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Log Calabi--Yau structure for endomorphisms on $\mathbf{P}^n$

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Pith's one-line read After iteration, the ramification of any self-map of the projective plane becomes log Calabi-Yau, completing the surface case of the log Calabi-Yau conjecture.

desk verdict Strong paper with a genuinely new technique; the surface case of Gongyo's conjecture is resolved conditional on an unpublished Chang–Zhang lemma. read the letter →

arxiv 2608.02114 v1 pith:RMTGOMNL submitted 2026-08-03 math.AG math.DS

Yujie Luo , Sheng Meng This is my paper · ORCID
classification math.AGmath.DS MSC 14E2214B0537F80
keywords JacobiandeterminantramificationdivisorpolarizedendomorphismvaluativetreelogcanonicalthresholdCalabi-Yaupaireigenvaluation
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

The paper establishes that a $q$-polarized endomorphism $f$ of the complex projective plane can be iterated so that the normalized ramification pair $(P^2, R_{f^s}/(q^s-1))$ is log canonical, meaning it has controlled singularities and is a log Calabi-Yau pair. Together with earlier work, this completes the proof of the log Calabi-Yau conjecture for all smooth projective surfaces, where the plane without totally invariant curves had been the last open case. The paper also shows that for a general endomorphism of $P^n$ the ramification pair is log canonical in every dimension, and that in dimension at least four the ramification divisor is singular for every endomorphism while in dimensions one to three it is smooth for a general one. Passing to an iterate is shown to be necessary, and the normalization coefficient $1/(q^s-1)$ is optimal.

What carries the argument

The load-bearing objects are eigenvaluations in the valuative tree of the superattracting surface germ and the equivariant extraction of such a valuation. For a superattracting germ $f$ with local degree equal to $c_\infty(f)^2 = q^2$, the theorem asserts a primitive divisorial valuation $v$ with $(f^2)^* v = q^2 v$; the proof splits according to a known trichotomy for superattracting two-dimensional germs. Depending on whether the unique eigenvaluation is an end, is quasi-monomial, or there is a fixed segment, the proof uses volume comparison or rationality to force $v$ to be divisorial. The Rees construction then extracts $v$: an $m_p$-primary ideal with $v$ as unique Rees valuation is normalized-blown up, producing a normal $\mathbb{Q}$-factorial model $Y$ with a finite $q$-polarized lift $g$ of $f$ and a totally invariant prime divisor $E$ with $\operatorname{ord}_E = v$ and $g^* E = q E$. Adjunction along $E$, followed by inversion of adjunction, transfers log canonicity from the induced curve endomorphism back to the plane pair.

What would settle it

Exhibit a $q$-polarized endomorphism $f$ of $P^2$ and a point $p \in E_2(f)$ for which $(P^2, R_{f^s}/(q^s-1))$ is not log canonical near $p$ for every $s \geq 1$; or, to attack the cited input directly, find a degree-$q$ endomorphism $h$ of a smooth projective curve and an lc divisor $\Theta$ such that $(C, \Theta + R_{\Theta,h^s}/(q^s-1))$ is not lc for any $s$.

Watch

Extended reading notes

Core claim

The central claim is that the only obstruction to log canonicity of the ramification pair of an endomorphism of $P^2$ is a finite, totally invariant set of points where the germ is superattracting and Jacobian orders grow at the maximal rate $q^s$. Away from this set, a uniform exponential bound with ratio $\rho < q$ makes the pair klt after sufficiently many iterations. At each exceptional point, the paper produces a primitive divisorial eigenvaluation, a valuation scaling by $q^2$ under pullback by the second iterate, and extracts it on an equivariant birational model as a totally invariant prime divisor. Adjunction to that divisor, inversion of adjunction, and a one-dimensional log-canonicity statement for the induced curve endomorphism then give local log canonicity at the point. The power map shows the normalized ramification divisor can be non-klt, so log canonical is the best possible conclusion.

Load-bearing premise

The proof at the superattracting points relies on a one-dimensional log-canonicity statement that is cited from an unpublished preprint and not proved in this paper; if that statement were false, Step 3 of the main theorem would fail at the points of $E_2(f)$.

Editorial extensions

If this is right

  • Every $q$-polarized endomorphism of a smooth projective surface satisfies the log Calabi-Yau conjecture: after some iterate $s$, the pair $(X, R_{f^s}/(q^s-1))$ is log canonical.
  • For $P^2$, the normalized ramification divisor defines a log Calabi-Yau pair with $K_{P^2} + R_{f^s}/(q^s-1)$ numerically equivalent to zero; the power map shows the klt version fails, so log canonical is sharp.
  • In arbitrary dimension, $E_{\mathrm{Jac}}(f)$ is a proper totally invariant algebraic set, and $(P^n, R_{f^s}/(q^s-1))$ is klt outside it for all sufficiently large $s$; for $n \geq 4$ the ramification divisor is singular for every $f$, while for $n \leq 3$ it is smooth for general $f$.
  • One cannot avoid iteration in general: there are endomorphisms for which the first iterate is not log canonical, and $\mathrm{lct}(P^2; R_{f^s}) \geq 1/(q^s-1)$ is the optimal lower bound.

Reading between the lines

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

  • A direct proof of the one-dimensional log-canonicity statement currently supplied by the unpublished preprint [7] would remove the most vulnerable external dependence from the argument.
  • The equivariant-extraction mechanism suggests a higher-dimensional strategy: if every superattracting fixed point admitted a quasi-monomial eigenvaluation with rational weight ratios, the same normalized Rees construction could produce a totally invariant divisor and the argument might extend to $P^n$.
  • Quantifying the trichotomy for superattracting germs would likely yield an effective bound $N(q)$ in the surface case; the paper proves existence of an iterate without giving such a bound.
  • The normalized-volume cocycle introduced in Section 6.3 points to a possible alternative route to log canonicity that avoids adjunction: if normalized volume never increases along the orbit of a valuation, the discrepancy inequality follows directly.
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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

2 major / 5 minor

Summary. The paper studies the singularities of ramification pairs (P^n, R_f) attached to q-polarized endomorphisms. The main results are: Theorem 1.1, asserting that for a general endomorphism f the pair (P^n, R_f) is log canonical and that R_f is smooth for general f when n <= 3 but singular for every f when n >= 4; Theorem 1.3, asserting that for every q-polarized endomorphism f of P^2 there is an iterate s such that (P^2, R_{f^s}/(q^s-1)) is log canonical and hence log Calabi-Yau; and Theorem 1.4, concerning the Jacobian exceptional set E_Jac(f) and klt-ness of the normalized ramification pair on its complement. The proof of Theorem 1.3 combines an estimate outside the exceptional set with a local analysis at superattracting points using the valuative tree, eigenvaluations, an equivariant extraction theorem, and an adjunction/inversion-of-adjunction argument.

Significance. If Theorem 1.3 is fully established, it completes Gongyo's conjecture for smooth projective surfaces, which is a central open problem in the birational geometry of polarized endomorphisms. The paper also proves a clean higher-dimensional generic statement (Theorem 1.1) and a structural result on the Jacobian exceptional set (Theorem 1.4). The surface proof is carefully organized and mostly self-contained: the local eigenvaluation theorem, the equivariant extraction, and the local-to-global descent in Step 3 of Theorem 1.3 fit together well. However, the proof of the key local log-canonicity statement, Proposition 4.7, depends on an unpublished arXiv preprint, and this dependency is load-bearing for the central theorem.

major comments (2)
  1. [§4.2, Proposition 4.7] The proof of local log canonicity at points of E_2(f) is not self-contained. After passing to the normalization of the extracted divisor and defining the one-dimensional pair (C, Theta), the proof invokes [7, Theorem 1.7] to conclude that (C, Theta + R_{Theta,h^s}/(q^s-1)) is lc for some s. This statement is load-bearing: without it, Step 3 of Theorem 1.3 fails at the points of E_2(f), because adjunction from (Y,E) gives no reason for Theta to vanish, so the parenthetical 'see also [4, Remark 2.8] when Theta = 0' does not cover the general case. Since [7] is an unpublished arXiv preprint, the manuscript as submitted does not independently establish Theorem 1.3; the authors should either supply a proof of the required one-dimensional statement inside the paper or replace [7] by a refereed published reference.
  2. [§4.2, Remark 4.8] The alternative proof of Proposition 4.7 advertised in Remark 4.8 depends on [7, Theorem 1.16(2)] applied to the g-pair (Y,0), which is an even heavier reliance on the same unpublished preprint. Remark 4.9 similarly suggests simplifying Step 3 by applying [7, Theorem 1.16(2)]. This reinforces the need either to make the local argument self-contained or to cite a published version of [7]; until then the claim that Theorem 1.3 'completes' Gongyo's conjecture for smooth projective surfaces remains conditional on an unrefereed external result.
minor comments (5)
  1. [§1, Abstract] The abstract states the log canonical threshold inequality for R_{f^s} as a global statement, but the proof establishes it pointwise via local log canonical thresholds; it would be helpful to phrase the inequality explicitly as a global consequence of the local estimates.
  2. [§2.3, Proposition 2.9] In Proposition 2.9(iii), the notation vol(nu) is defined by a lim sup, but the equality vol(nu) = alpha(nu)^{-1} is asserted from [13, Remark 3.33]; this could be stated as a citation rather than a definition to avoid ambiguity about the existence of the limit.
  3. [§4.2, Proof of Theorem 4.4] The phrase 'taking powers of these integrally equivalent ideals' is slightly nonstandard; it would be clearer to say 'taking powers of the equality of integral closures' and then to pass to integral closures again.
  4. [§4.3, Step 3] In Step 3, the statement 'Since df_p is nilpotent, by the chain rule d(f^2)_p = 0' is correct only because the germ is two-dimensional, where a nilpotent 2x2 matrix squares to zero. This dimension dependence could be mentioned explicitly to avoid confusion.
  5. [§5, Remark 5.2] Remark 5.2 introduces the functions eta_s(x) and asserts that they form an analytic submultiplicative cocycle, but no proof or reference is given for this assertion. Since the remark is not used later, this is a presentation issue, but it should be justified or explicitly labeled as a conjecture.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the proof is conditional on an external unpublished result [7], which is a completeness risk, not a self-referential deduction.

full rationale

I walked the derivation chain of Theorem 1.3. The global degree computation in Step 1, the reduction to the finite exceptional set E2(f) via Proposition 2.14(iii), the existence of the divisorial eigenvaluation in Theorem 4.2, the equivariant extraction in Corollary 4.5, and the discrepancy descent in Step 3 all use published external ingredients or arguments contained in the paper. The invocation of [32, Theorem 4.2] for the E1(f) nonempty case is a published theorem by the second author; it handles a disjoint case and is a legitimate dependency rather than a circular reuse of the claim being proved. The example from [31] is auxiliary. Proposition 4.7 does quote [7, Theorem 1.7], and Remark 4.8 quotes [7, Theorem 1.16(2)], from an unpublished preprint; this makes the proof conditional and externally dependent, but it does not make the argument circular, because those are external one-dimensional log-canonicity statements and the paper supplies the adjunction computation and the crepant identity that genuinely reduce the surface claim to them. No fitted parameter is renamed as a prediction, and no definition is made in terms of the target result. I therefore cannot exhibit any step in which the paper's own equations or self-citations force the conclusion by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests almost entirely on established theorems about valuations, singularities, and holomorphic dynamics; the paper's own contribution is the synthesis and the eigenvaluation-extraction method. No free parameters are fitted to data. The only fragile support is the unpublished Chang-Zhang theorem [7].

assumptions (8)
  • domain assumption Favre-Jonsson valuative tree theory: structure of V, skewness alpha(nu), volume identity vol(nu)=alpha(nu)^{-1}, and eigenvaluation theorems (Propositions 2.9 and 2.10)
    Foundational for Section 2.3 and Section 4.1; used in Theorem 4.2 to control cases and derive the volume contradiction.
  • domain assumption Gignac-Ruggiero trichotomy (Proposition 2.11): for a superattracting surface germ, either a unique end eigenvaluation, a unique quasi-monomial eigenvaluation, or a maximal fixed segment I with c(f^2,nu)=c_infty(f)^2 on I
    The three-case proof of Theorem 4.2 follows this trichotomy; the constancy on the fixed segment is essential in Case 3.
  • domain assumption Favre-Jonsson classification of E_1(g), E_2(g), and uniform bound ord_x(R_{g^s}) <= C rho^s off E_1 union E_2 (Proposition 2.14, from [12])
    Used in Step 2 of Theorem 1.3 to obtain klt on the complement of E_2(f).
  • domain assumption Zariski's theorem: every primitive divisorial valuation on a smooth surface germ is the unique Rees valuation of a simple complete m-primary ideal
    Used in Corollary 4.5 to make Theorem 4.4's Rees-valuation hypothesis automatic for surfaces.
  • domain assumption Docampo's determinantal threshold formula: (Mat_{n+1}, det=0) has nonnegative log discrepancies
    Used in Theorem 1.1(3) to prove (P^n,R_f) is lc for general f.
  • standard math Kawakita's inversion of adjunction [24, Theorem]: lc restriction to a reduced divisor implies local lc in a neighborhood
    Used in Proposition 4.7 to pass from the curve pair (C, Theta + R_{Theta,h^s}/(q^s-1)) to lc of (Y, E+B_s/(q^s-1)) along E.
  • domain assumption Chang-Zhang [7, Theorem 1.7] (unpublished preprint): for a q-polarized endomorphism h of a smooth curve C, the pair (C, Theta + R_{Theta,h^s}/(q^s-1)) is lc for some s under the stated adjunction setup
    Load-bearing in Proposition 4.7; not proved in this paper and depends on a preprint.
  • domain assumption Broustet-Horing [5, Corollary 3.3]: (Y,E) is log canonical when E is a reduced totally invariant divisor under a polarized endomorphism of a smooth variety
    Used in Proposition 4.7 to establish lc of (Y,E) before adjunction.

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Pith. "Pith review of Log Calabi--Yau structure for endomorphisms on $\mathbf{P}^n$." pith.science (2026). https://pith.science/paper/RMTGOMNL

@misc{pith2026260802114,
  author       = {Pith},
  title        = {Pith review of: Log Calabi--Yau structure for endomorphisms on $\mathbfP^n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RMTGOMNL}},
  note         = {Machine review of arXiv:2608.02114}
}
abstract

Let $f:\mathbf{P}^n\to\mathbf{P}^n$ be a $q$-polarized endomorphism, where $q>1$, and let $R_f$ be its ramification divisor. We study the singularities of the ramification pair $(\mathbf{P}^n,R_f)$. We show that, for a general $f$, the pair $(\mathbf{P}^n,R_f)$ is log canonical. When $n=2$, we prove that there exists an integer $s\geq1$ such that the log canonical threshold $\mathrm{lct}(\mathbf{P}^2;R_{f^s})\geq1/(q^s-1)$. The passage to an iterate is necessary in general, and the lower bound is optimal. In particular, $(\mathbf{P}^2,R_{f^s}/(q^s-1))$ is a log Calabi--Yau pair, completing the proof of Gongyo's conjecture for smooth projective surfaces.

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