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Geometry of Holomorphic One-forms on Smooth Projective Varieties

T0 review · 2 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read A morphism from a smooth projective variety to a simple abelian variety is smooth if and only if the pullback of some holomorphic 1-form has no zeros.

desk verdict The iff smoothness criterion for maps to simple abelian varieties is the core new claim, resting on a Sabbah-style lemma about Z-homology bundles, while the non-linear counterexample on zero loci stands out as a distinct addition. read the letter →

arxiv 2606.08185 v1 pith:NGG5FR6X submitted 2026-06-06 math.AG

Jiabin Du , Feng Hao , Haoyuan Li , Zichang Wang This is my paper · ORCID
classification math.AG
keywords holomorphicone-formssmoothprojectivevarietiessimpleabelianmorphismssmoothnesscriterionzerolocifibrebundlesblow-ups
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 proves an if-and-only-if criterion for smoothness of morphisms from smooth projective varieties to simple abelian varieties. The morphism is smooth precisely when there exists a holomorphic 1-form on the abelian variety whose pullback has no zeros. The argument rests on a technical result that Z-homology fibre bundle morphisms have no blow-ups in codimension zero. Further sections examine the structure of spaces of holomorphic 1-forms that do have zeros and construct an explicit counterexample to linearity in one case.

What carries the argument

The equivalence between smoothness of f and the existence of a holomorphic 1-form ω on A with f*ω nowhere zero, which follows from the no-blow-up property of Z-homology fibre bundle morphisms.

What would settle it

A morphism from a smooth projective variety to a simple abelian variety that is smooth yet every pullback of a holomorphic 1-form has a zero, or that is not smooth yet some pullback has no zero.

Watch

Extended reading notes

Core claim

Any morphism f from a smooth projective variety X to a simple abelian variety A is smooth if and only if there exists a holomorphic 1-form ω on A such that f*ω has no zero. This equivalence is obtained by showing that any Z-homology fibre bundle morphism is without blow-up in codimension 0 in the sense of Sabbah. The paper additionally shows that spaces of holomorphic 1-forms with zeros are linear for large classes of varieties, constructs a smooth projective subvariety of an abelian variety where such forms with positive-dimensional zero loci do not form a linear subset, and studies algebraic surfaces admitting holomorphic 1-forms with zeros that do not arise from cohomology jump loci.

Load-bearing premise

Z-homology fibre bundle morphisms have no blow-ups in codimension zero, which controls the geometry of the morphism f.

Editorial extensions

If this is right

  • Smoothness of morphisms to simple abelian varieties reduces to the existence of one non-vanishing pullback of a holomorphic 1-form.
  • Spaces of holomorphic 1-forms with zeros form linear subspaces for many varieties.
  • There exist smooth projective subvarieties of abelian varieties where holomorphic 1-forms with positive-dimensional zero loci fail to form a linear subset.
  • Algebraic surfaces exist that admit holomorphic 1-forms with zeros not coming from cohomology jump loci.

Reading between the lines

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

  • The criterion supplies a practical test for smoothness that could apply to classification of maps onto abelian varieties.
  • The counterexample to linearity shows that zero-locus geometry of 1-forms can be nonlinear even inside abelian varieties.
  • The surface examples suggest possible links between zero loci of 1-forms and the structure of irregular fibrations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper proves that for a morphism f from a smooth projective variety X to a simple abelian variety A, f is smooth if and only if there exists a holomorphic 1-form ω on A such that f*ω has no zeros. The central proof uses the key result that any ℤ-homology fibre bundle morphism has no blow-up in codimension 0 in Sabbah's sense. It additionally shows that spaces of holomorphic 1-forms with zeros are linear for large classes of varieties, constructs a counterexample subvariety of an abelian variety where such spaces are nonlinear, and studies algebraic surfaces admitting holomorphic 1-forms with zeros that do not arise from cohomology jump loci.

Significance. If the central iff criterion holds, it supplies a concrete geometric test for smoothness of morphisms to simple abelian varieties via pullbacks of 1-forms, potentially simplifying arguments about fibrations and zero loci in algebraic geometry. The extension of Sabbah's framework on homology bundles is a technical contribution that may apply more broadly. The explicit nonlinear example and the surface classification provide concrete data points that refine understanding of when linearity holds or fails.

major comments (2)
  1. [Key ingredient / proof of the main theorem] The key lemma asserting that every ℤ-homology fibre bundle morphism has no blow-up in codimension 0 (Sabbah sense) is load-bearing for the main iff theorem; the manuscript must supply a self-contained verification or explicit reduction showing independence from prior Sabbah results, as this step directly controls the geometry of f.
  2. [Section on the nonlinear example] In the construction of the delicate example (smooth projective subvariety of an abelian variety where 1-forms with positive-dimensional zero loci fail to form a linear subset), the argument that the zero loci are not linear must be checked against the definition of linearity used earlier in the paper; without explicit local equations or dimension counts, it is unclear whether the example is minimal or relies on special position.
minor comments (2)
  1. [Introduction] Notation for the pullback f*ω and the zero set should be introduced uniformly in the introduction and used consistently in all statements of theorems.
  2. [Section on algebraic surfaces] The final section on algebraic surfaces would benefit from a table summarizing the examples and which cohomology jump loci they avoid.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity and self-containedness.

read point-by-point responses

  1. Referee: [Key ingredient / proof of the main theorem] The key lemma asserting that every ℤ-homology fibre bundle morphism has no blow-up in codimension 0 (Sabbah sense) is load-bearing for the main iff theorem; the manuscript must supply a self-contained verification or explicit reduction showing independence from prior Sabbah results, as this step directly controls the geometry of f.

    Authors: We agree that the key lemma requires a fully self-contained presentation. While the manuscript includes a proof of the statement that every ℤ-homology fibre bundle morphism has no blow-up in codimension 0, we will expand this argument in the revised version by adding an explicit reduction directly from the definition of Sabbah's blow-up in codimension 0, without assuming additional prior results beyond the basic setup. This will make the independence clear and better control the geometry of the morphism f. revision: yes

  2. Referee: [Section on the nonlinear example] In the construction of the delicate example (smooth projective subvariety of an abelian variety where 1-forms with positive-dimensional zero loci fail to form a linear subset), the argument that the zero loci are not linear must be checked against the definition of linearity used earlier in the paper; without explicit local equations or dimension counts, it is unclear whether the example is minimal or relies on special position.

    Authors: We appreciate this point on the nonlinear example. In the revision, we will add explicit local equations defining the smooth projective subvariety inside the abelian variety, together with dimension counts for the relevant spaces of holomorphic 1-forms. These will be checked directly against the definition of linearity introduced earlier in the paper, confirming that the zero loci do not form a linear subset and clarifying that the construction does not rely on special position. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper states theorems establishing an if-and-only-if smoothness criterion for morphisms f: X → A (A simple abelian) via existence of a nowhere-vanishing pullback 1-form ω. The key technical claim—that every ℤ-homology fibre bundle morphism has no blow-up in codimension 0 in Sabbah's sense—is presented as a result shown in the paper itself. No equations, definitions, or steps reduce by construction to inputs, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation is self-contained once the stated lemma is granted, with no internal reductions to prior author work or ansatzes smuggled via citation.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper relies on standard results in algebraic geometry and complex geometry (e.g., properties of abelian varieties, pullbacks of forms) plus the technical lemma on Z-homology fibre bundles. No free parameters or invented entities are mentioned in the abstract.

assumptions (1)
  • domain assumption Any Z-homology fibre bundle morphism is without blow-up in codimension 0 in the sense of Sabbah.
    Invoked explicitly as the key ingredient for proving the smoothness criterion.

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Pith. "Pith review of Geometry of Holomorphic One-forms on Smooth Projective Varieties." pith.science (2026). https://pith.science/paper/NGG5FR6X

@misc{pith2026260608185,
  author       = {Pith},
  title        = {Pith review of: Geometry of Holomorphic One-forms on Smooth Projective Varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGG5FR6X}},
  note         = {Machine review of arXiv:2606.08185}
}
abstract

In this article, we show that any morphism $f$ from a smooth projective variety $X$ to a simple abelian variety $A$ is smooth, if and only if there exists a holomorphic 1-form $\omega$ on $A$ such that $f^*\omega$ has no zero. As the key ingredient in the proof, we show any $\mathbb{Z}$-homology fibre bundle morphism is without blow-up in codimension 0 in the sense of Sabbah. Furthermore, we investigate the structure of the spaces of holomorphic 1-forms with zeros, and show that they are linear for large classes of varieties. Also, we construct a delicate example of a smooth projective subvariety of an abelian variety for which the holomorphic 1-forms with positive dimensional zero loci do not form a linear subset. Finally, we study algebraic surfaces admitting holomorphic 1-forms that have zeros and do not arise from cohomology jump loci.

Figures

Figures reproduced from arXiv: 2606.08185 by the authors.

Figure 1
Figure 1. Assume dim Y = m and dim X = n, take a small open ball Uy0 together with complex analytic coordinates z1, . . . , zm around y0. Then dz1, . . . , dzm are nowhere vanishing pointwise linearly independent local holomorphic 1-forms on Uy0 . For any closed point x ∈ X◦ ∩ f −1 (Uy0 ) at which f is smooth, the cotangent map f ∗ x : T ∗ f(x) Y → T ∗ xX is injective (the dual of the surjective tangent map at x). Thus f ∗ x … view at source ↗

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Forward citations

Cited by 3 Pith papers

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

  1. Invisible singularities in complex algebraic geometry

    math.AG 2026-08 accept novelty 8.0 of 10

    Morphisms from smooth projective varieties to P^1 can have singular fibers that are topologically invisible, yielding counterexamples to the Fernandez de Bobadilla-Kollar, Kollar-Pardon, Kotschick, and Schreieder conjectures.

  2. 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.

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

    math.AG 2026-07 accept novelty 7.0 of 10

    There exist flat projective morphisms with smooth total space that are Z-homology fiber bundles but are neither smooth nor topological fiber bundles.

Reference graph

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