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The existence of valuative interpolation

Shijie Bao, Qi'an Guan, Zhitong Mi, Zheng Yuan

TL;DR

The paper investigates valuative interpolation for germs at the origin by linking valuations with relative types and Tian functions through tame maximal weights. It proves a sharp criterion: a valuation with prescribed values exists iff the relative type satisfies $\sigma(\log|F|,\varphi)=\sum_j a_j$, with $F=\prod f_j$ and $\varphi=\log\sum |f_j|^{1/a_j}$; in the isolated-zero case, a tame weight $\varphi_\nu$ realizing the valuation exists and matches jumping-number data via log canonical thresholds. It extends the framework of valuations to tame, Zhou, and quasimonomial settings, showing that every quasimonomial valuation arises as a relative type with respect to a tame weight and that Boucksom–Favre–Jonsson results carry over to quasimonomial Zhou valuations; convergence results for valuations and relative types under Tian-function analysis underpin the interpolation theory. The real-case translations via the P map yield analogous interpolation results for real-analytic germs. Overall, the work connects valuative interpolation with multiplier ideals, Lelong-like invariants, and the geometry of tame maximal weights, providing a robust toolkit for constructing valuations realizing prescribed data and clarifying the structure of valuations in complex and real analytic settings.

Abstract

In this article, using key tools including Zhou valuations, Tian functions and a convergence result for relative types, we establish necessary and sufficient conditions for the existence of valuative interpolations on the rings of germs of holomorphic functions and real analytic functions at the origin in $\mathbb{C}^{n}$ and $\mathbb{R}^{n}$, respectively. For the cases of polynomial rings with complex and real coefficients, we establish separate necessary conditions and sufficient conditions, which become both necessary and sufficient when the intersection of the zero sets of the given polynomials is the set of the origin in $\mathbb{C}^{n}$. Furthermore, we obtain a necessary and sufficient condition for a valuation to be of the form given by a relative type with respect to a tame maximal weight. We demonstrate a result of Boucksom--Favre--Jonsson on quasimonomial valuations also holds for quasimonomial Zhou valuations. Finally, we obtain a relationship between Zhou valuations and the differentiable points of Tian functions.

The existence of valuative interpolation

TL;DR

The paper investigates valuative interpolation for germs at the origin by linking valuations with relative types and Tian functions through tame maximal weights. It proves a sharp criterion: a valuation with prescribed values exists iff the relative type satisfies , with and ; in the isolated-zero case, a tame weight realizing the valuation exists and matches jumping-number data via log canonical thresholds. It extends the framework of valuations to tame, Zhou, and quasimonomial settings, showing that every quasimonomial valuation arises as a relative type with respect to a tame weight and that Boucksom–Favre–Jonsson results carry over to quasimonomial Zhou valuations; convergence results for valuations and relative types under Tian-function analysis underpin the interpolation theory. The real-case translations via the P map yield analogous interpolation results for real-analytic germs. Overall, the work connects valuative interpolation with multiplier ideals, Lelong-like invariants, and the geometry of tame maximal weights, providing a robust toolkit for constructing valuations realizing prescribed data and clarifying the structure of valuations in complex and real analytic settings.

Abstract

In this article, using key tools including Zhou valuations, Tian functions and a convergence result for relative types, we establish necessary and sufficient conditions for the existence of valuative interpolations on the rings of germs of holomorphic functions and real analytic functions at the origin in and , respectively. For the cases of polynomial rings with complex and real coefficients, we establish separate necessary conditions and sufficient conditions, which become both necessary and sufficient when the intersection of the zero sets of the given polynomials is the set of the origin in . Furthermore, we obtain a necessary and sufficient condition for a valuation to be of the form given by a relative type with respect to a tame maximal weight. We demonstrate a result of Boucksom--Favre--Jonsson on quasimonomial valuations also holds for quasimonomial Zhou valuations. Finally, we obtain a relationship between Zhou valuations and the differentiable points of Tian functions.
Paper Structure (20 sections, 50 theorems, 267 equations, 1 figure)

This paper contains 20 sections, 50 theorems, 267 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Valuative interpolation
  3. Applications on real cases
  4. Valuation and relative type
  5. Zhou valuation and quasimonomial valuation
  6. Valuations related to Tian functions
  7. Preliminaries
  8. Strong openness property of multiplier ideal sheaves
  9. Some results on plurisubharmonic functions, multiplier ideal sheaves and valuations
  10. Tian function
  11. Zhou weight
  12. Tame maximal weight
  13. Convergence results for valuations and for relative types
  14. Proofs of Theorem \ref{['thm:interpolation']} and its corollaries
  15. Proof of Theorem \ref{['thm:valuationexistsweight']}
  16. ...and 5 more sections

Key Result

Theorem 1.2

Given $m+1$ holomorphic functions $\{f_j\}_{0\le j\le m}$ near $o$, $a_0=0$ and $m$ positive numbers $\{a_j\}_{1\le j\le m}$, the following two statements are equivalent: Furthermore, if $o$ is an isolated point in $\cap_{1\le j\le m}\{f_j=0\}$ and $\sigma(\log|F|,\varphi)=\sum_{1\le j\le m}a_j$, then there exists a valuation $\nu$ and a tame maximal weight $\varphi_{\nu}$ such that $\nu(f_j)=a_j

Figures (1)

  • Figure 1: The valuative tree; see FJ05.

Theorems & Definitions (94)

  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Example 1.7
  • Corollary 1.8
  • Remark 1.9
  • Corollary 1.10
  • Theorem 1.11
  • ...and 84 more