{"id":"486895f9-ad7a-4431-a145-5afbea998d26","arxiv_id":"1908.11537","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every surjective self-map of a smooth rationally connected projective variety with an int-amplified endomorphism satisfies the Kawaguchi-Silverman conjecture.","lead":"A new proof establishes the Kawaguchi-Silverman conjecture, a central prediction about orbit growth, for all smooth rationally connected algebraic varieties that admit an int-amplified endomorphism. This extends previously known surface and threefold cases to arbitrary dimension and gives a template for using the minimal model program in arithmetic dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 5.6 passes to the base Y' of a fiber type contraction and runs K_MMP without justifying that Y' is Q-factorial klt; this is the load-bearing gap.","rationale":"The reader's weakest assumption correctly identifies the point in Theorem 5.6 where the proof reduces KSC for f to KSC for a lower-dimensional variety Y' and runs K_MMP there. This is the most load-bearing step because if Y' is not Q-factorial or does not admit an int-amplified endomorphism, Theorem 3.4 cannot be applied and the inductive argument collapses. The paper does not explicitly justify these properties for Y', so the written proof is incomplete. However, the missing facts are standard in the MMP literature: a fiber type contraction of a Q-factorial klt pair has no exceptional divisors, and the pushforward of a Cartier divisor on the total space is Cartier on the base; combined with Lemma 4.7 this gives Q-factoriality. Lemma 3.2(2) grants an int-amplified endomorphism on the base, and Lemma 4.10 grants klt. Thus the gap is fillable without changing the main argument. I agree with the reader's conditional verdict, and the concrete test above directly checks whether the missing justification holds.","tokens_in":18693,"tokens_out":21930,"duration_ms":162768,"concrete_test":"Verify the Q-factoriality of the base: For the fiber type contraction π: X -> Y in Proposition 5.1(3), prove that Y is Q-factorial by showing for any prime divisor D on Y, the pullback π^*D is Q-Cartier on X, and use Lemma 4.7 (no exceptional divisors) to descend Q-Cartierness to Y. Then confirm that the induced endomorphism on Y is int-amplified via Lemma 3.2(2), and that Lemma 4.10 gives Y klt. If these steps fail, exhibit a counterexample where a fiber type contraction from a Q-factorial klt variety has a non-Q-factorial base.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 5.6, case (3) of Proposition 5.1 produces a fiber type contraction X'_r -> Y' with δ_{f^m|_{X'_r}} = δ_{f^m|_{Y'}}. The proof then says 'run K_Y'-MMP' and invokes Theorem 3.4, which requires a Q-factorial klt variety. The paper does not prove Y' is Q-factorial, nor does it explain a Q-factorialization step that preserves the equality of dynamical degrees needed for Remark 2.11 to reduce KSC for f to KSC for the output. The same compressed argument appears when excluding Q-abelian outputs. This is a genuine gap in the written proof, although likely fillable: Lemma 4.7 shows that π has no exceptional divisors, and standard MMP (e.g., Kollár–Mori Lemma 3.38) then implies that the base of a fiber type contraction from a Q-factorial klt pair is Q-factorial. The paper should state and prove or cite this fact.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves the Kawaguchi-Silverman conjecture (KSC) for every surjective endomorphism on a smooth projective rationally connected variety over Q that admits an int-amplified endomorphism. The proof combines the equivariant minimal model program of Meng–Zhang with a new covering theorem (Theorem 4.4) that, under a horizontal ramification condition, produces a quasi-étale abelian cover. The main theorem is obtained by running an equivariant K_X-MMP, applying a key trichotomy (Proposition 5.1) to each Mori fiber space, and descending either to a Q-abelian variety or to a lower-dimensional base; properties of arithmetic/dynamical degrees (Remarks 2.10 and 2.11) then transfer KSC back to the original variety.","tokens_in":18889,"tokens_out":19226,"duration_ms":177303,"significance":"If the proof is correct, this is a substantial advance: it establishes KSC for a large class of higher-dimensional rationally connected varieties, going well beyond the surface case and the threefold results of Meng–Zhang. The paper is well structured and contains detailed arguments for the covering theorem and the MMP reduction, and the main theorem is a clear falsifiable statement. The strategy of excluding problematic fibrations via a covering theorem is original and likely to be influential.","major_comments":[{"comment":"When δ_{f^m|X'_r} = δ_{f^m|Y'}, the proof runs the K_{Y'}-MMP and invokes Theorem 3.4. This requires Y' to be Q-factorial klt. The paper does not prove that Y', the base of the fiber type contraction π1: X'_r → Y' arising from Proposition 5.1(3), is Q-factorial klt. Lemma 4.10 gives klt only under the hypothesis that −K_{X'_r} is π1-ample; in the δ_f > δ_g branch of Proposition 5.1, π1 is a (K_{X'_r}+Δ_r)-negative extremal ray contraction, and −K_{X'_r} is not shown to be π1-ample. The same issue occurs when the argument excludes Q-abelian outputs after running K_{Y'}-MMP. Since the induction on ρ and dim depends on passing to Y', this is a load-bearing gap. Please add a proof or a precise reference that Y' is Q-factorial klt in this setting, or modify the descent to avoid running an MMP on Y'.","section":"§5, Theorem 5.6, case (3), second subcase"}],"minor_comments":[{"comment":"When the MMP on X_r or Y' terminates in a Q-abelian variety that is a point, the statement 'Q-abelian case does not occur because endomorphisms on Q-abelian varieties satisfy (*)' is not correct: a point satisfies property (*) only vacuously, and the contradiction argument does not apply. This case is harmless (it leads to KSC via Proposition 5.1(1) with dim Y = 0), but it should be stated explicitly.","section":"§5, Theorem 5.6"},{"comment":"The assertion that every variety X' appearing in an MMP starting from a rationally connected variety satisfies N^1(X')_Q = Pic(X')_Q is used without proof or reference. This is standard but should be cited, for example to Debarre's book or a suitable statement on rationally connected varieties and the Picard group.","section":"§5, Theorem 5.6"},{"comment":"Theorem 4.4 is introduced as being taken from an upcoming paper by the second author, but a proof is included in the text. To avoid concerns about reliance on unpublished work, please add a sentence clarifying that the proof is self-contained and that the reference is only for provenance.","section":"§4, Theorem 4.4"},{"comment":"There are several typographical and notational slips in the text (e.g., 'there exits' in the statement of Proposition 5.1, and inconsistent subscripts X_1/X_r). The authors should proofread the manuscript carefully before publication.","section":"§5, Proposition 5.1"}],"recommendation":"major_revision","confidential_remarks":"The main gap identified in the proof of Theorem 5.6 is likely fillable by standard MMP techniques, so I do not recommend rejection. I would ask the authors to supply a complete justification, either by proving that the base Y' is Q-factorial klt or by restructuring the descent step. The provenance note on Theorem 4.4 should also be clarified, although the proof appears in the text. The paper is within the scope of math.AG and, after the revision, would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the theorem is real and important. They prove KSC for every smooth projective rationally connected variety over Q that admits an int-amplified endomorphism, for all surjective endomorphisms. This is the first all-dimensional result in that direction and subsumes the earlier surface and threefold cases. The architecture is sensible: run equivariant MMP, use a covering theorem to rule out bad fiber type contractions, then induct on dimension or Picard number. The covering theorem (Theorem 4.4) is proved in the text, not just deferred to the second author's upcoming paper. The statement credits the upcoming paper, but Section 4 contains a full argument; the provenance is a matter of attribution, not of missing proof.\n\nWhat is genuinely new: the covering theorem and the MMP induction around it. Proposition 5.1 cleanly separates the case where KSC follows from an anticanonical eigen-divisor with irreducible support from the case where one can descend to a smaller Picard number or dimension. Lemma 4.12, which shows the covering property (*) ascends along divisorial contractions, flips, and fiber type contractions, is the technical heart and it is done properly.\n\nWhere the soft spots are: first, in the final paragraph of Theorem 5.6, when Proposition 5.1(3) gives δ_{f^m|X'_r}=δ_{f^m|Y'}, the proof passes to Y' and runs K_{Y'}-MMP without justifying that Y' is Q-factorial klt with Pic=N^1. The klt part is covered by Lemma 4.10, and Pic=N^1 follows because Y' is still rationally connected. Q-factoriality is genuinely missing as a written statement. It should follow from Kollár–Mori 3.38 together with Lemma 4.7, but the paper should say so. This is a fillable gap, not a fatal one. It shows up again when excluding Q-abelian outputs in the same paragraph.\n\nSecond, the termination of the induction is handled in one sentence. The claim that Picard number or dimension strictly decreases is credible, but the bookkeeping for intermediate outputs of the MMP should be expanded.\n\nThe citation pattern is fine. The heavy use of Meng–Zhang and the authors' earlier papers is appropriate; there are no invented entities and no free parameters.\n\nThis paper deserves a serious referee. The result is significant and the proof is mostly rigorous, with one compressed technical step that a careful referee would want rewritten. My recommendation: send it out, and after the authors add the Q-factoriality justification and expand the termination paragraph, accept it.","headline":"The main theorem is an important all-dimensional KSC result and the proof is largely sound; the one real soft spot is a compressed base-Q-factoriality step that needs a citation but is almost certainly fillable.","tokens_in":19408,"tokens_out":5261,"would_cite":true,"duration_ms":52048,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Kawaguchi-Silverman conjecture proved for rationally connected varieties with an int-amplified endomorphism","keywords":["Kawaguchi-Silverman conjecture","arithmetic degree","dynamical degree","int-amplified endomorphism","rationally connected variety","equivariant minimal model program","Q-abelian variety"],"falsifier":"Construct a smooth projective rationally connected variety X over Q admitting an int-amplified endomorphism, and a surjective endomorphism f with a Zariski dense point x, such that the arithmetic degree alpha_f(x) is strictly less than the dynamical degree delta_f. According to the theorem, no such example can exist; finding one would disprove the claim. More locally, an explicit example where the base of a fiber-type contraction in the equivariant MMP is not Q-factorial or fails Pic = $N^{1}$ would test the weakest assumption: if such a base can occur while the rest of the proof's hypotheses hold, the reduction step would be invalid.","tokens_in":18495,"feed_emoji":"","tokens_out":1808,"duration_ms":18819,"temperature":0.7,"pith_summary":"This paper proves that the Kawaguchi-Silverman conjecture holds for every surjective endomorphism on a smooth projective rationally connected variety over Q that admits an int-amplified endomorphism. The conjecture predicts equality between the arithmetic degree of a point with Zariski dense orbit and the first dynamical degree of the map. The proof uses the equivariant minimal model program to reduce the problem to simpler varieties, while ruling out the appearance of certain fibrations that would block the reduction. This is the first case where the conjecture is established for all endomorphisms on a broad class of higher-dimensional varieties, going beyond surfaces and threefolds.","feed_headline":"KSC proved for rationally connected varieties with amplified maps","feed_subtitle":"A new theorem equates arithmetic and dynamical degrees for all surjective endomorphisms on a broad higher-dimensional class.","key_machinery":"The central mechanism is the equivariant minimal model program (MMP) developed by Meng and Zhang, together with a covering theorem (Theorem 4.4) proving that certain fiber-type contractions force the existence of a quasi-etale cover dominating an abelian variety. The covering theorem uses the index-1 cover of a log Calabi-Yau pair (Y, $\\Delta$) to construct a finite cover of the base, then forms the main component of the fiber product with X, and shows the resulting cover is quasi-etale using the horizontality of the ramification divisor. This cover then ascends along the steps of the MMP via Lemma 4.12, so if any step produced a Q-abelian output, the original rationally connected X would admit a quasi-etale cover dominating an abelian variety, which is impossible since X is algebraically simply connected.","core_discovery":"Theorem 1.1 states: if X is a smooth projective rationally connected variety over Q admitting an int-amplified endomorphism, then for every surjective endomorphism f of X and every x in X(Q) whose f-orbit is Zariski dense, the arithmetic degree alpha_f(x) equals the dynamical degree delta_f. The proof runs a KX-MMP equivariantly with respect to a power of f, and at each step applies a dichotomy: either the Kawaguchi-Silverman conjecture holds for the induced map on a birational model, or a certain fibration appears with controlled dynamical degrees. The authors show the problematic second case cannot occur for rationally connected X, because the fibration would produce a quasi-etale cover dominating an abelian variety, contradicting algebraic simple connectedness. The key technical ingredient is a covering theorem showing that when a fiber-type contraction has a log Calabi-Yau base, the whole fibration admits a quasi-etale cover by a variety that fibers over an abelian variety; this covering ascends along divisorial contractions, flips, and fiber-type contractions, so the contradiction propagates backward through the MMP.","pith_inferences":["If the equivariant MMP can be run on varieties that are not Q-factorial or have Pic(X)_Q != N^1(X)_Q, the same strategy could extend the conjecture beyond the rationally connected case, provided the abelian-covering obstruction can be ruled out.","The proof suggests that for any variety admitting an int-amplified endomorphism, the only potential obstruction to Kawaguchi-Silverman is the existence of a fiber-type contraction to a Q-abelian base; one could test this by looking for rationally connected threefolds whose MMP outputs are Q-abelian but which do not admit int-amplified endomorphisms.","The quasi-etale covering construction may be adaptable to prove statements about the non-existence of int-amplified endomorphisms on varieties that are algebraically simply connected, since it forces a cover by an abelian variety."],"forward_implications":["The Kawaguchi-Silverman conjecture now holds for all surjective endomorphisms on smooth projective rationally connected varieties that admit an int-amplified endomorphism, a class that includes many Fano-type and rationally connected threefolds and higher-dimensional examples.","The proof gives a template for proving the conjecture on broader classes: one only needs to rule out the appearance of fiber-type contractions with Q-abelian bases during the equivariant MMP.","Any future counterexample to the conjecture on a rationally connected variety must fail to admit an int-amplified endomorphism, since the theorem removes all such cases.","The covering theorem itself is a standalone structural result: certain fiber-type contractions with log Calabi-Yau bases are necessarily covered by fibrations over abelian varieties, which may be useful in other dynamical and birational questions."],"supporting_citations":[{"why":"Meng-Zhang's equivariant MMP theorem (Theorem 3.4 and Theorem 3.3 in the paper) supplies the equivariant sequence of divisorial contractions, flips, and fiber-type contractions used throughout the proof.","marker":"[28]"},{"why":"Meng-Zhang's paper on the Kawaguchi-Silverman conjecture for surjective endomorphisms is the source of Proposition 5.1, the key dichotomy step, and of Lemmas 5.2 and 5.3 used to handle the case of irreducible anti-canonical divisor.","marker":"[29]"},{"why":"Birkar-Cascini-Hacon-McKernan's existence of minimal models is invoked to run the KX-MMP and the additional p(KX+Delta)-MMP when needed.","marker":"[2]"},{"why":"Ambro's canonical bundle formula is used in Lemma 4.10 to conclude that the base of a fiber-type contraction is klt.","marker":"[1]"},{"why":"Gongyo's abundance theorem for numerically trivial log canonical divisors is cited to conclude that K_Y + Delta is Q-linearly equivalent to 0 in the proof of Theorem 4.4.","marker":"[12]"},{"why":"Meng's building-blocks results provide the int-amplified endomorphism properties, the existence of the abelian cover for Y1 (Theorem 5.2 in that paper), and the quasi-etale and totally invariant divisor lemmas used in Lemmas 4.5 and 4.12.","marker":"[26]"},{"why":"Meng-Zhang's earlier paper is cited for the flip equivariance and the property that rational singularities allow the rational map to A to be a morphism in the flip step of Lemma 4.12.","marker":"[27]"},{"why":"Kawaguchi-Silverman's work supplies the existence of the arithmetic degree limit and the upper bound alpha_f(x) <= delta_f, foundational for the conjecture.","marker":"[15]"},{"why":"Kawaguchi-Silverman's formulation of the conjecture and the criterion used in Proposition 2.13 (via the existence of a nef eigen-divisor with positive Iitaka dimension) are load-bearing.","marker":"[16]"}],"fun_headline_variants":["KSC holds for all endomorphisms on rationally connected varieties with amplified maps","Arithmetic degree equals dynamical degree on rationally connected varieties","New proof of Kawaguchi-Silverman for rationally connected varieties","KSC via equivariant MMP on rationally connected varieties","Amplified endomorphisms force arithmetic-dynamical degree equality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The base Y' obtained after the MMP step (and in the reduction to exclude Q-abelian outputs) is assumed to be Q-factorial klt with Pic(Y')_Q = $N^{1}$(Y')_Q, but the paper does not justify that the constructed fiber-type contraction has a base with these properties; a Q-factorialization step that preserves the reduction is not provided.","fun_headline_variants_meta":{"raw":{"variants":["KSC holds for all endomorphisms on rationally connected varieties with amplified maps","Arithmetic degree equals dynamical degree on rationally connected varieties","New proof of Kawaguchi-Silverman for rationally connected varieties","KSC via equivariant MMP on rationally connected varieties","Amplified endomorphisms force arithmetic-dynamical degree equality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000657,"raw_usage":{"total_tokens":2926,"prompt_tokens":785,"completion_tokens":2141,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":2055}},"tokens_in":401,"tokens_out":2141,"duration_ms":15124,"temperature":1.0,"reasoning_tokens":2055,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:13:41.379942+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a smooth projective rationally connected variety X over Q admitting an int-amplified endomorphism, and a surjective endomorphism f with a Zariski dense point x, such that the arithmetic degree alpha_f(x) is strictly less than the dynamical degree delta_f. According to the theorem, no such example can exist; finding one would disprove the claim. More locally, an explicit example where the base of a fiber-type contraction in the equivariant MMP is not Q-factorial or fails Pic = $N^{1}$ would test the weakest assumption: if such a base can occur while the rest of the proof's hypotheses hold, the reduction step would be invalid.","supporting_citations":[{"cited_title":"Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism","cited_arxiv_id":"1806.05828","evidence_quote":"Meng-Zhang's equivariant MMP theorem (Theorem 3.4 and Theorem 3.3 in the paper) supplies the equivariant sequence of divisorial contractions, flips, and fiber-type contractions used throughout the proof."},{"cited_title":"Kawaguchi-Silverman conjecture for certain surjective endomorphisms","cited_arxiv_id":"1908.01605","evidence_quote":"Meng-Zhang's paper on the Kawaguchi-Silverman conjecture for surjective endomorphisms is the source of Proposition 5.1, the key dichotomy step, and of Lemmas 5.2 and 5.3 used to handle the case of irreducible anti-canonical divisor."},{"cited_title":"D., and McKernan, J., Existence of minimal models for varieties of log general type , J","cited_arxiv_id":null,"evidence_quote":"Birkar-Cascini-Hacon-McKernan's existence of minimal models is invoked to run the KX-MMP and the additional p(KX+Delta)-MMP when needed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Ambro's canonical bundle formula is used in Lemma 4.10 to conclude that the base of a fiber-type contraction is klt."},{"cited_title":"Algebraic Geom","cited_arxiv_id":null,"evidence_quote":"Gongyo's abundance theorem for numerically trivial log canonical divisors is cited to conclude that K_Y + Delta is Q-linearly equivalent to 0 in the proof of Theorem 4.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Meng's building-blocks results provide the int-amplified endomorphism properties, the existence of the abelian cover for Y1 (Theorem 5.2 in that paper), and the quasi-etale and totally invariant divisor lemmas used in Lemmas 4.5 and 4.12."},{"cited_title":"and Zhang, D.-Q., Building blocks of polarized endomorphisms of normal projective varieties , Adv","cited_arxiv_id":null,"evidence_quote":"Meng-Zhang's earlier paper is cited for the flip equivariance and the property that rational singularities allow the rational map to A to be a morphism in the flip step of Lemma 4.12."},{"cited_title":"H., Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical hei ghts on abelian vari- eties, Trans","cited_arxiv_id":null,"evidence_quote":"Kawaguchi-Silverman's work supplies the existence of the arithmetic degree limit and the upper bound alpha_f(x) <= delta_f, foundational for the conjecture."},{"cited_title":"H., On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties , J","cited_arxiv_id":null,"evidence_quote":"Kawaguchi-Silverman's formulation of the conjecture and the criterion used in Proposition 2.13 (via the existence of a nef eigen-divisor with positive Iitaka dimension) are load-bearing."}],"review_version":1}