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Numerical action for endomorphisms

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Pith's one-line read Surjective endomorphisms of projective varieties are quasi-amplified exactly when they are cohomologically hyperbolic, and amplified exactly when every periodic subvariety is cohomologically hyperbolic; the paper proves both by computing…

desk verdict Serious paper with a real gap: the ample-cone spectrum is not proven because the final separation step in Theorem 6.1 is invalid; still deserves refereeing. read the letter →

arxiv 2502.04779 v1 pith:VM4EKNBM submitted 2025-02-07 math.DS math.AG

Junyi Xie This is my paper · ORCID
classification math.DSmath.AG MSC 37P5514J40
keywords endomorphismsofprojectivevarietiesdynamicaldegreescohomologicalLyapunovexponentsamplifiedquasi-amplifiedspectrumaninvariantconegeneratedpositivecyclesperiodicsubvarieties
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies how a surjective self-map $f$ of a projective variety acts on numerical classes of divisors. It introduces a general notion of spectrum for a linear endomorphism on a finite-dimensional real vector space once an open, salient invariant cone is fixed: the set of $\alpha$ for which no class $N$ satisfies $gN-\alpha N\in C$. For the big cone, the paper proves that this spectrum is exactly the set of cohomological Lyapunov exponents $\mu_i(f)=\lambda_i(f)/\lambda_{i-1}(f)$. For the ample cone, it proves that the spectrum is the union of the same exponents over all irreducible periodic subvarieties. This yields the paper's central equivalences: quasi-amplified means cohomologically hyperbolic, and amplified means every subsystem is cohomologically hyperbolic.

What carries the argument

The central new object is the $C$-spectrum $Sp(g,C)$, defined for an endomorphism $g$ and an open salient invariant cone $C$ as the set of $\alpha$ such that no $v$ satisfies $gv-\alpha v\in C$. Theorem 1.3 links this spectrum to generated eigenspaces: $E_S(C)$ meets $C$ exactly when $S$ contains $Sp(g,C)$, so spectral questions become linear-algebra questions about which invariant subspaces intersect the positivity cone. For the big cone, the recursive inequalities and mixed-degree computations from the paper's companion work compute that intersection directly. For the ample cone, the paper introduces generated positive cycles, algebraic analogues of positive closed currents; their atomic decomposition (an algebraic Siu-type decomposition) lets arbitrary pseudo-effective divisor classes be written as countable sums of atoms, reducing growth rates to those of periodic subvarieties.

What would settle it

Take an explicit endomorphism such as $(x,y)\mapsto (x^a,y^b)$ on $\mathbb P^1\times\mathbb P^1$, or a power map lifted to a blow-up of $\mathbb P^2$, and compute the eigenvalues of $f^*$ on $N^1(X)_{\mathbb R}$ together with the cohomological Lyapunov exponents $\mu_i(V,f)$ of every irreducible periodic subvariety. The theorem predicts the second collection is contained in the first; a single periodic subvariety whose $\mu_i(V,f)$ is not an eigenvalue of the global $f^*$ would refute Theorem 1.5, because the ample-cone spectrum must lie inside the eigenvalue set.

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Extended reading notes

Core claim

The paper's central claim is an exact computation of two positivity spectra. On the big cone, $Sp(f^*, \mathrm{Big}(X)) = \{\mu_i(f) \mid i=1,\dots,d\}$; on the ample cone, $Sp(f^*, \mathrm{Amp}(X)) = \bigcup_V \{\mu_i(V,f) \mid i=1,\dots,\dim V\}$, where $V$ runs over irreducible periodic subvarieties and $\mu_i(V,f)$ is the $i$-th cohomological Lyapunov exponent of $f^{r_V}|_V$, normalized by the period $r_V$. Geometrically, $\alpha$ is not in the big-cone spectrum exactly when some divisor class is stretched by $f^*$ by a factor bigger than $\alpha$, and the paper shows this happens precisely when $\alpha$ avoids the Lyapunov exponents. For the ample cone the same statement must hold after restriction to every periodic subvariety. The $\alpha=1$ cases give the equivalences with quasi-amplified and amplified endomorphisms, and the factor-inheritance statement follows from the relative dynamical degree product formula.

Load-bearing premise

In the ample-cone proof, the load-bearing step is the assertion that a nonzero class in the closure of effective divisors cannot be perpendicular to itself, so once the generated eigenspaces all lie in $Z^\perp$, the growth rate of $Z$ must fall outside the candidate spectrum; this is used without proof or citation.

Editorial extensions

If this is right

  • If the main theorems are correct, every surjective endomorphism $f$ is quasi-amplified if and only if $\mu_i(f)\neq 1$ for all $i$, so the existence of a big class with $f^*N-N$ big is equivalent to spectral hyperbolicity.
  • An endomorphism is amplified if and only if every subsystem is cohomologically hyperbolic, meaning every irreducible periodic subvariety $V$ has $\mu_i(V,f)\neq 1$ for all $i$.
  • Every cohomological Lyapunov exponent $\mu_i(f)$ is an eigenvalue of $f^*$ acting on $N^1(X)_{\mathbb R}$, since the big-cone spectrum is contained in the eigenvalue set.
  • Amplified and quasi-amplified properties pass to factors of endomorphisms.
  • The quantitative versions say $f$ is $\alpha$-quasi-amplified exactly when $\alpha$ is not a Lyapunov exponent, and $\alpha$-amplified exactly when each periodic subvariety is $\alpha^{r_V}$-quasi-amplified under $f^{r_V}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The spectrum construction is not specific to divisors: the same $C$-spectrum could be applied to invariant cones in other numerical groups, such as nef, movable, or pseudoeffective cones of higher codimension, whenever the needed recursive inequalities exist.
  • Generated positive cycles give a purely algebraic replacement for closed positive currents and are announced as the basis for follow-up work on the Kawaguchi–Silverman and Dynamical Mordell–Lang conjectures; a test of that program is whether the atomic decomposition survives in relative settings and arbitrary characteristic.
  • A cheap numerical check of the theorem is to compute, for a monomial endomorphism such as $(x,y)\mapsto (x^a,y^b)$ on $\mathbb P^1\times\mathbb P^1$, whether every exponent attached to a periodic subvariety is an eigenvalue of the global pullback; the theorem predicts this inclusion, so finding a periodic subvariety whose exponent is not a global eigenvalue would refute it.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies the numerical action of a surjective endomorphism f: X -> X of a projective variety, mainly on N^1(X)_R. It introduces an abstract notion of spectrum Sp(g,C) for a linear endomorphism g preserving an open salient cone C, and proves a characterization of this spectrum via generated eigenspaces (Theorem 1.3/2.10). It then computes Sp(f^*, Big(X)): Theorem 1.4 says this spectrum is exactly the set of cohomological Lyapunov exponents {mu_i(f)}, and Theorem 1.5 says Sp(f^*, Amp(X)) is the union, over all irreducible periodic subvarieties V, of the corresponding Lyapunov exponents mu_i(V,f). Consequences claimed include: f is quasi-amplified iff it is cohomologically hyperbolic; f is amplified iff every subsystem is cohomologically hyperbolic; and factors of amplified/quasi-amplified endomorphisms are amplified/quasi-amplified. The proof introduces generated positive cycles as an algebraic analogue of positive closed currents and proves an atomic-decomposition theorem for them.

Significance. If the results are correct, they are significant for algebraic dynamics: they give a numerical characterization of amplified and quasi-amplified endomorphisms and yield stability of these properties under factors. The abstract spectrum construction and the generated-positive-cycles machinery are interesting tools in their own right, and Theorem 5.9 and Theorem 6.5 are nontrivial. The paper does not provide machine-checked proofs; it relies heavily on two preprints by the same author, [Xie23] and [Xie24], for recursive inequalities and weak Dynamical Mordell-Lang, so the editor should verify that those references are complete and correct. As written, the proof of Theorem 1.5 contains a substantive gap in the final separation argument.

major comments (2)
  1. [Section 6.2, final paragraph of the proof of Theorem 6.1] The separation step is not valid as written. From E_S ∩ Amp(X) = ∅, Hahn-Banach applied in W = N^1(X)_R gives a nonzero linear functional φ in W^* = N_{d-1}(X)_R that is positive on Amp(X) and vanishes on E_S; this is a curve class, not automatically an element of Psef^1(X). Even if one identifies W with its dual, the next assertion 'we have β_f(Z) notin S' does not follow from E_S ⊆ Z^⊥. If β_f(Z) ∈ S, then Z ∈ E_S, and E_S ⊆ Z^⊥ only yields Z·Z = 0, which is possible for a nonzero pseudo-effective divisor. For example, take X = P^1 × P^1, f(z,w) = (z^2,w^3), S = {2}, and Z = H_1 = π_1^*O(1). Then E_S = span(Z), Z^2 = 0, E_S ⊆ Z^⊥, the functional φ = (·Z) is positive on the ample cone, and β_f(Z) = 2 ∈ S. To prove Theorem 1.5 one therefore needs either a curve-class version of Theorem 6.5 for f_* on N_{d-1}(X), or a proof that the separating functional can be represented by a divisor class whose growth is controlled by Theorem 6.5. Without such an argument, the nontrivial inclusion of Theorem 1.5 is unsupported.
  2. [Section 2, Lemma 2.8] The proof of Lemma 2.8 is garbled. It writes 'Pick c ∈ S \ Sp(g,C)' before defining S, and the final appeal to Lemma 2.3 does not establish the claimed inclusion. Since Theorem 2.10 uses Lemma 2.8 in its 'only if' direction, and Theorem 2.10 is applied in the proofs of both main theorems, the proof needs to be rewritten. The statement itself is elementary: if α ∉ Sp(g|_V, C∩V), then some N ∈ V satisfies gN − αN ∈ C∩V, hence α ∉ Sp(g,C), giving Sp(g,C) ⊆ Sp(g|_V, C∩V).
minor comments (6)
  1. [Section 2.1, Eq. (2.1)] In the speed-of-growth formula, the symbol f^n(v) should be g^n(v), since Section 2 works with a linear endomorphism g and f is reserved for the endomorphism of X.
  2. [Proof of Theorem 3.1] In the paragraph preceding (3.1), the text says 'we have α ∈ (µ_i, µ_{i+1})'; given the minimal index i with µ_{i+1} < α, the correct interval is α ∈ (µ_{i+1}, µ_i), and the subsequent choice of ε only makes sense with this ordering.
  3. [Theorems 1.5 and 6.1] The 'In other words' formulations in Theorem 1.5 and Theorem 6.1 differ: the former says f^{r_V}|_V is α-quasi-amplified, while the latter says α-amplified. Since the quasi-amplified version is the substantive equivalence and the amplified version follows trivially by taking V = X, the two statements should be aligned to avoid confusion.
  4. [Section 5.2, proof of Lemma 5.2] The alteration morphism is written 'q : Y → Vi', but V_i is not defined in the lemma; it should presumably be q : Y → Z_{x_l} or an explicitly defined V_l.
  5. [Section 5.3, proof of Theorem 5.9] In the atomic-decomposition proof, after writing ν^{Z_j}_{α_j} = v_j δ_{x_j} + β_j, the text then says 'where ǫ_j ∈ M(|Z_j|, Psef_i(Z_j))'; the symbol ǫ_j should be β_j.
  6. [Theorems 1.3 and 2.10, notation] The symbol C is used both for the good invariant cone and for the field of complex numbers; in the statement 'for every subset S ⊆ C' in Theorem 1.3/2.10, the first C should explicitly denote the complex numbers (or R_{>0} after Corollary 2.11), not the cone C. The current notation is confusing.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity (score 2): the two spectrum theorems are genuine derivations from [Xie24] inequality inputs plus new generated-cycle machinery; the author's self-citations are load-bearing but constitute real evidence whose assumptions exclude the target statements. Correctness caveat flagged separately: 'In particular, we have β_f(Z) ∉ S' in Theorem 6.1 is not implied by E_S ⊆ Z^⊥.

full rationale

The derivation chain for Theorem 1.4 is not circular: μ_i(f) is defined as the ratio λ_i/λ_{i-1} of dynamical degrees, while Sp(f^*, Big(X)) is defined via α-amplification, and the theorem is proved by applying [Xie24, Theorem 3.7] and [Xie24, Corollary 3.4] (statements about bigness of explicit classes and about mixed degrees, with assumptions that do not include the spectrum) to get α-quasi-amplification between consecutive μ_i, plus a new contradiction argument (Lemma 3.2) for the converse. Theorem 1.5's easy inclusion restricts to periodic subvarieties and applies Theorem 1.4; the hard inclusion is intended to follow from Theorem 6.5, which bounds β_f(v) for pseudo-effective v by the same union S of Lyapunov exponents of periodic subvarieties, via the decomposition of positive generated cycles into atoms (Theorem 5.9) and the limit inequality of Proposition 5.11. This chain is a genuine reduction: the cited [Xie23] results (published) and [Xie24] results (preprint) are parameter-free and do not contain Theorem 1.4 or 1.5, so per the review rules they count as real evidence; their heavy, same-author, load-bearing use is why the score is 2 rather than 0. I flag one step as a correctness risk, not as circularity: in Section 6.2, 'By Hahn-Banach theorem, there is Z ∈ Psef^1(X) \ {0} such that E_S ⊆ Z^⊥. In particular, we have β_f(Z) ∉ S.' The containment does not imply the conclusion: for X = P^1×P^1, f(z,w) = (z^2,w^3), S = {2}, Z = H_1, we have E_S = span(H_1) ⊆ Z^⊥ because H_1^2 = 0, yet β_f(Z) = 2 ∈ S; also β_f(Z) ∈ S does not force Z ∈ E_S when Z has additional components with smaller eigenvalues outside S. Thus the nontrivial inclusion in Theorem 1.5 is not fully supported as written without an additional positivity (e.g., big/ample) property of the separating class; this is an omitted proof, not a self-referential reduction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 2 invented entities

All main theorems are proven within the paper except that they import several results from the author's own prior works ([Xie23], [Xie24]) and standard theorems. There are no fitted numerical parameters; the only invented objects are formal definitions.

assumptions (7)
  • domain assumption Log-concavity of dynamical degrees: lambda_i(f)^2 >= lambda_{i-1}(f) lambda_{i+1}(f) for i=1,...,d-1
    Cited to [Tru20, Theorem 1.1(3)]; implies the sequence mu_i is decreasing, which is used throughout Section 3.
  • ad hoc to paper Recursive inequalities and mixed-degree asymptotics from [Xie24, Theorem 3.7 and Corollary 3.4]
    These are the engine of Theorem 3.1 and Proposition 6.3; the current paper does not prove them and they come from the author's own prior arXiv paper.
  • ad hoc to paper Atomicity of finite Borel measures on the constructible topology [Xie23, Theorem 1.12]
    Used in Section 4 to reduce generated cycles to countable sums of atoms in Theorem 5.9.
  • ad hoc to paper Weak Dynamical Mordell-Lang [Xie23, Theorem 1.17]
    Used in Proposition 5.11 to show an orbit spends asymptotically zero time in a divisor support.
  • standard math Hahn-Banach separation theorem
    Used in Lemma 2.3, Theorem 2.10 and Theorem 6.1 to separate cones from subspaces.
  • standard math Poincare recurrence theorem
    Used in Lemma 2.4, Lemma 3.2 and Proposition 5.11 to find recurrent subsequences of rotations.
  • standard math De Jong's alterations and Chow's moving lemma
    Used in Lemma 5.2 and Lemma 5.8 to produce effective cycles with prescribed classes in local cones.
invented entities (2)
  • Generated positive cycles G^+_i(X)
    purpose: An algebraic analogue of positive closed currents; used to prove atomic decomposition and to control growth rates of pseudo-effective classes under pullback.
    Defined in Section 5 from closures of effective cycles; no independent empirical handle outside the paper, but it is a mathematical construct rather than a physical postulate.
  • Spectrum Sp(g,C) for a good invariant cone
    purpose: A linear algebra object encoding which positive cone translations are possible; the main theorems compute it for big and ample cones.
    Definition 1.2; it is internal to the paper.

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Pith. "Pith review of Numerical action for endomorphisms." pith.science (2026). https://pith.science/paper/VM4EKNBM

@misc{pith2026250204779,
  author       = {Pith},
  title        = {Pith review of: Numerical action for endomorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VM4EKNBM}},
  note         = {Machine review of arXiv:2502.04779}
}
abstract

Let $f: X\to X$ be a surjective endomorphism of a projective variety of dimension $d$. The aim of this paper is to study the action of $f$ on the numerical group of divisors. For exmaple, I proved that $f$ is cohomologically hyperbolic if and only if it is quasi-amplified; and it is amplified if and only if every subsystem of $(X,f)$ is cohomologically hyperbolic. For the proofs, I introduced a notion of spectrum in linear algebra for an open and saliant invariant cone. I also introduce a notion of generated (positive) cycles as an algebraic analogy of (positive) closed current.

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Forward citations

Cited by 1 Pith paper

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  1. Arithmetic Degrees are Cohomological Lyapunov Multipliers

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    For surjective endomorphisms of normal projective varieties over characteristic zero fields, the arithmetic degree of any Zariski dense orbit must be a cohomological Lyapunov multiplier; if the first dynamical degree ...

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