{"id":"840df4ea-827a-4fa2-9709-872d107c30d6","arxiv_id":"2607.15102","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New explicit projective varieties disprove Kotschick's conjecture, the remaining implication of the Bobadilla–Kollár conjecture, and Schreieder's conjecture on zeros of holomorphic one-forms.","lead":"The paper constructs explicit high-dimensional complex algebraic varieties that violate topological properties previously conjectured to be equivalent: a variety can fiber smoothly over a circle even though every holomorphic one-form vanishes somewhere. A second variety satisfies a strong cohomological condition on all finite covers yet carries no real closed one-form without zeros.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.4's freeness assertion does not follow from the derived splitting in Corollary 4.3; it is load-bearing for Property (C).","rationale":"The reader's weakest assumption is exactly the freeness assertion in Corollary 4.4, and I agree it is the most load-bearing gap. The derivation from Corollary 4.3 is invalid: a splitting in the derived category of sheaves does not entail a splitting of the cup-product module structure. All of Property (C) in Theorem 1.4 depends on the exactness of the Aomoto complexes, which in the proof is obtained solely from the free-module decomposition. Other components of the paper, such as the simply-connectedness argument in Proposition 3.3 and the Mayer–Vietoris computation in Lemma 4.6, are plausible or have minor typographical issues, but they are not as load-bearing. A direct module computation would settle whether the gap is formal or substantive. Since the reader already assigned CONDITIONAL, my assessment does not change the verdict; the concern is addressable and does not by itself falsify the theorems.","tokens_in":125,"tokens_out":19870,"duration_ms":886335,"concrete_test":"Compute H*(X,Q) explicitly as an H*(E,Q)-module for the construction in §4.2 using the blow-up formula (2.1) and the known rational cohomology ring of S (an exterior algebra on four degree-one generators, with p:S→E making H*(S,Q) free over H*(E,Q)). Check whether the two degree-one generators of H*(E,Q) form a regular sequence on H*(X,Q): equivalently, write down a homogeneous basis and verify that multiplication by any nonzero α∈H^1(E,Q) gives an exact Koszul-type complex on H*(X,Q). If this check succeeds, Corollary 4.4 is true and the gap can be repaired by inserting this computation; if it fails, find a class u such that (H*(X,C), ∧u) has nonzero cohomology, which would disprove Proposition 4.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.4 rests on Proposition 4.1, which asserts the exactness of all Aomoto complexes on every finite étale cover. The argument reduces this to Corollary 4.4: H*(X,Q) is free as a graded H*(E,Q)-module. But Corollary 4.4 is not justified by Corollary 4.3. Corollary 4.3 only gives an isomorphism in the derived category D^b(E): Rh_*Q_X ≅ ⊕_{0≤i≤2N} Q_E[-i]^{⊕a_i}. This determines H*(X,Q) as a graded vector space (a direct sum of shifted copies of H*(E,Q)), but it does not determine the cup-product action of H*(E,Q) on H*(X,Q). An object-level splitting of R h_*Q_X does not imply an algebra-level splitting; the multiplication by pullback classes could have extensions or nonzero Massey products that are invisible to the derived decomposition. The 'therefore' in Corollary 4.4 is a non sequitur. The freeness is exactly what allows Proposition 4.1 to decompose the Aomoto complex (H*(X,C), ∧ω) into shifted copies of the exterior Koszul complex on H*(E,C), whose exactness is then immediate. Without freeness, the Aomoto complex may have nonzero cohomology even if the vector-space splitting holds. Hence Property (C), and with it the claimed counterexample to (C)⇒(A), is not established by the text as written. This is a concrete, fixable gap rather than a disproof, but it is the most load-bearing unsupported step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs two classes of smooth complex projective varieties. The first, a 7-fold X, has Albanese map f:X→E (E an elliptic curve) that is a homotopy fiber bundle but not a submersion, while X fibers smoothly over S^1 and yet every holomorphic 1-form has a zero. The second, a 5-fold X, is shown to satisfy a strong form of Property (C) — for every nonzero holomorphic 1-form on every finite étale cover, the Aomoto complex is exact — while X admits no nonzero real closed 1-form without zeros. The authors conclude that Kotschick's conjecture, the (ii)⇒(i) implication of the Bobadilla–Kollár conjecture, and Schreieder's (C)⇒(A) conjecture are false. The proofs combine a globalized version of the Corrêa–Kollár example with a blow-up trick, and a rational cohomology torus of Debarre–Jiang–Lahoz.","tokens_in":15683,"tokens_out":16597,"duration_ms":168739,"significance":"If the results are correct, they settle several open questions in the topology of algebraic varieties and 1-forms. The first construction gives a counterexample to the remaining implication in the Bobadilla–Kollár conjecture and to Kotschick's conjecture in dimension 7; notably, the verification that every real cohomology class is representable by a nowhere vanishing closed 1-form is done through Latour's criteria, and the construction is genuinely projective. The second construction gives a counterexample to the (C)⇒(A) conjecture, a property previously known to be implied by (B); the stronger exactness statement for all holomorphic 1-forms on all finite étale covers is a considerable strengthening. The paper also provides a self-contained sheaf-theoretic proof of the Corrêa–Kollár theorem (Theorem 3.2) and uses external results (CK26, DJL17) in a non-circular way; there are no fitted parameters. If the gaps identified below are repaired, this is a significant advance.","major_comments":[{"comment":"The freeness assertion does not follow from Corollary 4.3. A derived isomorphism R h_* Q_X ≅ ⊕ Q_E[-i]^{a_i} in D^b(E) determines H^*(X,Q) only as a graded vector space (or as the cohomology of a direct sum of shifted constant sheaves); it does not determine the cup-product action of H^*(E,Q) on H^*(X,Q). The displayed module isomorphism in the proof of Corollary 4.4 is therefore a non sequitur. This freeness is exactly what is used in Proposition 4.1 to decompose the Aomoto complex into shifted Koszul complexes on H^*(E,C). Without a proof of the module structure — e.g., an explicit blow-up computation of the cohomology ring of Bl_S(P^N×E), or a Leray–Hirsch argument using the classes from P^N and the exceptional divisor — the exactness of all Aomoto complexes, and hence Theorem 1.4, is not established by the text.","section":"§4.3, Corollary 4.4"},{"comment":"The topological decomposition of \\tilde E into U, V=∪V_b, and W is only described informally (\"as illustrated in the figure\", but no figure appears). The assertion that W can be chosen contractible with W∩(V∪U) a disjoint union of two contractible subsets is not automatic for an infinite strip with infinitely many removed neighborhoods; the proof of infinite-dimensionality of H^2(\\tilde S,F_2) depends on this decomposition. A precise construction of these sets (or a different argument) is needed before Proposition 4.5, and with it the nonexistence of a smooth S^1-fibration in Theorem 1.4, is established.","section":"Lemma 4.6"}],"minor_comments":[{"comment":"The simple-connectivity of the blow-up g^{-1}(U) is invoked without justification. Since the center has codimension at least 3, this is standard, but a reference or a one-line argument would be helpful.","section":"§3.2, Proposition 3.3(1)"},{"comment":"The sentence \"This implies that the Albanese map of S induces an isomorphism of rational cohomology groups. In particular, the rational cohomology ring...\" is logically imprecise: the ring statement does not follow from the cohomology-group isomorphism alone, though it is true for surfaces by Poincaré duality. Rephrase to separate the two statements.","section":"§4.1"},{"comment":"The notation H^2(\\tilde S,F_2) appears twice in the final sentence: the first is homology, the second cohomology. Please distinguish them (e.g., H_2 vs. H^2).","section":"Lemma 4.6"},{"comment":"The two \"as illustrated in the following figure\" passages in Lemma 4.6 refer to missing figures; the proof is difficult to follow without them.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main technical issue is the freeness gap in Corollary 4.4. It appears fixable by a direct computation of the H^*(E)-module structure on the blow-up cohomology, but as written it is a genuine gap in the proof of Theorem 1.4. The Lemma 4.6 decomposition also needs tightening. If the authors provide the missing arguments, the paper would be a strong contribution to the field."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should read this one. It kills Kotschick's conjecture, the remaining Bobadilla–Kollár implication, and Schreieder's (C)⇒(A) conjecture with explicit smooth projective counterexamples. The constructions are concrete, and the main arguments hold up. The reader's stress-test about Corollary 4.4 does not land.\n\nWhat's new: two constructions. Theorem 1.1 builds a 7-fold whose Albanese map is a homotopy fiber bundle but not a submersion, yet every nonzero real class is represented by a nonsingular closed 1-form while every holomorphic 1-form has a zero. Theorem 1.4 gives a 5-fold satisfying Property (C) but with no real closed 1-form without zeros. The blow-up/base-change trick controls fundamental groups while preserving the needed cohomological structure, and the Latour/Farrell checks are clean. The paper also re-derives the key Corrêa–Kollár statement, which is good practice.\n\nSoft spots are minor. The proof of Theorem 3.2 has a subscript typo (ξ₂ should be ξ₁ when discussing the fixed point in S_t). More substantively, Corollary 4.4 is stated with a one-line proof, and the stress-test claims the derived splitting doesn't imply freeness of the H*(E)-module. That's not right: an isomorphism in D(E) is an isomorphism of the object, and the H*(E)-module structure on H*(E, –) is functorial. If Rh_*Q_X is a direct sum of shifts of Q_E in the derived category, then H*(X,Q) is indeed free as a graded H*(E,Q)-module. The authors should add a sentence explaining this, but it is not a gap. Proposition 4.1's reduction to the Koszul complex then goes through.\n\nOne place that could use a footnote is Proposition 3.3(1), where they assert a small neighborhood deformation retracts onto a fiber. That's not automatic for arbitrary proper maps, but it is plausible here because of the local model and the blow-up construction; I would flag it as a request for clarification, not a substantive error.\n\nBottom line: this is a major, credible result, and it deserves a serious referee. I'd cite it and discuss it in reading group.","headline":"This paper disproves three conjectures with explicit projective counterexamples, and the reader's main worry about Corollary 4.4 is not a real gap—the derived splitting does give the claimed module freeness.","tokens_in":16192,"tokens_out":11475,"would_cite":true,"duration_ms":123251,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F45","32Q55","14D06","55R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Complex projective 7-fold can admit zero-free real 1-forms even though every holomorphic 1-form vanishes, while a 5-fold satisfies the strongest cohomological vanishing yet has no zero-free real 1-form.","keywords":["holomorphic one-forms","Albanese morphism","homotopy fiber bundles","fibrations over the circle","Aomoto complexes","blow-up construction","Kähler manifolds","projective varieties"],"falsifier":"For the fivefold X of (4.1) with N=4, take any surjection π_1(X)→Z, form the associated infinite cyclic cover, and compute its F_2-cohomology. If for some surjection the F_2-cohomology is finite-dimensional, then X fibers smoothly over S^1 and Theorem 1.4 is false.","tokens_in":15205,"feed_emoji":"","tokens_out":11312,"duration_ms":110333,"temperature":0.7,"pith_summary":"Compact complex projective varieties can exhibit nontrivial real one-form topology while all holomorphic one-forms vanish. The paper constructs a 7-fold whose Albanese map to an elliptic curve is a homotopy fiber bundle but not a submersion: the variety fibers smoothly over the circle, every nonzero real cohomology class is represented by a closed 1-form without zeros, yet every holomorphic 1-form has a zero. A second construction yields a 5-fold with the same Albanese-cohomology control: for every nonzero holomorphic 1-form on every finite étale cover, the Aomoto complex (cohomology wedged with the form) is exact, while no real closed 1-form is zero-free. Together these counterexamples disprove the conjecture linking real and holomorphic zero-free one-forms, the remaining implication of the conjecture about homotopy fiber bundles over the disc, and a further conjecture asserting that exact Aomoto complexes force a holomorphic one-form without zeros.","feed_headline":"Complex 7-fold disproves a 1-form conjecture","feed_subtitle":"Real closed 1-forms can be zero-free while every holomorphic 1-form vanishes; a 5-fold extends the break to the cohomological setting.","key_machinery":"The central mechanism is the blow-up trick. To construct X, one starts with a known 'bad' fibration over a base E (a homology fiber bundle that is not a homotopy fiber bundle, or a rational cohomology torus), embeds it into P^N × E, and blows up along it. The resulting variety has the same fundamental group as E (isomorphism via the Albanese map), while the exceptional divisors carry the controlled cohomology. For the sevenfold, this turns the known local homology fiber bundle into a global homotopy fiber bundle; for the fivefold, it turns the rational cohomology torus into a variety whose cohomology is a free module over the base cohomology, making all Aomoto complexes exact. The negative s","core_discovery":"The paper's first object is a smooth complex projective sevenfold X. Its Albanese morphism f:X→E to an elliptic curve is a homotopy fiber bundle — every finite cover pulled back from a cover of E, all higher direct images of the constant sheaf Z are trivial — and it induces an isomorphism on fundamental groups. Nevertheless f is not a submersion: the underlying construction contains a singular fiber of a known homology fiber bundle, and the blow-up that creates X does not remove that failure. Because f is a homotopy fiber bundle with trivial local systems, a general criterion shows every nonzero class in H^1(X,R) is represented by a nonsingular closed real 1-form, and the classical criterion","pith_inferences":["The dimensions 7 and 5 come from the embedding choices (using a generic projection lemma); it would be natural to search for lower-dimensional analogues, and the authors' Question 3.11 asks whether general-type examples exist.","The blow-up trick likely generalizes: one can take any singular fibration over an abelian base with controlled cohomology and blow up a subvariety that resolves the fundamental group to that of the base, suggesting a template for further counterexamples.","The proof for the fivefold hinges on a freeness statement for H^*(X,Q) over H^*(E,Q); a direct cup-product computation checking this freeness on the actual cohomology ring would test the robustness of the method.","The F_2-cohomology non-vanishing of infinite cyclic covers suggests a general obstruction: if a projective variety fibers smoothly over S^1, its infinite cyclic covers must be homotopy finite, so checking F_2-cohomology of such covers could be a practical test for non-fibering."],"forward_implications":["There is a smooth complex projective 7-fold X with b_1(X)>0 that fibers smoothly over S^1, yet every holomorphic 1-form on X has a zero; existence of a real zero-free 1-form does not imply existence of a holomorphic one.","The remaining implication of the conjecture about homotopy fiber bundles over the disc fails: a homotopy fiber bundle that is not a submersion exists among projective morphisms (by restricting the construction to a neighborhood of a critical value).","Property (C) — exactness of all Aomoto complexes on all finite étale covers — does not imply property (A) nor property (B); the fivefold satisfies (C) but admits no zero-free real 1-form.","Even in the presence of a smooth fibration over S^1, the harmonic representative of a nonzero class in H^1(X,R) always has a zero, so harmonic representatives do not inherit the nonvanishing property.","The examples have Kodaira dimension −∞ and are birational to P^N × E, so the phenomenon is not forced by positivity of the canonical bundle."],"fun_headline_variants":["7-fold disproves Kotschick's one-form conjecture","Holomorphic 1-forms always have zeros on new 7-fold","Real zero-free 1-forms exist, holomorphic ones don't on 7-fold","Smooth over circle yet every holomorphic 1-form has a zero"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of Theorem 1.4 depends on the assertion that the cohomology ring of the blown-up variety X is a free module over the cohomology ring of the elliptic curve E, so that wedging with a holomorphic one-form decomposes into exact exterior complexes; if that freeness fails for the actual cup product, the exactness of the Aomoto complexes is not established.","fun_headline_variants_meta":{"raw":{"variants":["7-fold disproves Kotschick's one-form conjecture","Holomorphic 1-forms always have zeros on new 7-fold","Real zero-free 1-forms exist, holomorphic ones don't on 7-fold","Smooth over circle yet every holomorphic 1-form has a zero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001005,"raw_usage":{"total_tokens":4068,"prompt_tokens":706,"completion_tokens":3362,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":3281}},"tokens_in":450,"tokens_out":3362,"duration_ms":25860,"temperature":1.0,"reasoning_tokens":3281,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T00:13:01.628768+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the fivefold X of (4.1) with N=4, take any surjection π_1(X)→Z, form the associated infinite cyclic cover, and compute its F_2-cohomology. If for some surjection the F_2-cohomology is finite-dimensional, then X fibers smoothly over S^1 and Theorem 1.4 is false.","supporting_citations":[],"review_version":1}