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On upper bounds of arithmetic degrees
American Journal of Mathematics ( IF 1.7 ) Pub Date : 2020-01-01 , DOI: 10.1353/ajm.2020.0045
Yohsuke Matsuzawa

Abstract:Let $X$ be a smooth projective variety defined over $\\overline\{\\Bbb\{Q\}\}$, and $f\\colon X\\dashrightarrow X$ be a dominant rational map. Let $\\delta_f$ be the first dynamical degree of $f$ and $h_X\\colon X(\\overline\{\\Bbb\{Q\}\})\\rightarrow [1,\\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds of the sequence $(h_X(f^n(P)))_\{n\\geq0\}$ where $P$ is a point of $X(\\overline\{\\Bbb\{Q\}\})$ whose forward orbit by $f$ is well defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; $\\overline\{\\alpha\}_f(P)\\leq\\delta_f$. Furthermore, we prove the canonical height functions of rational self-maps exist under certain conditions. For example, when the Picard number of $X$ is one, $f$ is algebraically stable (in the sense of Fornaess-Sibony) and $\\delta_f>1$, the limit defining canonical height $\\lim_\{n\\to\\infty\}h_X(f^n(P))\\big/\\delta_f^n$ converges.

中文翻译:

关于算术度的上限

摘要:设 $X$ 是定义在 $\\overline\{\\Bbb\{Q\}\}$ 上的平滑射影变体,$f\\colon X\\dashrightarrow X$ 是占主导地位的有理映射。设 $\\delta_f$ 是 $f$ 和 $h_X\\colon X(\\overline\{\\Bbb\{Q\}\})\\rightarrow [1,\\infty)$ 的第一个动态度数是 $X$ 上与 $X$ 上的充足除数相关联的 Weil 高度函数。我们证明了几个不等式,它们给出了序列 $(h_X(f^n(P)))_\{n\\geq0\}$ 的上限,其中 $P$ 是 $X(\\overline\{\ \Bbb\{Q\}\})$ 的前向轨道 $f$ 是明确定义的。作为推论,我们证明上算术度小于或等于第一动力度;$\\overline\{\\alpha\}_f(P)\\leq\\delta_f$. 此外,我们证明了理性自映射的规范高度函数在某些条件下存在。例如,当$X$的Picard数为1时,
更新日期:2020-01-01
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