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Automorphisms of bounded growth

Alexandra Kuznetsova

TL;DR

The paper classifies birational automorphisms with bounded growth on smooth projective varieties, showing that some power of an infinite order such automorphism either induces an infinite order translation on the Albanese variety or factors through an infinite order regular automorphism of a projective space, with a finite action on the remaining factor. It introduces a framework that avoids excessive minimal model program machinery by exploiting a big and nef line bundle when the Albanese action is finite and using Blanc’s conjugacy for reductions to $\mathbb{P}^n$, which forces invariant subvarieties to be cones. Consequently, for rationally connected threefolds with bounded-growth automorphisms, the variety is rational and ultimately birational to a regular automorphism of $\mathbb{P}^3$ after a suitable iterate. The work also provides a higher‑dimensional generalization of Blanc–Deserti and exhibits a higher-dimensional counterexample showing that not every such automorphism must be conjugate to a projective automorphism. Overall, it clarifies the rarity and structural constraints of bounded-growth automorphisms and their birational manifestations.

Abstract

We study birational automorphisms of algebraic varieties of bounded growth, i.e. such that the norms of the inverse images ${(f^n)}^* \colon \mathrm{NS}(X)\to \mathrm{NS}(X)$ of the powers of the automorphism $f\in\mathrm{Bir}(X)$ are bounded above for $n\geqslant 0$. We prove that some power of an infinite order automorphism of a variety $X$ with such property factors either through an infinite order translation on the Albanese variety of $X$ or through an infinite order regular automorphism of $\mathbb{P}^m$ for $m\geqslant 1$. We deduce from this that if a rationally connected threefold admits an infinite order automorphism whose growth is bounded then the threefold is rational and an iterate of the automorphism is birationally conjugate to a regular automorphism of $\mathbb{P}^3$, a generalization of Blanc and Deserti's result.

Automorphisms of bounded growth

TL;DR

The paper classifies birational automorphisms with bounded growth on smooth projective varieties, showing that some power of an infinite order such automorphism either induces an infinite order translation on the Albanese variety or factors through an infinite order regular automorphism of a projective space, with a finite action on the remaining factor. It introduces a framework that avoids excessive minimal model program machinery by exploiting a big and nef line bundle when the Albanese action is finite and using Blanc’s conjugacy for reductions to , which forces invariant subvarieties to be cones. Consequently, for rationally connected threefolds with bounded-growth automorphisms, the variety is rational and ultimately birational to a regular automorphism of after a suitable iterate. The work also provides a higher‑dimensional generalization of Blanc–Deserti and exhibits a higher-dimensional counterexample showing that not every such automorphism must be conjugate to a projective automorphism. Overall, it clarifies the rarity and structural constraints of bounded-growth automorphisms and their birational manifestations.

Abstract

We study birational automorphisms of algebraic varieties of bounded growth, i.e. such that the norms of the inverse images of the powers of the automorphism are bounded above for . We prove that some power of an infinite order automorphism of a variety with such property factors either through an infinite order translation on the Albanese variety of or through an infinite order regular automorphism of for . We deduce from this that if a rationally connected threefold admits an infinite order automorphism whose growth is bounded then the threefold is rational and an iterate of the automorphism is birationally conjugate to a regular automorphism of , a generalization of Blanc and Deserti's result.

Paper Structure

This paper contains 9 sections, 14 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Automorphisms of projective space
  3. Conjugacy classes of $\mathop{\mathrm{Aut}}\nolimits(\mathbb{P}^n)$ under the action of $\mathop{\mathrm{Bir}}\nolimits(\mathbb{P}^n)$
  4. Eigenpolynomials under the action of $\mathop{\mathrm{PGL}}\nolimits(n+1,\mathbb{K})$
  5. Infinite order automorphisms
  6. Invariant subvarieties of infinite order elements in $\mathop{\mathrm{Aut}}\nolimits(\mathbb{P}^n)$
  7. Action of automorphisms on $\mathop{\mathrm{Pic}}\nolimits^0(X)$ and $\mathop{\mathrm{Alb}}\nolimits(X)$
  8. Proofs
  9. Automorphism of a rational threefold which is not conjugate to $\mathop{\mathrm{Aut}}\nolimits(\mathbb{P}^3)$

Key Result

Theorem 1.1

Let $S$ be a rational surface over an algebraically closed field of characteristic $0$ and let $f$ be its infinite order automorphism whose growth is bounded. Then there exists a birational map $\alpha\colon S\dashrightarrow \mathbb{P}^2$ such that $\alpha\circ f\circ \alpha^{-1}$ is a regular autom

Theorems & Definitions (26)

  • Theorem 1.1: Blanc-Deserti
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: Blanc_conjugacy_of_PGL
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 16 more