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A high-codimensional Yuan's inequality and its application to higher arithmetic degrees

Jiarui Song

TL;DR

This work advances arithmetic dynamics in higher codimension by proving a high-codimensional Yuan-type inequality, which yields submultiplicativity and ensures the existence of higher arithmetic degrees $\alpha_k(f)$. It then establishes a sharp relative degree formula $\alpha_k(f)=\max\{\lambda_k(f),\lambda_{k-1}(f)\}$ via twisted-metric techniques, linking arithmetic and geometric growth. The paper further proves that subvariety-arithmetic degrees are uniformly bounded above by $\alpha_{k+1}(f)$ and are independent of the chosen arithmetic model, and it provides a counterexample to a proposed higher-dimensional Kawaguchi-Silverman conjecture, highlighting nuanced behavior in the arithmetic of subvarieties. Collectively, these results deepen our understanding of how high-dimensional arithmetic degrees evolve under dominant rational maps and clarify when equalities with dynamical degrees occur, with implications for canonical heights and dynamical Manin–Mumford phenomena.

Abstract

In this article, we consider a dominant rational self-map $f:X \dashrightarrow X$ of a normal projective variety defined over a number field. We study the arithmetic degree $α_k(f)$ for $f$ and $α_k(f,V)$ of a subvariety $V$, which generalize the classical arithmetic degree $α_1(f,P)$ of a point $P$. We generalize Yuan's arithmetic version of Siu's inequality to higher codimensions and utilize it to demonstrate the existence of the arithmetic degree $α_k(f)$. Furthermore, we establish the relative degree formula $α_k(f)=\max\{λ_k(f),λ_{k-1}(f)\}$. In addition, we prove several basic properties of the arithmetic degree $α_k(f, V)$ and establish the upper bound $\overlineα_{k+1}(f, V)\leq \max\{λ_{k+1}(f),λ_{k}(f)\}$, which generalizes the classical result $\overlineα_f(P)\leq λ_1(f)$. Finally, we discuss a generalized version of the Kawaguchi-Silverman conjecture that was proposed by Dang et al, and we provide a counterexample to this conjecture.

A high-codimensional Yuan's inequality and its application to higher arithmetic degrees

TL;DR

This work advances arithmetic dynamics in higher codimension by proving a high-codimensional Yuan-type inequality, which yields submultiplicativity and ensures the existence of higher arithmetic degrees . It then establishes a sharp relative degree formula via twisted-metric techniques, linking arithmetic and geometric growth. The paper further proves that subvariety-arithmetic degrees are uniformly bounded above by and are independent of the chosen arithmetic model, and it provides a counterexample to a proposed higher-dimensional Kawaguchi-Silverman conjecture, highlighting nuanced behavior in the arithmetic of subvarieties. Collectively, these results deepen our understanding of how high-dimensional arithmetic degrees evolve under dominant rational maps and clarify when equalities with dynamical degrees occur, with implications for canonical heights and dynamical Manin–Mumford phenomena.

Abstract

In this article, we consider a dominant rational self-map of a normal projective variety defined over a number field. We study the arithmetic degree for and of a subvariety , which generalize the classical arithmetic degree of a point . We generalize Yuan's arithmetic version of Siu's inequality to higher codimensions and utilize it to demonstrate the existence of the arithmetic degree . Furthermore, we establish the relative degree formula . In addition, we prove several basic properties of the arithmetic degree and establish the upper bound , which generalizes the classical result . Finally, we discuss a generalized version of the Kawaguchi-Silverman conjecture that was proposed by Dang et al, and we provide a counterexample to this conjecture.
Paper Structure (9 sections, 23 theorems, 158 equations)

This paper contains 9 sections, 23 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Definitions and main results
  3. Organization of the paper
  4. Notation and Terminology
  5. Acknowledgements
  6. The high-codimensional Yuan's inequality
  7. The relative degree formula
  8. Upper bounds of arithmetic degrees
  9. A generalized version of Kawaguchi-Silverman conjecture and its counterexample

Key Result

Theorem 1.2

Let $\mathcal{X}\rightarrow \mathrm{Spec\,}\mathbb{Z}$ be an arithmetic variety with relative dimension $d$, and $r$ and $k$ be positive integers with $r+k\leq d$. Let $\overline{\mathcal{L}}_1,\ldots, \overline{\mathcal{L}}_{k}, \overline{\mathcal{M}},\overline{\mathcal{N}}_1,\ldots,\overline{\math Here the symbol "$\leq$" will be explained in Lemma Siu1.

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2: Theorem \ref{['Siu2']}
  • Theorem 1.3: Theorem \ref{['Submulti2']}
  • Theorem 1.4: Theorem \ref{['RelaDeg2']}
  • Theorem 1.5: Theorem \ref{['UpBdd2']}
  • Theorem 2.1: dang_degrees_2020, Theorem 3.4.4
  • Theorem 2.2: qu_arithmetic_2024, Theorem 1.5
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 34 more