{"id":"fd8da1c1-0230-4a2e-b685-7db68951fd1c","arxiv_id":"1908.01605","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Kawaguchi-Silverman conjecture is proved for all projective surfaces and for rationally connected smooth threefolds with an int-amplified endomorphism.","lead":"The paper proves the Kawaguchi-Silverman conjecture for every projective surface, including singular ones, and for rationally connected smooth threefolds admitting an int-amplified endomorphism. A generalist may care because this settles a central conjecture in arithmetic dynamics for a natural class of spaces and reduces the remaining high-dimensional cases to one precise obstacle.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.11's reduction replaces f by f^k∘I while asserting Case TIR3 is preserved, but the preservation step is not proved and cannot follow from [28, Thm 1.4] alone.","rationale":"The reader identified reliance on external equivariant-MMP results as the weakest assumption. My stress-test sharpens this to a specific internal application: the replacement of f by f^k∘I in Theorem 1.11. This step is load-bearing because it is the bridge that lets Proposition 10.7 apply at all; without it, the argument only covers int-amplified endomorphisms, while the theorem claims all surjective endomorphisms. Since the preservation of Case TIR3 is asserted with a citation to a theorem that cannot literally know about this paper's Case TIR, the assertion needs a direct proof. The rest of the paper, including the surface theorem and the formal reduction to Case TIR, appears coherent, but this gap affects the main threefold theorem. I recommend CONDITIONAL rather than REJECT because the missing step is plausibly provable from the known properties of G, and no contradiction has been exhibited.","tokens_in":31047,"tokens_out":27210,"duration_ms":284484,"concrete_test":"Write out the preservation claim for Theorem 1.11: let f_r satisfy (A1)-(A4) and let I be int-amplified, both in the finite-index monoid G from [28, Theorem 1.4]. Using the simultaneous diagonalization or commutativity properties supplied by [28, Theorem 1.4], prove that for all sufficiently large k the map (f^k∘I)_r satisfies (A2) and (A4), computing δ_{f^k∘I} and δ_{(f^k∘I)|Y} explicitly. If this verification cannot be carried through from the stated hypotheses, the proof of Theorem 1.11 has an unproved step and the theorem should be marked conditional.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main threefold result rests on excluding Case TIR3 for the descended map f_r. The proof of Theorem 1.11 assumes f_r is in Case TIR3 and then says: 'Replacing f by f^k∘I for some k≫1, we may assume f is also int-amplified and f_r still satisfies Case TIR3 (cf. [28, Theorem 1.4]).' This is the pivotal step: Proposition 10.7 and Theorem 10.6 apply only to an int-amplified endomorphism with totally invariant ramification, so the whole contradiction depends on the replacement preserving conditions (A1)-(A4). But Case TIR is defined in this paper, not in [28], so [28, Theorem 1.4] cannot by itself certify that the new map still satisfies Case TIR3. In particular, condition (A2) requires (f^k∘I)^*D = δ_{f^k∘I} D; this needs the eigenvalue of the product on D to equal the first dynamical degree of the product, and a larger eigenvalue on another part of N1(X_r) would break the equality. Similarly, (A4) requires δ_{(f^k∘I)|Y} < δ_{f^k∘I}, which is plausible for large k but not demonstrated. A direct verification is missing, so the exclusion of Case TIR3 is not fully supported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves the Kawaguchi-Silverman conjecture (KSC) for every surjective endomorphism of a projective surface, including singular surfaces, and reduces KSC in higher dimensions to a well-specified ``Case TIR'' (totally invariant ramification case). Building on this reduction, it proves KSC for every surjective endomorphism of a rationally connected smooth projective threefold admitting an int-amplified endomorphism. The main ingredients are an equivariant minimal model program for surfaces and threefolds, effectivity of the anti-canonical divisor, and a toric characterization of pairs admitting an int-amplified endomorphism with totally invariant ramification.","tokens_in":31309,"tokens_out":15111,"duration_ms":158042,"significance":"If the threefold theorem is correct, the paper resolves KSC for all surfaces and for a substantial class of threefolds, going well beyond the previously known smooth-surface and Mori-dream-space cases. The surface argument is detailed and self-contained modulo standard results, and the product-type characterization in Theorem 5.2 is a genuine structural contribution. The toric criterion in Theorem 10.6 is also an interesting tool that could be useful beyond this paper. The high-dimensional reduction is conditional on the authors' earlier theorems on int-amplified endomorphisms and equivariant MMP, which are cited as published or to-appear; the main caveat is that the final step of the threefold proof contains a nontrivial gap in the preservation of Case TIR under an iterated composition.","major_comments":[{"comment":"The exclusion of Case TIR3 depends on replacing f by f^k∘I and asserting that ``fr still satisfies Case TIR3 (cf. [28, Theorem 1.4])''. This step is not justified. Case TIR is defined in this paper, and [28, Theorem 1.4] states only the existence of a finite-index submonoid equivariance for an MMP; it does not address conditions (A1)–(A4) for the composed map. In particular, condition (A2) for the descended map requires (f_r^k∘I_r)^*D = δ_{f_r^k∘I_r}D. Since f_r^*D = δ_{f_r}D and I_r^*D = λD for some integer λ>1, the displayed equality would force λ = δ_{I_r}. But δ_{I_r} is the spectral radius of I_r^* on N^1(X_r), and there is no reason that the eigenvalue on the extremal ray spanned by D attains this maximum; [28, Theorem 1.4] contains no statement to this effect. Without (A2), Proposition 10.7 and Theorem 10.6 cannot be applied to the modified map, so the contradiction proving Theorem 1.11 is not established. A direct verification of (A1)–(A4) for the iterated composition, or a different argument showing that some int-amplified endomorphism in the monoid realizes its first dynamical degree on D, is needed.","section":"Section 10, proof of Theorem 1.11 (last paragraph)"},{"comment":"The reduction ``By Theorem 1.7, it suffices to show that fr := f|Xr does not satisfy Case TIR3'' is not explained and appears insufficient. Theorem 1.7(2) requires KSC to hold for Case TIR for every f_i : Xi -> Xi appearing in any equivariant MMP starting from X, not only for the final model Xr. The proof rules out Case TIR3 only for fr. If some intermediate f_i (i<r) satisfied Case TIR3 with respect to a Fano contraction Xi -> P1 different from the chosen MMP's final contraction, the argument would not rule it out: Proposition 10.7 is stated only for a smooth rationally connected X, while Xi may be singular after birational MMP steps, and the manuscript does not prove that f_i being Case TIR3 forces fr to be Case TIR3. The reduction to fr must be justified, or the argument must be extended to all intermediate models.","section":"Section 10, proof of Theorem 1.11 (first paragraph)"}],"minor_comments":[{"comment":"The phrase ``Q-Goreinstein'' in Theorem 1.5 and Proposition 1.6 is a typo for ``Q-Gorenstein''; it appears several times.","section":"Section 1 and 7"},{"comment":"The line ``h0(X, ˆΩ1_X(log D) = dim(X)'' is missing a closing parenthesis; it should read ``h0(X, ˆΩ1_X(log D)) = dim(X)''.","section":"Section 10, proof of Theorem 10.6"},{"comment":"The cones ``NE(X)'' and ``PE1(X)'' are both defined as the pseudo-effective cone; using one symbol would avoid redundancy.","section":"Section 2, Notation"}],"recommendation":"major_revision","confidential_remarks":"The surface part of the paper appears sound and publishable. The threefold theorem is plausible but currently rests on a genuine gap: the replacement f -> f^k∘I does not provably preserve Case TIR3, and the reduction to the final model fr leaves intermediate MMP models untreated. These issues are local and may be fixable within the scope of the paper, for example by adding a lemma on common eigenvectors of int-amplified endomorphisms in Case TIR or by a more careful induction. The paper also relies heavily on the authors' own preceding results ([24], [28]); the editor may wish to verify that those statements cover exactly the forms used here."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the take: this is a genuinely strong paper. Theorem 1.3 resolves KSC for every projective surface, including singular ones, which was open. The threefold result, Theorem 1.11, is the first for a nontrivial class beyond Mori dream spaces, and the reduction to Case TIR is a useful structural framework. The effectiveness of -K_X (Theorem 1.5) is a nice strengthening of earlier results.\n\nThe surface part is the cleanest. The proof via the equivariant MMP, with the double-cover argument in Theorem 5.2 showing the only problematic Fano contraction is of product type, is convincing. I did not find a gap there. The paper also credits prior work properly; the dependence on [22, 24, 27, 28] is heavy but the cited results are real theorems with published or arXiv proofs.\n\nNow the soft spot, and it is real but not fatal. In the proof of Theorem 1.11, after assuming fr satisfies Case TIR3, the authors say: replace f by f^k ∘ I for k≫1, so that f is int-amplified and fr still satisfies Case TIR3, citing [28, Theorem 1.4]. The reference does not establish this. Case TIR3 is defined in this paper, and conditions (A2)-(A4) depend on f. A direct verification is needed. The good news is that it looks true. Since Xr is a threefold with a Fano contraction to P1, N1(Xr) is two-dimensional, spanned by D and π^*N1(Y). Both f_r and I_r preserve this decomposition, with eigenvalues (δ_fr, δ_{f|Y}) and (a, a_Y). After replacing f by f^k∘I, the eigenvalue on D is a δ_fr^k and on π^*N1(Y) is a_Y δ_{f|Y}^k. For k large, the first dominates, so (A2) and (A4) hold; (A3) holds by the ramification formula and Theorem 6.2. So the claim is likely correct, but the paper does not supply this argument. A referee should ask for it.\n\nOverall, this is a significant paper that deserves a serious referee. The surface theorem is a capstone, and the threefold result, once the missing verification is added, will be a landmark. Send it to peer review.","headline":"Strong paper that proves KSC for all projective surfaces and a new threefold class; the threefold proof has a small but real gap in the final reduction that is likely fixable.","tokens_in":31834,"tokens_out":5353,"would_cite":true,"duration_ms":51724,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37P55","14E30","08A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Kawaguchi-Silverman conjecture holds for every surjective endomorphism of any projective surface.","keywords":["Kawaguchi-Silverman conjecture","arithmetic degree","dynamical degree","int-amplified endomorphism","equivariant minimal model program","toric varieties","projective surfaces","rationally connected threefolds"],"falsifier":"Exhibit a $\\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism whose Albanese morphism is not surjective, or a surjective endomorphism of a projective surface with a Zariski-dense orbit whose arithmetic degree is not the first dynamical degree; either would directly contradict the paper's theorems.","tokens_in":1684,"feed_emoji":"📐","tokens_out":2277,"duration_ms":49366,"temperature":0.7,"pith_summary":"This paper resolves the Kawaguchi-Silverman conjecture for all surjective endomorphisms of projective surfaces, including singular ones, by showing that the arithmetic degree of any point with Zariski-dense orbit equals the first dynamical degree. In higher dimensions, it reduces the conjecture to a single special case, called Case TIR, and then proves that this case cannot arise for rationally connected smooth projective threefolds admitting an int-amplified endomorphism. If the reduction holds, the conjecture would follow for a broad class of higher-dimensional varieties from a single remaining scenario. The proof turns on the equivariant minimal model program, the effectiveness of the anti-canonical divisor, and a toric characterization of varieties with totally invariant ramification.","feed_headline":"Arithmetic degree equals dynamical degree for all surface maps","feed_subtitle":"Proof also covers rationally connected threefolds that admit an int-amplified endomorphism.","key_machinery":"The central object is 'Case TIR' (totally invariant ramification): a surjective endomorphism $f$ of a $\\mathbb{Q}$-factorial klt projective variety $X$ admitting an int-amplified endomorphism, with anti-Iitaka dimension $\\kappa(X,-K_X)=0$, with $-K_X$ numerically equivalent to a positive multiple of an irreducible effective divisor $D$ equal to the support of the ramification divisor, and with an $f$-equivariant Fano contraction whose dynamical degree is strictly smaller than $\\delta_f$. The machinery that carries the argument is the equivariant minimal model program, which lets the authors repeatedly replace $X$ by lower-dimensional models; the effectiveness theorem for $-K_X$; the anti-Iitaka fibration and Chow reduction, which show KSC follows once $f^*K_X \\equiv \\delta_f K_X$ and $\\kappa(X,-K_X)>0$; and a toric-pair criterion showing that a rationally connected smooth variety with an int-amplified endomorphism of totally invariant ramification and a suitable numerical eigenspace structure is toric.","core_discovery":"The paper establishes that KSC holds for every surjective endomorphism of any projective surface, removing the smoothness assumption that earlier work required. For higher dimensions, it proves that if $X$ is a $\\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism and KSC holds for all surjective endomorphisms whose ramification divisor is totally invariant and irreducible (Case TIR), then KSC holds for every surjective endomorphism of $X$. As a consequence, KSC holds for every surjective endomorphism of any rationally connected smooth projective threefold admitting an int-amplified endomorphism. The central mechanism is a reduction: after running an equivariant minimal model program, either KSC follows immediately, or the map falls into Case TIR; the authors then rule out Case TIR in the threefold setting using a characterization of toric pairs.","pith_inferences":["If the toric-pair criterion generalizes, a rationally connected smooth projective variety of any dimension admitting an int-amplified endomorphism with totally invariant ramification should be toric, which would eliminate Case TIR in all dimensions unless the anti-Iitaka dimension is positive.","The surface proof's identification of the 'troubled' Fano contraction as being of product type suggests that similar product structures may force the desired numerical eigenvector in higher-dimensional fibrations, potentially weakening the need for the full MMP induction.","A testable extension is to replace rational connectedness in the threefold theorem by uniruledness or by klt singularities; if the Albanese-surjectivity and equivariant-MMP assumptions remain valid, the same toric argument may apply."],"forward_implications":["For every projective surface and every surjective endomorphism, points with Zariski-dense orbit have arithmetic degree equal to the first dynamical degree.","For rationally connected smooth projective threefolds admitting an int-amplified endomorphism, the same equality holds for all surjective endomorphisms.","The reduction to Case TIR means that resolving KSC in higher dimensions reduces to eliminating or handling one special configuration, rather than analyzing all endomorphisms individually.","The effectiveness of $-K_X$ for varieties admitting an int-amplified endomorphism stands as an independent structural result with applications beyond KSC.","The proof recovers and extends the smooth-surface case without relying on it, so the surface statement is unconditional for singular surfaces as well."],"supporting_citations":[{"why":"Supplies the smooth-surface case that the singular proof reduces to for automorphisms and provides the three-case breakdown for non-isomorphic maps.","marker":"[22]"},{"why":"Shows that a normal projective surface admitting a non-isomorphic surjective endomorphism has log canonical singularities, the starting point for the surface MMP.","marker":"[40]"},{"why":"Classifies surfaces with pseudo-effective canonical divisor under a non-isomorphic endomorphism, used to handle the KSC case when $K_X$ is pseudo-effective.","marker":"[32]"},{"why":"Establishes KSC for abelian varieties, which the surface proof uses after reduction to a quasi-etale abelian or elliptic-product cover.","marker":"[38]"},{"why":"Provides the Albanese-surjectivity theorem and eigenvalue bounds for int-amplified endomorphisms, load-bearing for the higher-dimensional reduction.","marker":"[24]"},{"why":"Supplies the equivariant minimal model program for $\\mathbb{Q}$-factorial klt varieties admitting an int-amplified endomorphism, essential for the induction and for Theorem 1.11.","marker":"[28]"},{"why":"Gives the complexity criterion for toric varieties, used to conclude that a pair with free logarithmic forms is toric.","marker":"[3]"},{"why":"Provides the earlier toric characterization in the polarized case, whose strategy is adapted to the int-amplified case in Section 10.","marker":"[27]"}],"fun_headline_variants":["All projective surfaces: arithmetic degree equals dynamical degree","KSC proven for all surjective endomorphisms on singular surfaces","Toric characterization resolves Kawaguchi-Silverman for threefolds","From singular surfaces to rational threefolds: KSC solved"],"cache_read_input_tokens":33920,"weakest_assumption_plain":"The higher-dimensional reduction depends, without reproof, on two external results: that a $\\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism has a surjective Albanese morphism, and that it admits an equivariant minimal model program; if either fails, the reduction to Case TIR breaks.","fun_headline_variants_meta":{"raw":{"variants":["All projective surfaces: arithmetic degree equals dynamical degree","KSC proven for all surjective endomorphisms on singular surfaces","Toric characterization resolves Kawaguchi-Silverman for threefolds","From singular surfaces to rational threefolds: KSC solved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001575,"raw_usage":{"total_tokens":6238,"prompt_tokens":848,"completion_tokens":5390,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":5320}},"tokens_in":464,"tokens_out":5390,"duration_ms":34762,"temperature":1.0,"reasoning_tokens":5320,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:07:16.409712+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a $\\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism whose Albanese morphism is not surjective, or a surjective endomorphism of a projective surface with a Zariski-dense orbit whose arithmetic degree is not the first dynamical degree; either would directly contradict the paper's theorems.","supporting_citations":[{"cited_title":"Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties","cited_arxiv_id":"1902.06072","evidence_quote":"Supplies the smooth-surface case that the singular proof reduces to for automorphisms and provides the three-case breakdown for non-isomorphic maps."},{"cited_title":"Ueno, Classification theory of algebraic varieties and compac t complex spaces, Lecture Notes in Mathematics, Vol","cited_arxiv_id":null,"evidence_quote":"Shows that a normal projective surface admitting a non-isomorphic surjective endomorphism has log canonical singularities, the starting point for the surface MMP."},{"cited_title":"Nakayama, Ruled surfaces with non-trivial surjective endo morphisms, Kyushu J","cited_arxiv_id":null,"evidence_quote":"Classifies surfaces with pseudo-effective canonical divisor under a non-isomorphic endomorphism, used to handle the KSC case when $K_X$ is pseudo-effective."},{"cited_title":"Shokurov, 3-fold log models, Algebraic geometry, 4","cited_arxiv_id":null,"evidence_quote":"Establishes KSC for abelian varieties, which the surface proof uses after reduction to a quasi-etale abelian or elliptic-product cover."},{"cited_title":"Int-amplified endomorphisms on normal projective surfaces","cited_arxiv_id":"1902.06071","evidence_quote":"Provides the Albanese-surjectivity theorem and eigenvalue bounds for int-amplified endomorphisms, load-bearing for the higher-dimensional reduction."},{"cited_title":"Meng and D.-Q","cited_arxiv_id":null,"evidence_quote":"Supplies the equivariant minimal model program for $\\mathbb{Q}$-factorial klt varieties admitting an int-amplified endomorphism, essential for the induction and for Theorem 1.11."},{"cited_title":"Brown, J","cited_arxiv_id":null,"evidence_quote":"Gives the complexity criterion for toric varieties, used to conclude that a pair with free logarithmic forms is toric."},{"cited_title":"Meng and D","cited_arxiv_id":null,"evidence_quote":"Provides the earlier toric characterization in the polarized case, whose strategy is adapted to the int-amplified case in Section 10."}],"review_version":1}