An Upper Bound for Polynomial Volume Growth of Automorphisms of Zero Entropy

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^{\ast }{|}_{{\mathsf{N}}^1{(X)}_{\mathbf{R}}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f^*|_{{\textsf{N}}^1(X)_{\textbf{R}}}$$\end{document}

of f on the real Néron–Severi space

{\mathsf{N}}^1{(X)}_{\mathbf{R}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textsf{N}}^1(X)_{\textbf{R}}$$\end{document}

is unipotent and denote the index of the eigenvalue 1 by

k+1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k+1$$\end{document}

. We establish the following upper bound for the polynomial volume growth

\text{plov}(f)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{plov}(f)$$\end{document}

of f:

\begin{array}{c}\hfill \text{plov}(f)\le (k/2+1)d.\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \textrm{plov}(f) \le (k/2 + 1)d. \end{aligned}$$\end{document}

This inequality is optimal in certain cases. Moreover, we prove that

k\le 2(d-1)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k\le 2(d-1)$$\end{document}

, 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

\begin{array}{c}\hfill \text{plov}(f)\le d^2,\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \textrm{plov}(f) \le d^2, \end{aligned}$$\end{document}

that affirmatively answers the questions of Cantat–Paris-Romaskevich and Lin–Oguiso–Zhang.

Keywords

Automorphism / Zero entropy / Polynomial volume growth / Gelfand–Kirillov dimension / Quasi-unipotency / Restricted partition / Unimodality / Degree growth

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Funding

National Natural Science Foundation of China(12371045)

RIGHTS & PERMISSIONS

Peking University