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

2023/06/20 by Jiarui Song, Song, Jiarui
Computer Science · Mathematics · Social Sciences · #14G40 #37P15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2306.11591

openalex publication_date 2023/06/20 · openalex created_date 2023/06/22 · openalex updated_date 2026/07/28

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 α₁(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 α_k+1(f, V)≤ max\λ_k+1(f),λ_k(f)\, which generalizes the classical result α_f(P)≤ λ₁(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.

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