{"id":"36687103-2ec3-4736-a0ab-73ff90569e67","arxiv_id":"2608.10102","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"No codimension-one multiple fibers exist in Lagrangian fibrations over projective space; in Calabi-Yau fibrations on simply-connected K-trivial varieties, the only possible exception is one multiplicity-2 fiber over the projective line.","lead":"This paper proves that Lagrangian fibrations of compact hyper-Kähler manifolds over projective space have no multiple fibers in codimension one, and that Calabi-Yau fibrations on simply-connected K-trivial varieties have none except one sharp case. The result clears a long-standing technical obstruction and turns several conditional theorems into unconditional ones.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.1(1) is not proved as written: the normalization step can disconnect the cyclic cover, so the claimed connected étale cover contradicting simple connectivity is not obtained; this affects Theorem 4.1(2).","rationale":"The reader's weakest-assumption focuses on Kollár vanishing, but that step is sound: the line bundles O_X(j/m) are m-th roots of f^*O(j) with j > 0 ample, precisely the setting of [32, Cor 10.15.2]. The real soft spot is Proposition 2.1(1), which the reader also mentions in passing but does not isolate. Its proof's normalization argument is not valid, because the cyclic cover is connected only before normalization; after normalization it splits into p components, so the claimed connected étale cover does not exist. This does not threaten Theorem 3.4, which depends only on Proposition 2.1(2), but it does threaten the sharpness statement in Theorem 4.1(2), where the coprime assertion rules out two multiplicity-2 fibers. Since the main Lagrangian result stands and the flaw is confined to a supporting statement for the CY exceptional case, keeping the reader's CONDITIONAL verdict is appropriate. A concrete algebraic check can settle whether Proposition 2.1(1) admits a valid substitute proof or fails outright.","tokens_in":16080,"tokens_out":50294,"duration_ms":520539,"concrete_test":"Decide the validity of Proposition 2.1(1) by analyzing the local algebra: for p | m, compute the normalization of Spec(O_X[t]/(t^p - h^m)) and verify that it is Spec(∏_{ζ^p=1} O_X), a disjoint union of p copies of X. Then check whether a replacement argument can be made: the branches t = ζ h^{m/p} would give p sections of the connected degree-p cover B' → B after restriction to a general fiber, which is impossible; alternatively, exhibit a counterexample with gcd(deg D, m) > 1. If no replacement proof exists, Theorem 4.1(2)'s 'at most one' claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 2.1(1) is incomplete. When p divides both deg(D) and m, the local equation of the cyclic cover is t^p = h^m = (h^{m/p})^p, which factors into p branches t = ζ h^{m/p}. These branches meet along the divisor E, so the fiber product X ×_B B' is connected but reducible. Its normalization is the disjoint union of p copies of X, not a connected degree-p étale cover; hence simple connectivity of X gives no contradiction. The assertion that the normalization is connected because the base and its fibers are connected is false: normalization can disconnect fibers over the branch locus. This gap directly affects Theorem 4.1(2): the bound 'at most one multiplicity-2 fiber' is proved by applying the unproved coprime statement to D = b + b'. Theorem 3.4 is unaffected, since it uses only Proposition 2.1(2), whose uniqueness argument is valid for smooth X with torsion-free Picard group. The reader's separate worry about Kollár vanishing does not appear to land: L = O_X(j/m) satisfies L^m = f^*O(j) with j > 0, which is exactly the hypothesis of [32, Cor 10.15.2].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies codimension-one multiple fibers in Lagrangian fibrations of compact hyper-Kähler manifolds and in Calabi-Yau fibrations of simply connected K-trivial varieties. Its main theorem states that a Lagrangian fibration of a hyper-Kähler manifold over projective space has no codimension-one multiple fibers, with consequences for local sections, primitive embeddings, the Néron model action, and several previously conditional results. For Calabi-Yau fibrations, the paper proves absence of codimension-one multiple fibers for base dimension at least two or odd relative dimension, and in the remaining case n=1, r even it allows at most a single multiplicity-2 fiber. It also constructs examples realizing this exceptional case via Enriques-Calabi-Yau divisors and discusses singular generalizations. The proofs use a cyclic-cover argument to produce m-th roots of f^*O(1), Kollár's vanishing and Horrocks's criterion to split higher direct image sheaves, and a Beilinson-norm argument on K_0(P^n) in the Lagrangian case.","tokens_in":16189,"tokens_out":28414,"duration_ms":316925,"significance":"If fully established, the results resolve central questions in the theory of Lagrangian fibrations: the primitivity of f^*Pic(P^n), local triviality in codimension one, and the unconditional status of previously conditional tcf, isotriviality, and dominability results. The technical apparatus is inventive and likely reusable: the combination of cyclic covers, Kollár vanishing, Horrocks splitting, and the Beilinson norm gives a genuinely new mechanism for excluding multiple fibers. The Calabi-Yau part is also sharp, with a clean construction of the exceptional case using Enriques-Calabi-Yau divisors. The paper is well organized, mostly self-contained, and careful about non-flat fibrations and singular total spaces. However, as detailed below, one load-bearing proof step is incomplete as written.","major_comments":[{"comment":"The proof of Proposition 2.1(1) is not valid as written. In the local computation t^p = f^m = (f^{m/p})^p, the fiber product X ×_B B' is reducible: the normalization is the disjoint union of the p branches t = ζ^r f^{m/p}, each isomorphic to X, rather than a connected scheme. Therefore X' is disconnected and X' → X is not a connected étale cover; simple connectivity of X gives no contradiction. The assertion that X' is connected because B' and its fibers are connected is incorrect for a reducible scheme whose fibers over the branch locus become disconnected after normalization. This gap is load-bearing: Proposition 2.1(1) is used to justify the gcd condition before Proposition 2.1(2) can be applied, and it is used directly in the proof of Theorem 4.1(2) for D = b + b' with m = 2 and deg D = 2. In particular, I do not agree that Theorem 3.4 is automatically safe: for a multiple fiber over a divisor D of degree d with gcd(d,m) > 1, the existence of O_X(1/m) is not obtained without part (1). The proposition may be true, but the current proof must be replaced by a correct argument, for example one based on the orbifold fundamental group of (B, D, m).","section":"§2, Proposition 2.1(1)"},{"comment":"The final step of Theorem 4.1(2) is unsupported as written. The sentence 'if there exist two such multiple fibers over distinct points b, b' ∈ P^1, then this violates Proposition 2.1 applied to a degree 2 reduced divisor D = b + b'' relies precisely on Proposition 2.1(1), whose proof is incomplete. No independent argument is supplied to rule out two multiplicity-2 fibers when n = 1 and r is even. Since the exceptional case in the main theorem is claimed to be optimal and the at-most-one statement is part of that optimality, this needs a rigorous proof either by repairing Proposition 2.1 or by a direct argument.","section":"§4.1, proof of Theorem 4.1(2)"}],"minor_comments":[{"comment":"Please add a sentence explaining why q(O_X((1+km)/m)) = 0; the reader must infer that this line bundle is an m-th root of a pullback from P^n and hence has zero Beauville-Bogomolov-Fujiki square.","section":"§3, proof of Theorem 3.4"},{"comment":"The application of [32, Cor. 10.15.2] should state the hypotheses explicitly: X is smooth and K-trivial, and L^m = f^*M with M = O_B(j) ample. The concern that the fractional line bundles are too negative does not appear to be justified, but spelling out the hypotheses would remove ambiguity.","section":"§2, Theorem 2.5"},{"comment":"In the claim that the constructed fibration has 'exactly one multiple fiber 2V', the exclusion of further multiple fibers relies implicitly on Theorem 4.1(2) or on the same multiplicity argument; please state this dependence.","section":"§4.2, Proposition 4.6"},{"comment":"There are a few typographical errors: 'Huybrecths–Riemann–Roch' in Remark 3.5 and 'Verbisky' in reference [25] should be corrected.","section":"Throughout"},{"comment":"The assertion that p(x) has nonnegative coefficients is not immediate from the displayed formula; a one-line verification would improve readability.","section":"§4.1, Lemma 4.2"}],"recommendation":"major_revision","confidential_remarks":"The flaw in Proposition 2.1(1) is a genuine gap in the written proof, not a stylistic issue, and it affects the main theorems as they are currently proved. The rest of the argument, including the Beilinson norm computation, the K-theory lemmas, and the Hilbert-polynomial obstructions for Calabi-Yau fibrations, appears coherent and promising. I would not reject the paper: the statements are likely correct and the methods are strong, but the authors must repair or replace the cyclic-cover proof of the coprime claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe main results are likely true, but the paper has a gap in the foundational step. Theorem 3.4 (no codimension-one multiple fibers for Lagrangian fibrations over P^n) was announced independently by Kamenova–Verbitsky, and the Calabi–Yau version with the sharp multiplicity-2 exception is genuinely new. The Beilinson-norm argument, the Horrocks splitting, and the singular-space generalizations are all nice. If the proof is fixed, this will be an important paper.\n\nThe problem is Proposition 2.1(1). The argument claims that the normalization of X ×_B B' is a connected étale cover of X. But when p divides both deg(D) and m, the local equation t^p = f^m factors into p branches t = ζ f^{m/p} that meet along E. The normalization is the disjoint union of p copies of X, not a connected cover. So simple connectivity gives no contradiction. This is not a minor technicality: part (2) of the same proposition constructs O_X(1/m) by solving 1 = ad+bm, which requires gcd(d,m)=1. Without the coprime claim, the m-th root line bundle is not known to exist, and the proofs of both Theorem 3.4 and Theorem 4.1 rely on it. The stress-test note is too optimistic when it says Theorem 3.4 is unaffected.\n\nThe reader's worry about Kollár vanishing, by contrast, does not land: the hypothesis L^m = f^*O(j) with j>0 is exactly what the cited result needs.\n\nI would not desk-reject this. The content is significant and the authors are clearly serious. But the gap in Proposition 2.1 is load-bearing, and a referee should ask for a correct proof of the coprimality claim, or a different way to construct O_X(1/m). If that can be supplied, the rest of the machine works. I'd bring this to reading group; even with the flaw, there's a lot to learn from the technique.\n\nRecommendation: send to peer review, but flag the Proposition 2.1 issue prominently.","headline":"A genuinely new technique and likely-true headline result, but a real gap in Proposition 2.1 currently blocks both main theorems as written.","tokens_in":16849,"tokens_out":9069,"would_cite":true,"duration_ms":94171,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D06","14J42","14J27","14J32","14F08"],"pacs":[],"model":"deepseek-v4-flash","headline":"Compact hyper-Kähler Lagrangian fibrations over projective space have no codimension-one multiple fibers.","keywords":["hyper-Kähler manifolds","Lagrangian fibrations","codimension one multiple fibers","Calabi-Yau fibrations","higher direct images","Beilinson norm","K3 surfaces","Enriques–Calabi–Yau manifolds"],"falsifier":"Find a compact hyper-Kähler manifold X with a Lagrangian fibration f:X→P^n and a prime divisor D such that $f^{{-1}}$(D)=mE with m>1. A more targeted check is to compute h^p(P^n, R^i f_* O_X(j/m)) for some p>0, 0≤i≤n, 0<j<m and obtain a nonzero value, contradicting equation (2.4); that would remove the splitting and the contradiction. For the Calabi–Yau statement, search for a simply-connected smooth projective K-trivial X with a Calabi–Yau fibration over P^n (n≥2) carrying a codimension-one multiple fiber, or with two double fibers over $P^{1}$ and even fiber dimension.","tokens_in":15753,"feed_emoji":"🧮","tokens_out":9550,"duration_ms":92745,"temperature":0.7,"pith_summary":"The paper proves that a Lagrangian fibration of a compact hyper-Kähler manifold over projective space cannot have a fiber that is multiple along a divisor of the base: no codimension-one multiple fibers exist. It proves the analogous statement for Calabi–Yau fibrations of simply-connected K-trivial varieties, with one sharp exception: along a curve base with even-dimensional fibers, a single double fiber can occur, and examples exist. The proof assumes such a fiber exists, uses the cyclic covering trick to produce a fractional line bundle, and then shows its higher direct images must split as sums of line bundles; a norm inequality on the class of the derived pushforward contradicts the Euler-characteristic computation. A corollary is that the pullback of the hyperplane class is primitive in Pic(X), and several previously conditional results about Lagrangian fibrations become unconditional.","feed_headline":"No multiple fibers in codimension one for hyper-Kähler fibrations","feed_subtitle":"A norm-counting proof also leaves Calabi–Yau fibrations with a single double fiber as the only exception.","key_machinery":"The engine is a chain of algebraic inputs. Assuming a multiple fiber of multiplicity m, the cyclic covering trick and simple-connectedness produce a unique line bundle O_X(1/m) whose m-th power is f^*O(1). Kollár's vanishing theorem is applied to the higher direct images Ω^i_j = R^i f_* O_X(j/m), making them locally free with vanishing cohomology; Horrocks's criterion then forces each Ω^i_j to split as a sum of line bundles of degrees in [-n,0]. In the Lagrangian case the class α=[Rf_*O_X(1/m)] in K_0(P^n) must equal (n+1)[O_p] because its Euler characteristic with every twist is n+1 by the Beauville–Bogomolov–Fujiki form and Huybrechts–Riemann–Roch. The Beilinson norm, the sum of absolute values of coefficients in the Beilinson basis [O(-n)],...,[O], gives ∥α∥=(n+1)2^n, while the splitting and the ranks, computed from the general fiber being an abelian variety, give ∥α∥≤Σ C(n,i)=2^n. That contradiction is the theorem.","core_discovery":"The central claim is Theorem 3.4: a Lagrangian fibration f:X→P^n of a compact hyper-Kähler manifold has no codimension-one multiple fibers, meaning there is no prime divisor D⊂P^n with $f^{{-1}}$(D)=mE for an integer m>1. The paper also proves the companion Theorem 4.1: if X is simply-connected, smooth projective, and K-trivial, and f:X→P^n is a Calabi–Yau fibration with general fiber a Calabi–Yau manifold, then no codimension-one multiple fibers exist when n≥2 or the relative dimension r is odd; when n=1 and r is even, fibers of multiplicity greater than 2 are impossible and at most one double fiber can occur. The exceptional double-fiber case is realized by fibrations obtained from an Enriques–Calabi–Yau divisor, generalizing the Borisov–Nuer examples.","pith_inferences":["One testable extension is to replace P^n by other bases with a full exceptional collection; the paper's reliance on Horrocks splitting suggests the Beilinson-norm argument is not automatic there.","The unique surviving double-fiber case suggests a structural characterization: an odd-dimensional simply-connected K-trivial variety admits a double-fiber fibration over P^1 exactly when it contains an Enriques–Calabi–Yau divisor; proving the converse would sharpen Theorem 4.1.","The unverified positivity in the vanishing step means the main theorem should be regarded as resting on a standard-but-unstated hypothesis; a counterexample to the vanishing would not necessarily produce a multiple fiber, but it would invalidate this proof.","For singular irreducible symplectic varieties, the paper's factorial version already removes codimension-one multiple fibers, so the same obstruction mechanism may extend to larger classes of singular K-trivial varieties."],"forward_implications":["Any Lagrangian fibration of a compact hyper-Kähler manifold over P^n admits local sections over a big open subset of the base, since absence of codimension-one multiple fibers is equivalent to that by known results the paper cites.","The torsion-and-cotorsion-free property of R^1 f_* Z_X, the isotriviality results, and holomorphic dominability of X by C^{2n}, which earlier papers stated conditionally on Theorem 1.1, are now unconditional.","The Néron model action extends to a big open subset of the base for Lagrangian fibrations over P^n.","For hyper-Kähler manifolds with Picard number 1 (non-projective) or 2 (projective), general singular fibers of a Lagrangian fibration are reduced and of Kodaira type I, II, III, or IV, a step toward Sawon's semistability conjecture.","The pullback f^*(Pic(P^n)) is a primitive sublattice of Pic(X), and a simply-connected K-trivial Calabi–Yau fibration over P^n with n≥2 or odd relative dimension cannot have multiple fibers; over P^1 with even fiber dimension, at most one double fiber can occur.","The exceptional double-fiber possibility is actually realized, so the Calabi–Yau statement is optimal as formulated."],"supporting_citations":[{"why":"Supplies Kollár's vanishing theorem (Cor. 10.15.2) used to get h^p(P^n, R^i f_* O_X(j/m))=0, the step that makes Horrocks splitting possible.","marker":"[32]"},{"why":"Establishes torsion-freeness of higher direct images of dualizing sheaves, used in proving local freeness of the pushforwards.","marker":"[30]"},{"why":"Provides the splitting of Rg_* ω_X̃ into cohomology sheaves and the maximal Cohen–Macaulay argument for local freeness.","marker":"[31]"},{"why":"Gives the degenerate twistor deformation and projectivity reduction, plus the Huybrechts–Riemann–Roch formula used to compute χ=n+1.","marker":"[16]"},{"why":"Beauville's result that the Euler characteristic is a degree-n polynomial in the Beauville–Bogomolov–Fujiki form with constant term n+1.","marker":"[4]"},{"why":"Shows Ω^i_0 ≃ Ω^i_{P^n}, identifying the ranks of the higher direct images used in the norm bound.","marker":"[36]"},{"why":"Produces the flat twistor family of Lagrangian fibrations used to reduce the proof to the projective case.","marker":"[47]"},{"why":"Shows fibers are biholomorphic across the twistor deformation, so the presence of multiple fibers is preserved.","marker":"[50]"},{"why":"Provides the Borisov–Nuer examples of simply-connected Calabi–Yau threefolds containing an Enriques surface, which the paper generalizes for the exceptional double-fiber case.","marker":"[6]"}],"fun_headline_variants":["Hyper-Kähler fibrations: no codim-one multiple fibers","Codim-one multiple fibers impossible for hyper-Kähler fibrations","CY fibrations allow at most one double fiber","Only exception: a single double fiber for CY fibrations","No multiple fibers in codim one for hyper-Kähler fibrations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that Kollár's vanishing theorem applies to the fractional line bundles O_X(j/m), which are trivial on the general fiber and hence only mildly positive; the paper invokes the vanishing h^p(P^n, R^i f_* O_X(j/m))=0 without verifying the positivity hypotheses, and if that vanishing fails the Horrocks splitting—and with it the norm contradiction—does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Hyper-Kähler fibrations: no codim-one multiple fibers","Codim-one multiple fibers impossible for hyper-Kähler fibrations","CY fibrations allow at most one double fiber","Only exception: a single double fiber for CY fibrations","No multiple fibers in codim one for hyper-Kähler fibrations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001645,"raw_usage":{"total_tokens":6496,"prompt_tokens":869,"completion_tokens":5627,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":5541}},"tokens_in":485,"tokens_out":5627,"duration_ms":42897,"temperature":1.0,"reasoning_tokens":5541,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:17:22.108208+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a compact hyper-Kähler manifold X with a Lagrangian fibration f:X→P^n and a prime divisor D such that $f^{{-1}}$(D)=mE with m>1. A more targeted check is to compute h^p(P^n, R^i f_* O_X(j/m)) for some p>0, 0≤i≤n, 0<j<m and obtain a nonzero value, contradicting equation (2.4); that would remove the splitting and the contradiction. For the Calabi–Yau statement, search for a simply-connected smooth projective K-trivial X with a Calabi–Yau fibration over P^n (n≥2) carrying a codimension-one multiple fiber, or with two double fibers over $P^{1}$ and even fiber dimension.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Kollár's vanishing theorem (Cor. 10.15.2) used to get h^p(P^n, R^i f_* O_X(j/m))=0, the step that makes Horrocks splitting possible."},{"cited_title":"Higher direct images of dualizing sheaves","cited_arxiv_id":null,"evidence_quote":"Establishes torsion-freeness of higher direct images of dualizing sheaves, used in proving local freeness of the pushforwards."},{"cited_title":"Higher direct images of dualizing sheaves","cited_arxiv_id":null,"evidence_quote":"Provides the splitting of Rg_* ω_X̃ into cohomology sheaves and the maximal Cohen–Macaulay argument for local freeness."},{"cited_title":"Compact hyperk¨ ahler manifolds","cited_arxiv_id":null,"evidence_quote":"Gives the degenerate twistor deformation and projectivity reduction, plus the Huybrechts–Riemann–Roch formula used to compute χ=n+1."},{"cited_title":"Vari´ et´ es K¨ ahleriennes dont la premi` ere classe de Chern est nulle.J","cited_arxiv_id":null,"evidence_quote":"Beauville's result that the Euler characteristic is a degree-n polynomial in the Beauville–Bogomolov–Fujiki form with constant term n+1."},{"cited_title":"Higher direct images of dualizing sheaves of Lagrangian fibrations.Amer","cited_arxiv_id":null,"evidence_quote":"Shows Ω^i_0 ≃ Ω^i_{P^n}, identifying the ranks of the higher direct images used in the norm bound."},{"cited_title":"Degenerate twistor spaces for hyperk¨ ahler manifolds.J","cited_arxiv_id":null,"evidence_quote":"Shows fibers are biholomorphic across the twistor deformation, so the presence of multiple fibers is preserved."},{"cited_title":"Borisov and Howard J","cited_arxiv_id":null,"evidence_quote":"Provides the Borisov–Nuer examples of simply-connected Calabi–Yau threefolds containing an Enriques surface, which the paper generalizes for the exceptional double-fiber case."}],"review_version":1}