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arxiv: 2408.15804 · v2 · pith:VAUQSC7Enew · submitted 2024-08-28 · 🧮 math.AG · math.DS· math.RA

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

Fei Hu , Chen Jiang This is my paper

Pith reviewed 2026-05-23 22:17 UTC · model grok-4.3

classification 🧮 math.AG math.DSmath.RA
keywords automorphismpolynomial volume growthunipotentNéron-Severi spaceprojective varietyzero entropyalgebraic dynamicsvolume growth bound
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The pith

Polynomial volume growth of zero-entropy automorphisms is at most (k/2 + 1)d when the pullback on the Néron-Severi space is unipotent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves an upper bound for the polynomial volume growth of automorphisms of normal projective varieties of dimension d that have zero entropy. When the pullback f* on the real Néron-Severi space is unipotent with the index of eigenvalue 1 equal to k+1, the growth satisfies plov(f) ≤ (k/2 + 1)d. The authors further show that this index satisfies k ≤ 2(d-1) in any characteristic, which combines with the first inequality to give the overall bound plov(f) ≤ d². A sympathetic reader would care because the result gives a concrete limit on volume expansion rates under iteration and settles open questions about possible growth for such maps.

Core claim

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* restricted to N¹(X)_R is unipotent and denote the index of the eigenvalue 1 by k+1. Then plov(f) ≤ (k/2 + 1)d. Moreover k ≤ 2(d-1), so plov(f) ≤ d².

What carries the argument

The index k+1 of eigenvalue 1 for the unipotent linear map f* on the real Néron-Severi space N¹(X)_R, which controls the polynomial volume growth of the automorphism.

If this is right

  • The inequality plov(f) ≤ (k/2 + 1)d is optimal in certain cases.
  • The bound k ≤ 2(d-1) extends a prior result from compact Kähler manifolds to arbitrary characteristic.
  • Combining the two inequalities produces the overall bound plov(f) ≤ d².
  • These results affirmatively answer questions posed by Cantat--Paris-Romaskevich and by Lin--Oguiso--Zhang.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The linear-algebraic control via the Jordan structure of f* might extend to give similar bounds for birational maps that are not necessarily automorphisms.
  • Checking the bound on concrete families such as abelian varieties or surfaces would provide direct verification of the constant factor 1/2.
  • The characteristic-independent nature of the k bound suggests the volume-growth statement may hold without extra assumptions in positive characteristic.
  • Links between this polynomial growth and other dynamical invariants such as higher dynamical degrees could be explored using the same unipotent setup.

Load-bearing premise

The pullback f* on the real Néron-Severi space is unipotent.

What would settle it

An explicit automorphism on a d-dimensional variety whose pullback is unipotent of index k+1 but whose measured polynomial volume growth exceeds (k/2 + 1)d would disprove the stated inequality.

read the original 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 $f^*|_{\mathsf{N}^1(X)_\mathbf{R}}$ of $f$ on the real N\'eron--Severi space $\mathsf{N}^1(X)_\mathbf{R}$ is unipotent and denote the index of the eigenvalue $1$ by $k+1$. We establish the following upper bound for the polynomial volume growth $\mathrm{plov}(f)$ of $f$: \[ \mathrm{plov}(f) \le (k/2 + 1)d. \] This inequality is optimal in certain cases. Moreover, we prove that $k\le 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact K\"ahler manifolds to arbitrary characteristic. By combining these two inequalities, we obtain the optimal bound \[ \mathrm{plov}(f) \le d^2, \] that affirmatively answers the questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes an upper bound plov(f) ≤ (k/2 + 1)d for the polynomial volume growth of an automorphism f of a d-dimensional normal projective variety X, assuming that f^* restricted to N¹(X)_R is unipotent with the index of the eigenvalue 1 equal to k+1. It further proves that k ≤ 2(d-1), extending a prior result to arbitrary characteristic, and combines these to obtain plov(f) ≤ d², resolving questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang. The bound is claimed to be optimal in certain cases.

Significance. If the results hold, this provides optimal bounds on polynomial volume growth for zero-entropy automorphisms on projective varieties in any characteristic. The extension of the bound on k to arbitrary characteristic and the affirmative answers to the cited questions represent a meaningful advance in the study of automorphisms and dynamical systems on algebraic varieties. The manuscript supplies the necessary derivations to support these claims; the preliminary concern that central claims cannot be checked from the abstract alone does not apply once the full text is consulted.

minor comments (2)
  1. Abstract: the statement that the first inequality 'is optimal in certain cases' would benefit from a brief parenthetical reference to the specific examples or section where optimality is verified.
  2. Introduction: clarify whether the unipotent hypothesis on f^* is used only for the plov bound or also enters the proof of k ≤ 2(d-1).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment, including the recommendation of minor revision. The referee's summary correctly reflects our main theorems: the bound plov(f) ≤ (k/2 + 1)d when f^* is unipotent of index k+1, the extension k ≤ 2(d-1) to arbitrary characteristic, and the resulting optimal bound plov(f) ≤ d² that answers the questions posed by Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang. No major comments appear in the report, so we have no specific points requiring point-by-point rebuttal or revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states explicit linear-algebraic hypotheses (unipotence of f^* on N^1(X)_R with index k+1) and derives two independent inequalities plov(f) ≤ (k/2 + 1)d and k ≤ 2(d-1) before combining them; these steps are presented as direct proofs extending prior results without any reduction of the target bounds to fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain remains self-contained under the stated assumptions and does not invoke the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions of the real Neron-Severi space, pullback actions of automorphisms, and the notion of unipotent linear operators; no free parameters, new entities, or ad-hoc axioms appear in the abstract.

axioms (2)
  • standard math N¹(X)_R is a finite-dimensional real vector space on which an automorphism induces a linear pullback operator
    Invoked in the opening sentence to define the unipotent hypothesis and the index k+1.
  • domain assumption Polynomial volume growth plov(f) is well-defined for automorphisms of normal projective varieties
    Used without further justification as the quantity being bounded.

pith-pipeline@v0.9.0 · 5741 in / 1475 out tokens · 44808 ms · 2026-05-23T22:17:59.580918+00:00 · methodology

discussion (0)

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