cdot \operatorname{ord}_0(f) $, where $ A $ depends only on the model $ X $ and the metric on $ \Delta $.
• The ratio $ v(f)/\operatorname{ord}_0(f) $ is uniformly Lipschitz continuous in the weight parameters defining monomial valuations, with a bound independent of $ f $.
• The volume of a valuation is a Lipschitz continuous function of the weight vector defining the monomial valuation, with the Lipschitz constant bounded by a multiple of the maximal intersection number $ (L_1 \cdot M^{n-|J|-1} \cdot E_J) $.
• The extremal points of the Newton polyhedron $ \operatorname{Nw}(f,J) $ have norm at most $ AC \cdot \operatorname{ord}_0(f) $, where $ C $ is a geometric constant depending on the norm and $ A $ is the Lipschitz constant from Theorem A.
• The proof establishes that the Lipschitz constant of $ v \mapsto v(f) $ is controlled by the geometry of the dual complex and the positivity of the pullback of a hyperplane section under a birational morphism.