Zeros of one-forms and the topology of algebraic maps

Botong Wang; Stefan Schreieder

Zeros of one-forms and the topology of algebraic maps

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.15102 v2 pith:E5JDE2GC submitted 2026-07-16 math.AG math.CV

Stefan Schreieder , Botong Wang This is my paper · ORCID

classification math.AGmath.CV MSC 14F4532Q5514D0655R10

keywords holomorphicone-formsAlbanesemorphismhomotopyfiberbundlesfibrationsoverthecircleAomotocomplexesblow-upconstructionKählermanifoldsprojectivevarieties

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What'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.

Soft 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.

One 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.

Bottom line: this is a major, credible result, and it deserves a serious referee. I'd cite it and discuss it in reading group.

Referee Report

2 major / 4 minor

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.

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 (2)

[§4.3, Corollary 4.4] 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.
[Lemma 4.6] 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.

minor comments (4)

[§3.2, Proposition 3.3(1)] 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.
[§4.1] 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.
[Lemma 4.6] 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).
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constructions reduce to external results ([CK26], [DJL17], Latour, Farrell), and the paper disproves rather than relies on the first author's earlier conjecture.

full rationale

The derivation chain is not circular. Theorem 1.1 starts from the external Corrêa–Kollár homology fiber bundle that is not a homotopy fiber bundle (re-derived in Theorem 3.2), globalizes it, applies a blow-up construction, and verifies Latour's conditions using standard group-ring and spectral-sequence facts; none of the verified properties is assumed as an input. Theorem 1.4 starts from the external Debarre–Jiang–Lahoz rational cohomology torus and proves Property (C) by a cohomological computation over all finite étale covers; the non-fibration statement is proved by a separate F2-cohomology infinite-dimensionality argument. The only overlapping-author citations ([Schr21], [SY25]) supply the definition of Property (C) and motivational context; the paper's main theorems disprove, rather than rely on, the first author's conjecture, so those citations are not load-bearing in the direction of the claimed result. The skeptic's concern about Corollary 4.4 — that freeness of H*(X,Q) over H*(E,Q) does not formally follow from the derived splitting in Corollary 4.3 — is a legitimate mathematical gap in the written proof, but it is not a circularity: Corollary 4.4 is not obtained by renaming an input, fitting a parameter, or importing the conclusion through a self-citation. For circularity scoring, no step reduces to its own input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; this is pure mathematics. The central claims depend on prior/standard theorems: the blow-up formula and decomposition theorem, Ehresmann's theorem, Whitehead's theorem, Tischler's theorem, Latour's theorem, Farrell's theorem, the vanishing of Whitehead groups of Z^n, plus the two prior examples (Corrêa–Kollár, Debarre–Jiang–Lahoz) used as building blocks. None of these are invented ad hoc for this paper.

assumptions (6)

standard math Latour's theorem [Lat94, Thm 1']: for closed manifolds of dimension ≥6, a real class u is representable by a nonsingular closed 1-form iff stability, vanishing Novikov homology, and vanishing Whitehead–Franz–Reidemeister torsion hold.
Used in Proposition 3.6 and Theorem 1.1 to show every nonzero class in H^1(X,R) has a zero-free representative.
standard math Tischler's theorem [Ti70]: a compact manifold admitting a closed nonsingular 1-form fibers smoothly over S^1.
Used in Theorem 1.4 to pass from a zero-free real one-form to a smooth fibration over the circle.
standard math Farrell's theorem [Far71, Thm 6.4] on the obstruction to fibering a manifold over S^1.
Used in Proposition 3.4 to produce a smooth S^1-fibration for the Theorem 1.1 example.
standard math Blow-up formula for pushforwards of constant sheaves (2.1) and decomposition theorem [BBD82].
Used in Proposition 2.1 and Corollary 4.3 to control Z- and Q-homology fiber bundle properties under blow-up.
domain assumption Corrêa–Kollár example [CK26]: a Z-homology fiber bundle V→P^1 that is not a submersion and locally not a homotopy fiber bundle.
Starting point for Theorem 1.1; the paper gives a direct proof of the needed case in Theorem 3.2.
domain assumption Debarre–Jiang–Lahoz rational cohomology torus S [DJL17, Ex 1.11]: p:S→E is a Q-homology fiber bundle with R^i p_*Q trivial local systems but R^2 p_*Z not local.
Starting point for Theorem 1.4; recalled with proof sketch in Section 4.1.

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Pith. "Pith review of Zeros of one-forms and the topology of algebraic maps." pith.science (2026). https://pith.science/paper/E5JDE2GC

@misc{pith2026260715102,
  author       = {Pith},
  title        = {Pith review of: Zeros of one-forms and the topology of algebraic maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5JDE2GC}},
  note         = {Machine review of arXiv:2607.15102}
}

abstract

We construct a smooth complex projective variety whose Albanese morphism is a homotopy fiber bundle but not a submersion. The same variety fibers smoothly over the circle, although every holomorphic one-form on it has a zero. A second construction yields smooth complex projective varieties $X$ such that the Aomoto complex of every nonzero holomorphic one-form on every connected finite \'etale cover of $X$ is exact, while $X$ admits no real closed one-form without zeros. The two constructions build, respectively, on a homology fiber bundle of Corr\^ea--Koll\'ar that is not a homotopy fiber bundle and on a rational cohomology torus constructed by Debarre--Jiang--Lahoz. Consequently, we disprove Kotschick's conjecture, the remaining implication in the Bobadilla--Koll\'ar conjecture, and a conjecture of the first-named author.

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