{"id":"3393ecc4-be95-429b-9f2f-ecd21241bab7","arxiv_id":"2502.04779","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The spectrum of the pullback action on the big and ample divisor cones is exactly the cohomological Lyapunov exponents of the map and of its periodic subvarieties.","lead":"This paper introduces a spectrum for invariant cones and computes it for the big and ample cones of divisor classes under endomorphisms of projective varieties. It matters because it links amplified endomorphisms to cohomological hyperbolicity and builds tools aimed at open problems in algebraic dynamics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 6.1 has an unjustified separation step: the claim 'β_f(Z) not in S' is not implied by E_S ⊆ Z^⊥ and is false for self-orthogonal boundary pseudo-effective classes.","rationale":"The reader's weakest assumption identifies the same final step of Theorem 6.1, and the concern is real: the implication from E_S ⊆ Z^⊥ to β_f(Z) ∉ S is not justified. I partially agree because my reading adds that the separating object is naturally a functional in the dual space N_{d-1}(X)_R, not an element of Psef^1(X), so there is a space mismatch on top of the self-intersection gap. This is load-bearing: Theorem 1.5 is one of the paper's central claims, and its proof's only nontrivial inclusion relies on this separation argument. I do not see evidence that the theorem itself is false; rather, this version lacks a proof of a necessary positivity or duality statement. Hence the reader's CONDITIONAL verdict remains appropriate, and I recommend no change to the verdict. The concrete test above would settle whether the gap is merely cosmetic (fillable by an easy duality argument) or substantive (requiring new ideas).","tokens_in":28134,"tokens_out":10018,"duration_ms":124113,"concrete_test":"Check the final paragraph of Theorem 6.1 by making the dual space explicit: replace the claimed Z ∈ Psef^1(X) with the separating functional φ ∈ (N^1(X)_R)^* = N_{d-1}(X)_R and verify whether an analogue of Theorem 6.5 holds for f_* on N_{d-1}(X). Independently, run the explicit diagonal counterexample X = P^1×P^1, f(z,w) = (z^2, w^3), S = {2}, Z = H_1: confirm that E_S ⊆ Z^⊥ holds, yet β_f(Z) = 2 ∈ S, so the inference fails. If the check succeeds in producing the counterexample, the proof requires an additional argument not present in the manuscript.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the final paragraph of Section 6.2, the proof supposes Sp(f^*, Amp(X)) ⊄ S and applies Theorem 2.10 to obtain E_S ∩ Amp(X) = ∅. Hahn-Banach separation on W = N^1(X)_R with C = Amp(X) yields a functional in W^* = N_{d-1}(X)_R, i.e., a curve class, not an element of Psef^1(X). Even if one identifies N^1 with its dual (via Poincaré duality or an auxiliary inner product), the next sentence 'In particular, we have β_f(Z) not in S' does not follow from E_S ⊆ Z^⊥. If β_f(Z) ∈ S, then Z ∈ E_S, so E_S ⊆ Z^⊥ only gives Z·Z = 0; a nonzero pseudo-effective divisor can be self-orthogonal. 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 f^*Z = 2Z, so β_f(Z) = 2 ∈ S; moreover Z^2 = 0, and the line spanned by Z is contained in Z^⊥. The linear functional given by Z is positive on the ample cone and vanishes on E_S = span(Z), precisely matching the separation setup. Thus the asserted inference is false in the paper's own framework. The contradiction with Theorem 6.5 therefore collapses unless an additional positivity property of Z is proved (for instance, that the separating class can be chosen big/ample, or that the correct dual cone of curve classes is used with an analogue of Theorem 6.5 for f_* on N_{d-1}(X)). This is the only step establishing the nontrivial inclusion of Theorem 1.5, so the central claim is not fully supported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28637,"tokens_out":17338,"duration_ms":162937,"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":[{"comment":"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.","section":"Section 6.2, final paragraph of the proof of Theorem 6.1"},{"comment":"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).","section":"Section 2, Lemma 2.8"}],"minor_comments":[{"comment":"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.","section":"Section 2.1, Eq. (2.1)"},{"comment":"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.","section":"Proof of Theorem 3.1"},{"comment":"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.","section":"Theorems 1.5 and 6.1"},{"comment":"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.","section":"Section 5.2, proof of Lemma 5.2"},{"comment":"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.","section":"Section 5.3, proof of Theorem 5.9"},{"comment":"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.","section":"Theorems 1.3 and 2.10, notation"}],"recommendation":"major_revision","confidential_remarks":"The central claims are potentially important, but the proof of Theorem 1.5 has a real gap in the Hahn-Banach step, and Lemma 2.8's proof is not usable as printed. The paper also depends on two preprints by the same author ([Xie23] and [Xie24]) for load-bearing results; I have not verified those, and the editor should ensure they are in stable, correct form. A revision that supplies a curve-class analogue of Theorem 6.5, or otherwise justifies the separation step, may make the paper publishable, but the current version is not."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a serious paper with a real gap in the proof of Theorem 1.5/6.1. The spectrum framework and the generated-cycle machinery are new and likely useful, and the big-cone result (Theorem 1.4) is plausible modulo the author's earlier preprint. But the final Hahn-Banach step in Section 6.2 is wrong as written, and the paper currently does not prove the ample-cone spectrum. It still deserves a serious referee.\n\nWhat's good: the definition of Sp(g,C) for a good invariant cone is clean, and Theorem 1.3/2.10 is a nice linear-algebra fact. The application to Big(X) is conceptually simple: cohomological Lyapunov exponents are exactly the spectrum against the big cone. The generated positive cycles G^+_i(X) are a genuinely new object, and the atomic decomposition (Theorem 5.9) is a strong tool, a believable algebraic analogue of positive closed currents and Siu's decomposition. The periodic-subvariety statement in Theorem 1.5 is the right guess, and Corollary 1.6 on factors is a clean consequence.\n\nSoft spots: (1) The gap. After E_S ∩ Amp(X) = ∅, Hahn-Banach on W = N^1(X)_R gives a functional in the dual, i.e., a curve class, not a divisor class in Psef^1(X). Even if you identify the dual with N^1 via a polarization, the inference 'β_f(Z) ∉ S' does not follow from E_S ⊆ Z^⊥. The stress-test example is correct: on P^1×P^1 with f = (z^2, w^3), S = {2}, Z = H_1, we have E_S = span(Z), Z^2 = 0, Z is positive on Amp, and β_f(Z) = 2 ∈ S. The contradiction with Theorem 6.5 collapses. A fix needs either a dual statement for f_* on curve classes or an argument that the separating class can be chosen big. (2) Lemma 2.8's proof is garbled: it picks c ∈ S \\ Sp(g,C) with S undefined and does not prove the claimed inclusion. Theorem 2.10 may still be true, but the proof as printed is not. (3) Theorem 3.1 has a reversed interval (μ_i, μ_{i+1}) and leans on [Xie24] for two key inequalities, so the big-cone theorem is conditional on that preprint. There is also a notational blur between Psef^1 classes and their representatives in G^+_1 in Section 6; this is probably fixable.\n\nBottom line: the framework is good, the main theorems are probably true, but this version does not fully support Theorem 1.5. Send it out; a referee should focus on Section 6.2.","headline":"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.","tokens_in":29078,"tokens_out":8408,"would_cite":true,"duration_ms":95035,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37P55","14J40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["endomorphisms of projective varieties","dynamical degrees","cohomological Lyapunov exponents","amplified endomorphisms","quasi-amplified endomorphisms","spectrum of an invariant cone","generated positive cycles","periodic subvarieties"],"falsifier":"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.","tokens_in":27912,"feed_emoji":"📐","tokens_out":12842,"duration_ms":126491,"temperature":0.7,"pith_summary":"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.","feed_headline":"Big and ample cones have Lyapunov spectra","feed_subtitle":"Cohomological Lyapunov exponents and periodic subvarieties decide which divisor classes can be stretched.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $f^{r_V}$."],"supporting_citations":[{"why":"Supplies the recursive inequalities used to show that every $\\alpha$ outside the Lyapunov set is quasi-amplified.","marker":"[Xie24, Theorem 3.7]"},{"why":"Computes the mixed degrees used in the big-cone proof and the growth-rate lemmas.","marker":"[Xie24, Corollary 3.4]"},{"why":"Gives the log-concavity of dynamical degrees that makes the Lyapunov exponents decreasing.","marker":"[Tru20, Theorem 1.1(3)]"},{"why":"Provides the big and pseudo-effective cone results used to ensure local pseudo-effective cones have nonempty interior.","marker":"[Dan20, Theorem 3.3.3]"},{"why":"De Jong alteration theorem is used to construct cycles with prescribed numerical classes in the generated-cycle machinery.","marker":"[dJ96]"},{"why":"Gives the atomic description of measures on the constructible topology underlying the atomic decomposition of generated cycles.","marker":"[Xie23, Theorem 1.12]"},{"why":"Weak dynamical Mordell-Lang is used to rule out negative intersection with divisors along dense orbits.","marker":"[Xie23, Theorem 1.17]"},{"why":"Relative dynamical degree product formula is used to prove that amplification and quasi-amplification pass to factors.","marker":"[DN11]"}],"fun_headline_variants":["Cone spectra reveal Lyapunov exponents","Divisor spectra match Lyapunov exponents","Lyapunov spectra characterize amplified maps","Numerical action: spectra on big and ample cones"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Cone spectra reveal Lyapunov exponents","Divisor spectra match Lyapunov exponents","Lyapunov spectra characterize amplified maps","Numerical action: spectra on big and ample cones"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000314,"raw_usage":{"total_tokens":1751,"prompt_tokens":884,"completion_tokens":867,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":811}},"tokens_in":500,"tokens_out":867,"duration_ms":9382,"temperature":1.0,"reasoning_tokens":811,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T21:31:38.883021+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}