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Numerical spectrums control Cohomological spectrums

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For any surjective self-map of a smooth projective variety, the growth rates on numerical cycle classes and on l-adic cohomology coincide, forcing all eigenvalues of polarized maps to have the predicted size.

desk verdict A major theorem with one missing continuity argument in the general finite-correspondence statement; the endomorphism corollaries and the Tate conjecture application look solid and deserve refereeing. read the letter →

arxiv 2412.01216 v2 pith:EMGFJU6Q submitted 2024-12-02 math.AG

Junyi Xie This is my paper · ORCID
classification math.AG MSC 14C2514F2014G15
keywords numericalequivalencel-adiccohomologyspectralradiuscohomologicalcorrespondencesdynamicaldegreesendomorphismsfixedpointcountingpolarized
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 establishes that, for a smooth projective variety over any field, the numerical spectrum of a surjective endomorphism controls its cohomological spectrum: the spectral radius of $f^*$ on the $i$-th numerical group $N^i(X)$ equals the spectral radius on the $2i$-th $l$-adic cohomology group, and the analogous equality holds for the smallest eigenvalues. The proof works for a broader class of objects—finite cohomological correspondences, meaning effective dimension-$d$ cycles in $X\times X$ whose projection to the first factor is finite, viewed through their cohomology classes—and the conclusion is that the numerical polygon and the cohomological polygon, the concave hulls of the log spectral radii, are identical. The payoff is a generalization of the Riemann hypothesis over finite fields: if $f$ is $q$-straight, meaning its numerical radii are $1,q,\ldots,q^d$, then every eigenvalue of $f^*$ on $H^j$ has modulus $q^{j/2}$, settling a conjecture proposed in 1964. This also yields asymptotic formulas for counting fixed points of iterates and for a moving-target variant. The result matters because it derives purely cohomological information from numerical intersection data, which are easier to compute and are defined over any field.

What carries the argument

The argument is carried by the two polygons together with a tilting trick. After a standard reduction to the case of a finite ground field, the proof perturbs a finite correspondence $c$ to $c_m=c+m^{-1}\Delta$, where $\Delta$ is the diagonal, so that every numerical spectral radius is positive. It then considers twisted correspondences $c_{s,t}=\Phi_q^s\circ c^t$, where $\Phi_q$ is the correspondence induced by the $q$-Frobenius. Because Frobenius eigenvalues have size $q^{j/2}$, the polygons transform linearly: $NP_{c_{s,t}}(x)=\frac12 s(\log q)x+t\,NP_c(x)$ and $CP_{c_{s,t}}(x)=\frac12 s(\log q)x+t\,CP_c(x)$. Choosing the slope $s/t$ makes a prescribed vertex of $CP_c$ the unique maximum of $CP_{c_{s,t}}$; a trace estimate then identifies the maximum of $CP_{c_{s,t}}$ with the maximum of $NP_{c_{s,t}}$, forcing equality at every vertex of the original polygons. The other load-bearing tool is the formula $\beta_i(c)=\lim_{n\to\infty}((c^n)_*L^i\cdot L^{d-i})^{1/n}$, which converts numerical spectra into intersection numbers and enables the pseudo-effective cone estimates that control growth.

What would settle it

Take a concrete finite correspondence $c$ on a smooth projective variety over a finite field and compute the characteristic polynomials of $c^*$ on $N^i(X)\otimes\mathbb{R}$ and on $H^{2i}(X_{\bar k},\mathbb{Q}_l)$ for some $i$; if the spectral radius on $H^{2i}$ is strictly larger than that on $N^i$, the theorem is false. The random-product example $c=(f_1+f_2)/2$ on projective space, with $f_1,f_2$ of distinct algebraic degrees, is a natural test case since the paper shows its numerical radii are not log-concave.

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Extended reading notes

Core claim

The central discovery is the equality of two polygons attached to a cohomological correspondence $c$ on a smooth projective variety $X$ of dimension $d$. The cohomological polygon $CP_c$ is the minimal concave function on $[0,2d]$ that lies above all numbers $\log\alpha_j(c)$, where $\alpha_j(c)$ is the spectral radius of $c^*$ on $H^j(X_{\bar k},\mathbb{Q}_l)$; the numerical polygon $NP_c$ is the minimal concave function lying above $\log\beta_i(c)$ at the even points $2i$, where $\beta_i(c)$ is the spectral radius on the numerical group $N^i(X)\otimes\mathbb{R}$. Since cohomological equivalence implies numerical equivalence, one always has $NP_c\le CP_c$. The theorem proves that for every finite correspondence $c$ the two polygons are equal, $NP_c=CP_c$. For an endomorphism $f$, this gives $\log\beta_i(f)=\log\alpha_{2i}(f)$ for every $i$, together with the corresponding equality for the minimal eigenvalues. Consequently, if $f$ is $q$-straight—its numerical radii are $1,q,\ldots,q^d$, as happens for $q$-polarized endomorphisms—then every eigenvalue of $f^*$ on $H^j$ has absolute value $q^{j/2}$ for every $j=0,\ldots,2d$. The paper reads this as a proof of a 1964 conjecture on algebraic cycles, and it derives from the polygon equality asymptotic formulas for counting fixed points, including a moving-target version.

Load-bearing premise

The load-bearing premise is that the polygon equality survives the perturbation limit $c_m=c+m^{-1}\Delta\to c$ and that the established Frobenius eigenvalue bounds over finite fields hold; if either gives way, the equality $NP_c=CP_c$ for the original correspondence is unsupported.

Editorial extensions

If this is right

  • For a surjective endomorphism of a smooth projective variety over any field, the spectral radii on $N^i$ and on $H^{2i}$ coincide for every $i$, and so do the minimal spectral radii; numerical data determine the cohomological spectrum in even degrees.
  • If $f$ is $q$-straight, then every eigenvalue of $f^*$ on $H^j$ has modulus $q^{j/2}$ for every $j$, extending the classical Riemann hypothesis over finite fields from Frobenius to arbitrary maps with this numerical growth.
  • For an int-amplified endomorphism (one with $\beta_d>\beta_{d-1}$), the number of fixed points of $f^n$ is $\beta_d^n+O((\beta_d\beta_{d-1})^{n/2})$; in the straight case this is $q^{dn}+O(q^{(d-1/2)n})$.
  • The moving-target estimate holds: if a sequence of maps $h_n$ grows more slowly than $\beta_d/\beta_{d-1}$, then the number of solutions to $f^n(x)=h_n(x)$ is $q^{dn}+o((\beta_d\beta_{d-1})^{(1+\epsilon)n})$.

Reading between the lines

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

  • Editorial inference: the polygon equality may hold for correspondences that are only generically finite rather than everywhere finite; the perturbation argument via the diagonal suggests finiteness is a convenience rather than the essential condition, but the paper does not prove this.
  • Editorial inference: in the random-product example, the theorem predicts that the cohomological spectral radii obey the same non-log-concave pattern forced by the numerical radii; computing explicit examples in small dimension would test how sharp the polygon statement is.
  • Editorial inference: the methods point toward semisimplicity and equidistribution questions for the action of correspondences on cohomology, but the paper does not address those; a natural next step is to ask whether the polygon equality plus log-concavity implies a Hodge-theoretic or motivic refinement.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies, for a smooth projective variety X over a field k and a finite cohomological correspondence c, the relation between the numerical spectral radii β_i(c) on N^i(X)⊗R and the l-adic cohomological spectral radii α_j(c) on H^j(X, Q_l)⊗C. The main theorem (Theorem 1.14 = Theorem 2.12) asserts that the least concave majorants of the sequences log β_i(c) and log α_j(c), called the numerical and cohomological polygons, coincide for every finite correspondence. For a surjective endomorphism f, this yields Corollary 2.16 comparing the even-degree cohomological spectral radii with the numerical ones, and for odd degrees gives two-sided bounds in terms of the numerical polygon. Consequences include: for a q-straight endomorphism (in particular, a q-polarized endomorphism), every eigenvalue of f^* on H^j has modulus q^{j/2}, generalizing Deligne's Weil conjecture and proving a conjecture of Tate; estimates for the number of fixed points of int-amplified endomorphisms; and a 'moving target' fixed-point count. The proof reduces to finite fields, introduces a perturbation c_m = c + m^{-1}Δ, verifies the equality for the perturbed correspondences, and then asserts that one may assume β_i(c)>0. The central derivation otherwise relies on Deligne's theorem for Frobenius, Truong's log-concavity theorem, and standard étale cohomology facts, with no free parameters or circular definitions.

Significance. If the main theorem is correct, it is a substantial advance: it provides a general numerical control of cohomological spectra for finite correspondences in arbitrary characteristic, recovers and generalizes known results over C, and proves a long-standing conjecture of Tate for polarized endomorphisms. The fixed-point counting consequences are also of interest to arithmetic dynamics. The paper is transparent about its reliance on Deligne's Weil conjectures, and the structure of the argument is elegant, using the Frobenius correspondence to rotate the polygon and a perturbation to force positivity of numerical radii. The examples of random products and extensions illustrate the scope of the correspondence formalism. However, the proof of the main theorem as written contains a gap at the perturbation step, and an application statement contains an undefined symbol, so the current version does not fully support all of its claims.

major comments (3)
  1. [§2.5, Theorem 2.12] The perturbation argument is not closed. The proof introduces c_m := c + m^{-1}Δ, proves β_i(c_m) > 0, and then states 'after replacing c by c_m, we may assume β_i(c) > 0'. The subsequent argument proves the polygon equality for the perturbed c_m, not for the original c. The theorem is stated for every finite c, including cases with β_i(c) = 0 (e.g., c = cl(X × {p}) on P^1). To conclude NP_c = CP_c, one must supply a limiting or continuity argument showing that NP_{c_m} → NP_c and CP_{c_m} → CP_c as m → ∞. This is plausible because the spectral radii α_j(c_m) and β_i(c_m) are spectral radii of matrices depending continuously on m, and the least concave majorant of finitely many points is continuous in the data; but the proof neither states nor proves this. This gap affects the central claim and must be repaired.
  2. [§2.6, Proposition 2.17] The statement of Proposition 2.17 uses the symbol q in the asymptotic formula '#{f^n(x) = h(x)} = qdn + o(...)' without defining q. In context, q appears to denote β_d(f) or deg(f), but this is not said. In the proof, the leading term is written as μ_d^n with μ_d := β_d(f)/β_{d-1}(f), which is inconsistent with the claimed qdn and with Corollary 1.7, where the leading growth is β_d^n. The notation and the leading term must be corrected and clarified.
  3. [§2.5, proof of Theorem 2.12] The invocation of Lemma 2.9 to show β_i(c_m) ≥ β_i(m^{-1}Δ) is not literally valid: Lemma 2.9 assumes both correspondences are bi-finite, while c_m = c + m^{-1}Δ is only known to be finite, not bi-finite, when c is merely finite. The inequality is still likely true by a direct argument using Lemma 2.8 and the effectivity of c_m^n - (m^{-1}Δ)^n, but this needs to be written out explicitly rather than imported from Lemma 2.9.
minor comments (5)
  1. [Abstract] There is a typo: 'arbitary' should be 'arbitrary'.
  2. [§2.5, proof of Theorem 2.12] The phrase 'spreading our argument' should be 'spreading out argument'.
  3. [§2.6, Proposition 2.17] There are typos: 'int-amplified ampliefied endmorphism' should be 'int-amplified endomorphism'.
  4. [§2.5, proof of Theorem 2.12] The statement 'After replacing c by c_m, we may assume β_i(c) > 0' is misleading because the proof never returns to the original c; it would be clearer to say that the equality is first proved under the additional positivity assumption and then extended by a limiting argument.
  5. [§2.6, proof of Proposition 2.17] The final bound 'by Corollary 2.16, for every i = 0, ..., 2d−1, we get α_i(f) ≤ (β_{d−1}β_d)^{1/2}' uses the concavity of the numerical polygon for the odd-indexed values, not the literal inequalities of Corollary 2.16 for all i; the reasoning should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof derives NP_c=CP_c from Deligne's Weil conjecture, the Lefschetz trace formula, and external log-concavity results; the perturbation step is a proof gap, not a circular reduction.

full rationale

Theorem 2.12 is a genuine derivation from external results. The comparison between numerical and cohomological spectra is one-sided from the start (NP_c <= CP_c because numerical groups are sub-quotients of cohomology), and the reverse inequality is obtained by twisting c with the q-Frobenius Phi_q, whose eigenvalues are known by Deligne [Del74], then applying Corollary 2.11 (max alpha = max beta under Condition (A), via Lemmas 2.3 and 2.10) to shifted correspondences c_{s,t}. No fitted parameter, renormalization, or defining identity forces the equality. The log-concavity of beta_i(f) is cited from Truong and Dang and is used only to identify NP_f(2i) with log beta_i(f), not to assume the theorem. Corollary 1.5 is not circular: Deligne's theorem controls Frobenius, while the conclusion concerns arbitrary q-straight endomorphisms. The one genuine issue in the manuscript is the perturbation step in the proof of Theorem 2.12: the text says 'We may approximate c by c_m := c + m^{-1} Delta ... After replacing c by c_m, m >= 1, we may assume that beta_i(c) > 0' but supplies no explicit m -> infinity limit or continuity argument returning from c_m to the original c. That is an omitted proof of a likely repairable analytic step, not a reduction of the conclusion to its own input, so it does not constitute circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

All inputs are established theorems or standard facts; the paper introduces no fitted constants and no new entities. The main external inputs are Deligne's proof of the Weil conjectures, Truong's log-concavity theorem for numerical spectral radii, and standard etale cohomology reductions. The perturbation c_m near the diagonal is a proof device, not a free parameter.

assumptions (4)
  • domain assumption Deligne's Weil conjecture [Del74]: Frobenius eigenvalues on H^j(X, Q_l) have modulus q^{j/2}.
    Used in Theorem 2.12 to compute CP and NP of the Frobenius-twisted correspondences c_{s,t} and to get Condition (A). It is an established theorem, not an ad hoc postulate.
  • domain assumption Log-concavity of numerical spectral radii β_i(f) (Truong [Tru20, Theorem 1.1(3)]).
    Used in Section 2.6 to assert endomorphisms are numerically log-concave and to derive the bounds on odd cohomological radii in Theorem 1.2.
  • standard math Smooth-proper base change and spreading out reduce the statement from arbitrary k to finite fields.
    Invoked in Fact 2.1 and Theorem 2.12; standard in etale cohomology but not proved in the paper.
  • standard math Poincaré duality and the projection formula relate spectra of f, its transpose, and deg f.
    Gives Proposition 2.15 and the lower spectral radii formulas.

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Pith. "Pith review of Numerical spectrums control Cohomological spectrums." pith.science (2026). https://pith.science/paper/EMGFJU6Q

@misc{pith2026241201216,
  author       = {Pith},
  title        = {Pith review of: Numerical spectrums control Cohomological spectrums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EMGFJU6Q}},
  note         = {Machine review of arXiv:2412.01216}
}
abstract

Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $\tau: \mathbb{Q}_l\to \mathbb{C}$ be any field embedding. Let $f: X\to X$ be a surjective endomorphism. We show that for every $i=0,\dots,2d$, the spectral radius of $f^*$ on the numerical group $N^i(X)\otimes \mathbb{R}$ and on the $l$-adic cohomology group $H^{2i}(X_{\overline{\mathbf{k}}},\mathbb{Q}_l)\otimes \mathbb{C}$ are the same. As a consequence, if $f$ is $q$-polarized for some $q>1$, we show that the norm of every eigenvalue of $f^*$ on the $j$-th cohomology group is $q^{j/2}$ for all $j=0,\dots, 2d.$ This generalizes Deligne's theorem for Weil's Riemann Hypothesis to arbitary polarized endomorphisms and proves a conjecture of Tate. We also get some applications for the counting of fixed points and its ``moving target" variant. Indeed we studied the more general actions of certain cohomological coorespondences and we get the above results as consequences in the endomorphism setting.

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