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Homology fiber bundles of varieties, that are not topological fiber bundles

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Pith's one-line read Smooth total spaces can form Z-homology fiber bundles that fail to be smooth or topological fiber bundles.

desk verdict 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. read the letter →

arxiv 2607.05603 v1 pith:JAFR3AJ6 submitted 2026-07-06 math.AG math.CVmath.GT

Maurício Corrêa , János Kollár This is my paper · ORCID
classification math.AGmath.CVmath.GT MSC 14D0514B0532S5055R10
keywords homologyfiberbundletopologicalequisingularityMilnormonodromyabelianvarietyquotientprojectivemorphism
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper constructs flat projective morphisms Y o riangle 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 o riangle 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 imes X)/(τ, γ) by a translation τ of the same order on an abelian variety A. Theorem 3 asserts that Y o riangle 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.

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.

minor comments (5)
  1. [Construction 11] 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.
  2. [§12] 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.
  3. [Construction 13] 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.
  4. [Introduction] The notation “Z2r” for the fundamental groups of the fibres should be clarified (e.g., as ℤ/2rℤ or (ℤ/2ℤ)^r).
  5. Minor typographical issues: “c ith” (should be “c-th”), missing space in “forn≥2”, and inconsistent formatting of group-order notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: direct geometric quotient construction plus classical monodromy, with self-citation only of the conjecture being disproved.

full rationale

The derivation of Theorem 3 proceeds by an explicit quotient Y = (A imes X)/( au, au) of a projective morphism with isolated critical points by a finite-order diagonal action, followed by a Mayer-Vietoris decomposition of the fibers (paragraph 8) that reduces the Z-homology-fiber-bundle claim to the acyclicity of a local system L_T on the circle (Lemma 9 and paragraph 10). The latter is an elementary computation: H^*(S^1, L_T) vanishes precisely when au_T(1) = au1. Concrete examples (Constructions 11/13, Example 4, Example 14) verify the monodromy condition by invoking the classical eigenvalue lists of Pham and Brieskorn for sums of powers (via Thom-Sebastiani), which are external and parameter-free. The only self-citation is the 2012 conjecture of the second author that is being disproved; it supplies the target statement, not a load-bearing premise of the proof. There is no fitting of parameters, no redefinition of the target quantity by construction, no uniqueness theorem imported from prior joint work, and no ansatz smuggled via citation. The argument is therefore self-contained against external classical benchmarks.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is pure algebraic geometry / topology of singularities. It relies on standard tools (Mayer-Vietoris, Künneth, Leray spectral sequence, Thom-Sebastiani, classical monodromy of Pham-Brieskorn) and on the topological fact that abelian varieties are homeomorphic to tori. No free parameters are fitted; the only numerical choices are the tuples of exponents c_i that make the monodromy condition hold. No new physical or geometric entities are postulated.

assumptions (5)
  • standard math Thom-Sebastiani theorem: monodromy of a sum of powers is the tensor product of the individual monodromies.
    Used in Section 12 to identify the eigenvalues of T_c with sums of fractions a_i/c_i.
  • standard math Pham-Brieskorn computation of monodromy eigenvalues for x^r = t (roots of unity exp(2 pi i a/r) for a=1..r-1).
    Cited as [Pha65, Bri66]; supplies the input eigenvalues for the characteristic polynomial evaluation at 1.
  • standard math An abelian variety of complex dimension m is homeomorphic to (S^1)^{2m}.
    Used in Lemma 9 to reduce the quotient (A x M)/(tau,gamma) to a product involving a single circle.
  • standard math For a local system L_T on S^1 with monodromy T in GL_r(Z), H^0 vanishes iff 1 is not an eigenvalue and H^1 vanishes iff det(T-I)=plus or minus 1.
    Elementary computation recorded in paragraph 10; converts chi_T(1)=plus or minus 1 into acyclicity.
  • domain assumption The weighted projective hypersurface admits a mu_c-equivariant simultaneous resolution along the divisor w=0.
    Stated as 'not hard to check; we leave this to the reader' in Construction 13; needed so that the only critical point of the resolved family is the origin of the sum of powers.

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Pith. "Pith review of Homology fiber bundles of varieties, that are not topological fiber bundles." pith.science (2026). https://pith.science/paper/JAFR3AJ6

@misc{pith2026260705603,
  author       = {Pith},
  title        = {Pith review of: Homology fiber bundles of varieties, that are not topological fiber bundles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAFR3AJ6}},
  note         = {Machine review of arXiv:2607.05603}
}
abstract

We construct flat, projective morphisms that are $\mathbb Z$-homology fiber bundles, have a smooth total space, but are not smooth. This disproves one of the conjectures of the second author and Fern\'andez de Bobadilla.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Zeros of one-forms and the topology of algebraic maps

    math.AG 2026-07 conditional novelty 8.0 of 10

    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.

Reference graph

Works this paper leans on

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