{"id":"abb30e55-074a-49b3-89e6-843a4702190a","arxiv_id":"2607.05603","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"There exist flat projective morphisms with smooth total space that are Z-homology fiber bundles but are neither smooth nor topological fiber bundles.","lead":"The paper builds flat projective maps with smooth total space that are Z-homology fiber bundles but not smooth (hence not topological fiber bundles). This disproves half of a 2012 conjecture of Fernández de Bobadilla and Kollár on when homology triviality forces smoothness.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates the monodromy computation as the only external input that could fail. That input is standard classical material (Pham 1965, Brieskorn 1966) and is applied under the explicit pairwise-coprimeness hypothesis that the constructions satisfy. The remainder of the argument (Mayer-Vietoris decomposition of the quotient, Künneth reduction to a circle local system, and the elementary criterion det(T-I)=\tau1) is self-contained and elementary. Consequently no load-bearing concern survives scrutiny; the Reader's ACCEPT verdict with high confidence and low correctness risk is unchanged.","tokens_in":6635,"tokens_out":562,"duration_ms":4133,"concrete_test":"Independently recompute the monodromy eigenvalues of ∑x_i^{c_i}=t for the pairs (c_1,c_2)=(2,3) and (2,3,5) via the Pham formula or a computer-algebra implementation of the Thom-Sebastiani product; verify that none equal a c-th root of unity and that the resulting characteristic polynomial satisfies \tau(1)=\tau1. If the values match the paper, the local-system acyclicity (and therefore Theorem 3) stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 3 is that the quotient construction Y=(A\times X)/(\tau,\tau) yields a Z-homology fiber bundle that is neither smooth nor a topological fiber bundle, provided the monodromy characteristic polynomials satisfy \tau_i(1)=\tau1. The proof reduces via Mayer-Vietoris (Section 8) to the acyclicity of the local system L_T on the circle (Lemma 9 and paragraph 10), which holds precisely when det(T-I)=\tau1. The monodromy input for the concrete families (Construction 11/13 and Section 12) is classical: Thom-Sebastiani applied to Pham-Brieskorn eigenvalues for sums of powers. When the c_i are pairwise relatively prime the listed eigenvalues are never c-th roots of unity (n≥2), so the characteristic polynomial divides an expression that evaluates to 1 at t=1; thus \tau_c(1)=\tau1. The same conclusion is available for the E_8 case (Example 14) because the Coxeter transformation of the E_8 lattice is known to be unimodular after subtracting the identity. No hidden gap appears in the reduction or in the classical monodromy data used by the examples.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs flat projective morphisms Y \to \triangle with smooth total space that are ℤ-homology fiber bundles but are not smooth (hence not topological fiber bundles), disproving one direction of Conjecture 1 of Fernández de Bobadilla–Kollár. Starting from a projective morphism X \to \triangle with finitely many critical points that admits a finite-order automorphism γ fixing those points and acting on the Milnor fibers so that the monodromy characteristic polynomials satisfy χ_i(1) = ±1, one forms the quotient Y = (A \times X)/(τ, γ) by a translation τ of the same order on an abelian variety A. Theorem 3 asserts that Y \to \triangle is then a ℤ-homology fiber bundle that is neither smooth nor a topological fiber bundle; the proof reduces via Mayer–Vietoris to the vanishing of the cohomology of the associated local systems on the circle (Lemma 9 and paragraph 10). Concrete examples are supplied by Pham–Brieskorn singularities (sums of powers) and the E_8 singularity, yielding families of surfaces and higher-dimensional varieties, some with normal or canonical singularities.","tokens_in":6893,"tokens_out":1094,"duration_ms":97402,"significance":"The result cleanly separates the ℤ-homology fiber-bundle property from smoothness (and from topological local triviality) for projective morphisms with smooth total space, giving a definitive negative answer to half of the 2012 conjecture. The construction is short, geometric, and uses only standard tools (Mayer–Vietoris, Leray spectral sequences for local systems on the circle) together with classical monodromy computations of Pham, Brieskorn and Thom–Sebastiani; the examples are completely explicit. The observation that the homology-fiber-bundle property fails to be stable under finite étale covers is a useful byproduct. The paper therefore constitutes a significant contribution to equisingularity theory and the topology of algebraic families, while clearly isolating the remaining open question on homotopy fiber bundles.","major_comments":[],"minor_comments":[{"comment":"The pairwise relative primeness of the exponents c_i is used in an essential way in the monodromy argument of §12 but is never stated in Construction 11; it should be added to the hypotheses of the construction.","section":"Construction 11"},{"comment":"The sentence “none of these are c ith roots of unity for n≥2” is garbled (almost certainly a typographical error). It should be rephrased to say that none of the sums ∑ a_i/c_i is an integer, so that 1 is not an eigenvalue of the monodromy. In addition, the displayed rational function is not a priori a polynomial; under the coprimeness hypothesis it simplifies to a monic polynomial of degree ∏(c_i-1) that coincides with χ_c (as can be verified directly for (2,3) and similar small tuples). A one-sentence clarification that the expression is in fact equal to the characteristic polynomial would make the evaluation χ_c(1)=±1 immediate and self-contained.","section":"§12"},{"comment":"The existence of a μ_c-equivariant simultaneous resolution of the singularities along w=0 is left to the reader. While the claim is standard for these quasi-homogeneous singularities, a brief indication (or a reference) that the resolution introduces no new critical points of π_X would be helpful.","section":"Construction 13"},{"comment":"The notation “Z2r” for the fundamental groups of the fibres should be clarified (e.g., as ℤ/2rℤ or (ℤ/2ℤ)^r).","section":"Introduction"},{"comment":"Minor typographical issues: “c ith” (should be “c-th”), missing space in “forn≥2”, and inconsistent formatting of group-order notation.","section":null}],"recommendation":"minor_revision","confidential_remarks":"High-quality short paper that settles a natural question with a clean geometric construction. After the purely expository fixes in the monodromy section it is ready for acceptance; suitable for a top journal in algebraic geometry or singularity theory. The fact that the second author co-formulated the original conjecture is unproblematic and strengthens the force of the counter-example."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper gives the first flat projective morphisms with smooth total space that are Z-homology fiber bundles but not topological fiber bundles (hence not smooth). That settles the (1.1) ⇔ (1.3) half of Conjecture 1 from FdBK12 in the negative; earlier counterexamples all had singular total spaces.\n\nThe construction is short and transparent. Start with a projective family X → Δ having finitely many critical points and a finite-order automorphism γ that fixes them and acts on the Milnor fibers with characteristic polynomials satisfying χ_i(1) = ±1. Form the diagonal quotient Y = (A × X)/(τ, γ) by an abelian variety A with a translation of the same order. Theorem 3 then follows from Mayer–Vietoris plus the elementary fact (Lemma 9 + §10) that a local system on the circle with monodromy T is acyclic precisely when det(T − I) = ±1. The monodromy input for the concrete families (sums of powers, weighted models, E8) is classical Pham–Brieskorn via Thom–Sebastiani; when the exponents are pairwise coprime the eigenvalues never hit c-th roots of unity, so the condition holds. Concrete equations are written down (Example 4, Construction 13, Example 14), so the examples are reproducible by hand.\n\nSoft spots are minor and already flagged by the authors. The fibers always admit a finite étale cover that is a product with an abelian variety, so they are never of general type; the paper notes this and leaves open the question of ample canonical class. The homotopy-versus-smoothness half of the conjecture remains untouched, again correctly. The monodromy black box is standard citation, not a gap.\n\nThis is for people working on equisingularity, topological triviality of algebraic fibrations, or the topology of vanishing cycles. The argument is elementary once the monodromy data are granted, the citations are appropriate, and the result is permanent. It deserves a serious referee and should be accepted after ordinary polishing.","headline":"Clean counterexamples with smooth total space that kill the homology of vanishing cycles via a diagonal abelian quotient, disproving one half of the 2012 FdB–Kollár conjecture while correctly leaving the homotopy case open.","tokens_in":7524,"tokens_out":522,"would_cite":true,"duration_ms":3911,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D05","14B05","32S50","55R10"],"pacs":[],"model":"grok-4.5","headline":"Smooth total spaces can form Z-homology fiber bundles that fail to be smooth or topological fiber bundles.","keywords":["homology fiber bundle","topological fiber bundle","equisingularity","Milnor fiber","monodromy","abelian variety quotient","projective morphism"],"falsifier":"Compute the monodromy characteristic polynomial for one of the explicit sums of powers (for example the E8 singularity x^2+y^3+z^5) and check whether its value at 1 is really plus or minus 1; if it is not, the local system is not acyclic and the homology fiber bundle claim fails.","tokens_in":7516,"feed_emoji":"🔢","tokens_out":961,"duration_ms":7027,"temperature":0.7,"pith_summary":"The paper settles half of a conjecture about when projective morphisms of complex manifolds behave like fiber bundles. Smooth morphisms are always differentiable fiber bundles, and those are always homotopy and homology fiber bundles, but the converse was open. Earlier counterexamples to the topological side always had singular total spaces. Here the authors build flat projective maps whose total space is smooth and whose fibers give the same integral homology, yet the maps are not smooth and the fibers are not all homeomorphic. The construction takes a family with isolated critical points, multiplies by an abelian variety, and quotients by a diagonal finite-order action that cancels the vanishing-cycle contribution in homology. The resulting families have fundamental group Z/2^r and are never homotopy fiber bundles, so the remaining half of the conjecture (homotopy versus smooth) stays open.","feed_headline":"Smooth spaces can still fail to be fiber bundles","feed_subtitle":"Z-homology fiber bundles with smooth total space need not be smooth or topologically trivial","key_machinery":"The diagonal quotient Y=(A times X)/(tau,gamma). Lemma 9 shows that projection from the quotient of A times a Milnor fiber induces a cohomology isomorphism precisely when the monodromy characteristic polynomial evaluates to plus or minus 1 at 1; that acyclicity of the local system over the circle erases the homology contribution of the vanishing cycles, so Mayer-Vietoris yields a homology fiber bundle.","core_discovery":"If a projective family X to a disc has finitely many critical points fixed by a finite-order automorphism gamma whose monodromy characteristic polynomials satisfy chi_i(1)=plus or minus 1, and if A is an abelian variety with a translation of the same order, then the quotient Y=(A times X)/(tau,gamma) to the disc is a Z-homology fiber bundle whose total space is smooth, yet Y is neither smooth nor a topological fiber bundle.","pith_inferences":["Because every fiber admits a finite etale cover that is a product with an abelian variety, the construction cannot produce fibers of general type; an entirely different source of monodromy would be needed for that.","The same local-system acyclicity criterion could be tested on other classical monodromy operators (e.g., other isolated hypersurface singularities) to generate further examples without abelian factors.","If a homotopy-fiber-bundle counterexample with smooth total space is later found, it will almost certainly have to avoid the abelian-product structure used here, since that structure forces nontrivial fundamental groups."],"forward_implications":["Smoothness of the total space does not force a projective Z-homology fiber bundle to be smooth or topologically trivial.","The remaining open half of the conjecture is whether a projective homotopy fiber bundle with smooth total space must itself be smooth.","Being a Z-homology fiber bundle is not preserved by finite etale covers: the product A times X is never a homology fiber bundle, yet the quotient is.","Concrete surface and higher-dimensional families exist (including ones with canonical or terminal singularities) that realize the phenomenon."],"fun_headline_variants":["Smooth total spaces need not make homology fiber bundles smooth","Z-homology fiber bundles with smooth totals can fail smoothness","Projective homology bundles that are not topological fiber bundles","Nonsmooth Z-homology fiber bundles whose total space is smooth","Homology fiber bundles can have smooth total space yet not be smooth"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The monodromy eigenvalues on the Milnor fiber of a sum of powers with pairwise coprime exponents never include a root of unity of order equal to the product of the exponents, so the characteristic polynomial equals plus or minus 1 at 1.","fun_headline_variants_meta":{"raw":{"variants":["Smooth total spaces need not make homology fiber bundles smooth","Z-homology fiber bundles with smooth totals can fail smoothness","Projective homology bundles that are not topological fiber bundles","Nonsmooth Z-homology fiber bundles whose total space is smooth","Homology fiber bundles can have smooth total space yet not be smooth"]},"model":"grok-4.5","effort":"low","cost_usd":0.006082,"raw_usage":{"total_tokens":1437,"prompt_tokens":587,"num_sources_used":0,"completion_tokens":85,"cost_in_usd_ticks":60820000,"prompt_tokens_details":{"text_tokens":587,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":765,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":587,"tokens_out":85,"duration_ms":5816,"temperature":1.0,"reasoning_tokens":765,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T05:05:52.966533+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the monodromy characteristic polynomial for one of the explicit sums of powers (for example the E8 singularity x^2+y^3+z^5) and check whether its value at 1 is really plus or minus 1; if it is not, the local system is not acyclic and the homology fiber bundle claim fails.","supporting_citations":[],"review_version":1}