An upper bound for polynomial volume growth of automorphisms of zero entropy

2024/08/28 by Fei Hu, Hu, Fei, Jiang, Chen · 1 citation
Mathematics · #05E14 #14J50 #16P90 #16S38 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2408.15804

openalex publication_date 2024/08/28 · openalex created_date 2024/09/22 · openalex updated_date 2026/08/01

Abstract

Let X be a normal projective variety of dimension d over an algebraically closed field and f an automorphism of X. Suppose that the pullback f^*|_N¹(X)R of f on the real Néron--Severi space N¹(X)_R is unipotent and denote the index of the eigenvalue 1 by k+1. We establish the following upper bound for the polynomial volume growth plov(f) of f: plov(f) ≤ (k/2 + 1)d. This inequality is optimal in certain cases. Moreover, we prove that k≤ 2(d-1), extending a result of Dinh--Lin--Oguiso--Zhang for compact Kähler manifolds to arbitrary characteristic. By combining these two inequalities, we obtain the optimal bound plov(f) ≤ d², that affirmatively answers the questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang.

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