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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2306.11591 |
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| _version_ | 1866909577076801536 |
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| author | Song, Jiarui |
| author_facet | Song, Jiarui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11591 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A high-codimensional Yuan's inequality and its application to higher arithmetic degrees Song, Jiarui Number Theory Algebraic Geometry Dynamical Systems 37P15, 14G40 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. |
| title | A high-codimensional Yuan's inequality and its application to higher arithmetic degrees |
| topic | Number Theory Algebraic Geometry Dynamical Systems 37P15, 14G40 |
| url | https://arxiv.org/abs/2306.11591 |