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The existence of valuative interpolation at a singular point

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

TL;DR

The paper addresses valuative interpolation at singular points of irreducible analytic varieties by extending Zhou weights, Tian functions, and Zhou valuations to the singular setting. It develops a precise criterion: a valuation matching prescribed values on a finite family of germs exists if and only if $\sigma(\log|F|,\varphi)=\sum a_j$, with $F=\prod f_j$ and $\varphi=\log(\sum|f_j|^{1/a_j})$, and it extends this to weakly holomorphic functions and to quotient polynomial rings over $\mathbb{C}$ and $\mathbb{R}$. Central tools include the singular-version local Zhou weights, the connection between relative types and valuations via Tian functions, and resolution-of-singularities techniques to relate singular and regular data. The results provide necessary and sufficient conditions for valuative interpolation in both holomorphic and real-algebraic settings, with insights into how the zero-set geometry governs interpolation and the role of multiplier-ideal-type concepts in the singular context.

Abstract

The present paper studies the existence of valuative interpolation on the local ring of an irreducible analytic subvariety at singular points. We firstly develop the concepts and methods of Zhou weights and Tian functions near singular points of irreducible analytic subvarieties. By applying these tools, we establish the necessary and sufficient conditions for the existence of valuative interpolations on the rings of germs of holomorphic functions and weakly holomorphic functions at a singular point. As applications, we characterize the existence of valuative interpolations on the quotient ring of the ring of convergent power series in real variables. We also present separated necessary and sufficient conditions for the existence of valuative interpolations on the quotient ring of polynomial rings with complex coefficients and real coefficients. Furthermore, we show that the conditions become both necessary and sufficient under certain conditions on the zero set of the given polynomials.

The existence of valuative interpolation at a singular point

TL;DR

The paper addresses valuative interpolation at singular points of irreducible analytic varieties by extending Zhou weights, Tian functions, and Zhou valuations to the singular setting. It develops a precise criterion: a valuation matching prescribed values on a finite family of germs exists if and only if , with and , and it extends this to weakly holomorphic functions and to quotient polynomial rings over and . Central tools include the singular-version local Zhou weights, the connection between relative types and valuations via Tian functions, and resolution-of-singularities techniques to relate singular and regular data. The results provide necessary and sufficient conditions for valuative interpolation in both holomorphic and real-algebraic settings, with insights into how the zero-set geometry governs interpolation and the role of multiplier-ideal-type concepts in the singular context.

Abstract

The present paper studies the existence of valuative interpolation on the local ring of an irreducible analytic subvariety at singular points. We firstly develop the concepts and methods of Zhou weights and Tian functions near singular points of irreducible analytic subvarieties. By applying these tools, we establish the necessary and sufficient conditions for the existence of valuative interpolations on the rings of germs of holomorphic functions and weakly holomorphic functions at a singular point. As applications, we characterize the existence of valuative interpolations on the quotient ring of the ring of convergent power series in real variables. We also present separated necessary and sufficient conditions for the existence of valuative interpolations on the quotient ring of polynomial rings with complex coefficients and real coefficients. Furthermore, we show that the conditions become both necessary and sufficient under certain conditions on the zero set of the given polynomials.
Paper Structure (15 sections, 54 theorems, 184 equations)

This paper contains 15 sections, 54 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Zhou weights and Zhou valuations near singular points
  3. Interpolation problem
  4. Preparation
  5. Basic knowledge
  6. Concavity
  7. Tian functions and Zhou numbers
  8. Relation between relative types and valuations
  9. Jumping number in the singular case
  10. Convergence results for valuations and for relative types
  11. Proof of Remark \ref{['rem:max_existence']}
  12. Proof of Theorem \ref{['thm:expression of relative types']}
  13. Proof of Theorem \ref{['thm:valu-jump']}
  14. Proofs of Theorem \ref{['thm:interpolation']}, Theorem \ref{['thm:interpolation weakly']} and their Corollaries
  15. The universal denominators for algebraic varieties

Key Result

Theorem 1.6

Let $\Phi_{z_0,\max}$ be a local Zhou weight related to $|f_{0}|^{2}e^{-2\varphi_{0}}$ near $z_0$. Then for any plurisubharmonic function $\psi$ near $z_0$ satisfying $\psi\le c\log|z|+O(1)$ near $z_0$ for some $c>0$, we have

Theorems & Definitions (121)

  • Definition 1.2
  • Remark 1.3
  • Definition 1.4: see demailly-book
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.9
  • Theorem 1.10
  • ...and 111 more