{"id":"21bb07bb-f3f7-44c9-bece-44704dd5c355","arxiv_id":"2608.02114","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every q-polarized endomorphism of P^2, some iterate (P^2, R_{f^s}/(q^s-1)) is log canonical, settling Gongyo's conjecture for smooth projective surfaces.","lead":"The paper proves that for any degree-q endomorphism of the complex projective plane, a suitable iterate of its ramification divisor, normalized, forms a log Calabi-Yau pair, completing Gongyo's conjecture for smooth projective surfaces. The proof blends the valuative tree of local valuation dynamics with birational extraction of eigenvaluations, a method that should apply to higher-dimensional problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The surface-case proof is conditional on an unpublished preprint: Proposition 4.7 invokes [7, Thm 1.7], and the alternative proof in Remark 4.8 depends on [7, Thm 1.16(2)]; until those statements are verified or made self-contained, Theorem 1.3 is not independently established.","rationale":"The reader's weakest-assumption analysis identifies exactly the load-bearing vulnerability I find. I read the construction of the divisorial eigenvaluation (Theorem 4.2), the equivariant extraction (Theorem 4.4 and Corollary 4.5), and the descent to P^2 in Step 3 as coherent, and the published ingredients [12, 14, 16, 32] support them. The only unresolvable gap is the invocation of [7] in Proposition 4.7, with Remark 4.8 offering no independent escape because it relies on [7, Theorem 1.16(2)] from the same preprint. I found no circularity, no data-fitting, and no internal inconsistency. Therefore the appropriate verdict remains CONDITIONAL: the theorem is plausible and largely self-contained, but its completion claim depends on an unrefereed external statement that should be verified or proved in an appendix before the result can be accepted unconditionally.","tokens_in":20208,"tokens_out":22397,"duration_ms":212135,"concrete_test":"Verify [7, Theorem 1.7] directly from arXiv:2509.17927v1, and independently re-derive the special case needed in Proposition 4.7: for h:C->C of degree q and an effective lc R-divisor Theta, prove that there exists s>=1 with (C, Theta + R_{Theta,h^s}/(q^s-1)) log canonical, using only the Hurwitz formula and standard curve-pair discrepancy computations. If this derivation cannot be supplied, or if [7, Theorem 1.7] itself depends on the unpublished surface theorem, then Proposition 4.7 is unsupported and Theorem 1.3 should remain CONDITIONAL (or be marked UNVERDICTED) until the authors include a self-contained proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central proof of Theorem 1.3 is not self-contained at the key local step. Proposition 4.7, which supplies local log canonicity at the superattracting points p in E2(f), finishes by invoking [7, Theorem 1.7]: for a degree-q endomorphism h of a smooth curve C and an lc divisor Theta, the pair (C, Theta + R_{Theta,h^s}/(q^s-1)) is lc for some s. Without that statement, Step 3 of Theorem 1.3 fails at points of E2(f), so Theorem 1.3 is unproved. The alternative offered in Remark 4.8 relies on [7, Theorem 1.16(2)] for the surface (Y,0), which is an even heavier dependence on the same unpublished preprint. The parenthetical 'see also [4, Remark 2.8] when Theta = 0' does not cover the general case, because the Theta produced by adjunction from (Y,E) need not vanish. This is not an internal inconsistency in the authors' own construction: the eigenvaluation extraction (Theorem 4.2), equivariant extraction (Corollary 4.5), and the descent inequalities in Step 3 are coherent and use published ingredients. But the claim that Theorem 1.3 'completes' Gongyo's conjecture for smooth projective surfaces is contingent on an unrefereed external result that is neither proved in the paper nor otherwise available to the reader.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20534,"tokens_out":6971,"duration_ms":66312,"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":[{"comment":"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.","section":"§4.2, Proposition 4.7"},{"comment":"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.","section":"§4.2, Remark 4.8"}],"minor_comments":[{"comment":"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.","section":"§1, Abstract"},{"comment":"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.","section":"§2.3, Proposition 2.9"},{"comment":"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.","section":"§4.2, Proof of Theorem 4.4"},{"comment":"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.","section":"§4.3, Step 3"},{"comment":"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.","section":"§5, Remark 5.2"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the dependence of Theorem 1.3 on the unpublished preprint [7]. The rest of the manuscript is strong and the internal arguments are coherent. If [7] is already accepted or if the authors can provide a self-contained proof of the one-dimensional lemma, the paper would be a solid candidate for publication. The referee report does not question the correctness of the authors' own constructions, only the external dependency."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi,\n\nHere's my take on arXiv:2608.02114. The paper probably resolves Gongyo's conjecture for smooth projective surfaces, and it does so with a genuinely new technique. But the proof as written is not self-contained at one load-bearing point.\n\nThe new technical core is Theorem 4.2: for a superattracting surface germ with local degree q^2, the square has a primitive divisorial eigenvaluation. The proof uses the Gignac–Ruggiero trichotomy plus a volume comparison (Lemma 4.1) to exclude the end and irrational cases. That's elegant and seems correct. Corollary 4.5, extracting that valuation via normalized blow-up with a q-polarized lift, is also well done. Step 3 of Theorem 1.3—passing from local log canonicity near each point of E2(f) to the global pair via the pushforward discrepancy inequality—is carefully structured and checks out.\n\nThe side results are solid too. Theorem 1.1 on generic smoothness of R_f for n <= 3, forced singularity for n >= 4, and generic log canonicity uses a clean universal Jacobian argument. Theorem 1.4 on the Jacobian exceptional set is a nice general-dimension statement.\n\nThe soft spot is Proposition 4.7. It is the only reason Theorem 1.3 holds at points of E2(f), and it delegates the entire content to [7, Theorem 1.7], an unpublished Chang–Zhang preprint. The statement needed—that for a degree-q endomorphism of a smooth curve and an lc divisor Theta, the pair (C, Theta + R_{Theta,h^s}/(q^s-1)) is lc for some s—is neither proved here nor otherwise available to the reader. The parenthetical 'see also [4, Remark 2.8] when Theta = 0' does not cover the general case, since the Theta produced by adjunction is not generally zero. The alternative in Remark 4.8 leans on [7, Thm 1.16(2)], an even heavier reliance on the same preprint. So the headline 'completes Gongyo's conjecture' is contingent until that dependency is resolved.\n\nThat said, there is no sign of circular reasoning or fitting; the internal steps are coherent and use published ingredients. The paper is honest about what it uses. I think this deserves a serious referee: the method is valuable and the result is likely correct, but a referee should either verify the Chang–Zhang statements or ask the authors to prove the curve lemma themselves before acceptance. I'd cite this paper in my own work on polarized endomorphisms.\n\nBest","headline":"Strong paper with a genuinely new technique; the surface case of Gongyo's conjecture is resolved conditional on an unpublished Chang–Zhang lemma.","tokens_in":21090,"tokens_out":2913,"would_cite":true,"duration_ms":25031,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E22","14B05","37F80"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Jacobian determinant","ramification divisor","polarized endomorphism","valuative tree","log canonical threshold","log Calabi-Yau pair","eigenvaluation","log canonical"],"falsifier":"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 $s$.","tokens_in":19972,"feed_emoji":"🔁","tokens_out":11024,"duration_ms":86364,"temperature":0.7,"pith_summary":"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.","feed_headline":"Iterating any plane endomorphism yields log Calabi-Yau structure","feed_subtitle":"Proof completes the log Calabi-Yau conjecture for smooth projective surfaces by taming ramification through iteration.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the exceptional sets $E_1, E_2$ and supplies the uniform growth bound $\\operatorname{ord}_x(R_{g^n}) \\leq C \\rho^n$ off them, the global estimate that makes the pair klt away from $E_2$.","marker":"[12]"},{"why":"Proves the $E_1(f) \\neq \\emptyset$ case of the surface theorem, which Theorem 1.3 relies on as its first reduction.","marker":"[32]"},{"why":"Establishes existence and properties of eigenvaluations for superattracting germs, including the pullback scaling by the asymptotic attraction rate used in Theorem 4.2.","marker":"[14]"},{"why":"Supplies the trichotomy for superattracting surface germs (end, quasi-monomial, or fixed segment) that Theorem 4.2 splits into.","marker":"[16]"},{"why":"An unpublished preprint giving the one-dimensional log-canonicity statement $(C, \\Theta + R_{\\Theta,h^s}/(q^s-1))$ lc for some $s$, invoked in Proposition 4.7; this is the load-bearing external input.","marker":"[7]"},{"why":"The criterion that every primitive divisorial valuation on a smooth surface is the unique Rees valuation of a simple complete ideal, enabling the equivariant extraction in Corollary 4.5.","marker":"[41]"},{"why":"Inversion of adjunction on log canonicity, used to transfer lc from the exceptional divisor to the blown-up surface and then down to $P^2$.","marker":"[24]"},{"why":"General log canonical machinery, including log discrepancies, Bertini for pairs, and the pushforward formula for discrepancies used in the local-to-global step.","marker":"[25]"},{"why":"The determinantal threshold formula showing the affine determinant hypersurface is log canonical, used to prove Theorem 1.1(3) via smooth pullback.","marker":"[10]"}],"fun_headline_variants":["Iterating plane endomorphisms yields log CY pairs","Optimal log threshold via iteration on P^2","Gongyo conjecture proven: iteration tames ramification","Log canonicity achieved after iteration for surfaces","Plane endomorphisms: iterated ramification is log CY"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $E_2(f)$.","fun_headline_variants_meta":{"raw":{"variants":["Iterating plane endomorphisms yields log CY pairs","Optimal log threshold via iteration on P^2","Gongyo conjecture proven: iteration tames ramification","Log canonicity achieved after iteration for surfaces","Plane endomorphisms: iterated ramification is log CY"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000652,"raw_usage":{"total_tokens":2972,"prompt_tokens":907,"completion_tokens":2065,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":1986}},"tokens_in":523,"tokens_out":2065,"duration_ms":13748,"temperature":1.0,"reasoning_tokens":1986,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:03:51.660965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $s$.","supporting_citations":[{"cited_title":"Favre and M","cited_arxiv_id":null,"evidence_quote":"Defines the exceptional sets $E_1, E_2$ and supplies the uniform growth bound $\\operatorname{ord}_x(R_{g^n}) \\leq C \\rho^n$ off them, the global estimate that makes the pair klt away from $E_2$."},{"cited_title":"Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms","cited_arxiv_id":"2204.11244","evidence_quote":"Proves the $E_1(f) \\neq \\emptyset$ case of the surface theorem, which Theorem 1.3 relies on as its first reduction."},{"cited_title":"Favre and M","cited_arxiv_id":null,"evidence_quote":"Establishes existence and properties of eigenvaluations for superattracting germs, including the pullback scaling by the asymptotic attraction rate used in Theorem 4.2."},{"cited_title":"Gignac and M","cited_arxiv_id":null,"evidence_quote":"Supplies the trichotomy for superattracting surface germs (end, quasi-monomial, or fixed segment) that Theorem 4.2 splits into."},{"cited_title":"Chang and D.-Q","cited_arxiv_id":null,"evidence_quote":"An unpublished preprint giving the one-dimensional log-canonicity statement $(C, \\Theta + R_{\\Theta,h^s}/(q^s-1))$ lc for some $s$, invoked in Proposition 4.7; this is the load-bearing external input."},{"cited_title":"Zariski and P","cited_arxiv_id":null,"evidence_quote":"The criterion that every primitive divisorial valuation on a smooth surface is the unique Rees valuation of a simple complete ideal, enabling the equivariant extraction in Corollary 4.5."},{"cited_title":"Kawakita,Inversion of adjunction on log canonicity, Invent","cited_arxiv_id":null,"evidence_quote":"Inversion of adjunction on log canonicity, used to transfer lc from the exceptional divisor to the blown-up surface and then down to $P^2$."},{"cited_title":"Koll´ ar and S","cited_arxiv_id":null,"evidence_quote":"General log canonical machinery, including log discrepancies, Bertini for pairs, and the pushforward formula for discrepancies used in the local-to-global step."},{"cited_title":"Docampo,Arcs on determinantal varieties, Trans","cited_arxiv_id":null,"evidence_quote":"The determinantal threshold formula showing the affine determinant hypersurface is log canonical, used to prove Theorem 1.1(3) via smooth pullback."}],"review_version":2}