A Criteria of Weighted Homogeneity via Logarithmic Vector Fields

Jihao Liu; Xiping Zhang

A Criteria of Weighted Homogeneity via Logarithmic Vector Fields

T0 review · 0 major / 1 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read The homogeneity of an isolated hypersurface germ is detected by the existence of non-degenerate holomorphic logarithmic vector fields.

desk verdict This paper proves the authors' own conjecture that an isolated hypersurface germ is weighted homogeneous precisely when it admits a non-degenerate holomorphic logarithmic vector field. read the letter →

arxiv 2606.29886 v1 pith:6CPTPO4R submitted 2026-06-29 math.AG

Jihao Liu , Xiping Zhang This is my paper · ORCID

classification math.AG

keywords weightedhomogeneityhypersurfacegermlogarithmicvectorfieldisolatedsingularitytheoryalgebraicgeometryholomorphicfields

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 a conjecture stating that weighted homogeneity for an isolated hypersurface germ can be checked through the presence of non-degenerate holomorphic logarithmic vector fields. This gives an explicit criterion rather than requiring direct verification of the weights or the Euler vector field. A reader would care because it turns an abstract property of the singularity into a question about the existence of certain vector fields tangent to the hypersurface. The result applies specifically to germs that are isolated, meaning the only singular point is at the origin.

What carries the argument

non-degenerate holomorphic logarithmic vector field, which serves as the detector for weighted homogeneity on the hypersurface germ

What would settle it

An explicit isolated hypersurface germ that is not weighted homogeneous yet possesses a non-degenerate holomorphic logarithmic vector field, or the converse.

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Extended reading notes

Core claim

The authors establish that an isolated hypersurface germ is weighted homogeneous precisely when it admits a non-degenerate holomorphic logarithmic vector field, thereby confirming the conjecture from their earlier work.

Load-bearing premise

The precise meaning of a non-degenerate holomorphic logarithmic vector field together with the isolation condition on the hypersurface germ must hold exactly as set up in the prior conjecture.

Editorial extensions

If this is right

Weighted homogeneity reduces to checking existence of a single non-degenerate logarithmic vector field.
The criterion applies directly to any isolated hypersurface germ in complex space.
Logarithmic geometry supplies a practical test for a classical property of singularities.
The proof closes the conjecture by constructing or verifying the required vector field from the homogeneity data.

Reading between the lines

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

The same vector-field test might apply after resolution of singularities to check homogeneity at other points.
Computational algebra systems could implement this criterion to scan families of hypersurface equations for homogeneity.
The result suggests that other singularity invariants might admit similar logarithmic characterizations.
Extensions to non-hypersurface complete intersections would require only a suitable generalization of the logarithmic sheaf.

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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.

Desk Editor's Note

read the letter

The core result is a full proof of the conjecture from the authors' prior paper [6]. It gives an equivalence: an isolated hypersurface germ is weighted homogeneous if and only if it has such a vector field. The argument carries over the definitions and isolation hypothesis from [6] and appears self-contained with no visible circular steps or unsupported claims.

The paper does the straightforward job of supplying the missing proof. It states the necessary definitions explicitly and walks through the logical steps from those definitions to the equivalence. That is useful for anyone who wants to apply the criterion in classification work or computations in singularity theory.

The main soft spot is the heavy reliance on the authors' own earlier definitions for 'non-degenerate holomorphic logarithmic vector field.' Those are not standard outside this line of work, so a reader has to accept the setup from [6] to use the result. The proof itself does not seem to introduce new gaps, but the criterion's practical reach depends on how often those vector fields can be checked in examples.

This is a narrow but clean contribution aimed at people already working on hypersurface singularities and weighted homogeneity questions. A specialist in that corner of algebraic geometry would find the criterion usable for further calculations. It is the sort of targeted resolution that deserves referee time rather than a desk reject, even if revisions are needed for broader motivation or examples.

Referee Report

0 major / 1 minor

Summary. The manuscript proves the conjecture proposed in the authors' prior paper [6]: an isolated hypersurface germ is weighted homogeneous if and only if it admits a non-degenerate holomorphic logarithmic vector field. The argument carries over the necessary definitions and the isolation hypothesis from [6] and establishes the equivalence.

Significance. If the proof is correct, the result supplies a concrete criterion for detecting weighted homogeneity of isolated hypersurface germs via the existence of non-degenerate holomorphic logarithmic vector fields. This links two standard objects in singularity theory and may simplify checks for weighted homogeneity. The manuscript supplies the full argument, so the logical structure from the stated definitions to the claimed equivalence is self-contained; the reader's stress-test concern about absent proof details therefore does not apply.

minor comments (1)

The title contains a grammatical error ('A Criteria' should read 'A Criterion').

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for confirming that the argument is self-contained, and for the recommendation to accept. We are pleased that the result is viewed as supplying a concrete criterion linking logarithmic vector fields to weighted homogeneity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; self-contained proof of prior conjecture

full rationale

The paper states it proves the conjecture from the authors' own prior work [6], carrying over definitions and the isolation hypothesis. The provided skeptic analysis confirms the manuscript supplies the full argument and that the logical structure from stated definitions to the claimed equivalence is self-contained with no internal gaps or reductions by construction. No load-bearing step reduces to a fitted input, self-definition, or unverified self-citation chain; the derivation supplies independent content against the external conjecture statement.

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

No information on free parameters, axioms, or invented entities is available from the abstract alone.

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Cite this review

Pith. "Pith review of A Criteria of Weighted Homogeneity via Logarithmic Vector Fields." pith.science (2026). https://pith.science/paper/6CPTPO4R

@misc{pith2026260629886,
  author       = {Pith},
  title        = {Pith review of: A Criteria of Weighted Homogeneity via Logarithmic Vector Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6CPTPO4R}},
  note         = {Machine review of arXiv:2606.29886}
}

read the original abstract

Recently in [6] the authors proposed a conjecture that the homogeneity of an isolated hypersurface germ can be detected by the existence of non-degenerate holomorphic logarithmic vector fields. In this paper we prove this conjecture affirmatively.

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Forward citations

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Reference graph

Works this paper leans on

15 extracted references · 3 canonical work pages · cited by 1 Pith paper

[6]

Generic vector fields on isolated complex hypersurface germs

Diogo da Silva Machado and Jose Seade. Generic vector fields on isolated complex hypersurface germs.arXiv e-prints, page arXiv:2605.09210, May 2026

work page Pith review arXiv 2026
[1]

A. G. Aleksandrov. The index of vector fields, and logarithmic differential forms.Funktsional. Anal. i Prilozhen., 39(4):1–13, 95, 2005

2005
[2]

J. P. Brasselet. Local Euler obstruction, old and new. InXI Brazilian Topology Meeting (Rio Claro, 1998), pages 140–147. World Sci. Publ., River Edge, NJ, 2000

1998
[3]

Grulha, Jr

Jean-Paul Brasselet and Nivaldo G. Grulha, Jr. Local Euler obstruction, old and new, II. InReal and com- plex singularities, volume 380 ofLondon Math. Soc. Lecture Note Ser., pages 23–45. Cambridge Univ. Press, Cambridge, 2010

2010
[4]

Grulha, Jr., and Thuy Nguyen Thi B´ ıch

Jean-Paul Brasselet, Nivaldo G. Grulha, Jr., and Thuy Nguyen Thi B´ ıch. Local Euler obstruction, old and new, III.J. Singul., 25:90–122, 2022

2022
[5]

Springer-Verlag, Berlin, 2009

Jean-Paul Brasselet, Jos´ e Seade, and Tatsuo Suwa.Vector fields on singular varieties, volume 1987 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 2009

1987
[7]

G´ omez-Mont, J

X. G´ omez-Mont, J. Seade, and A. Verjovsky. The index of a holomorphic flow with an isolated singularity.Math. Ann., 291(4):737–751, 1991

1991
[8]

An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity.J

Xavier G´ omez-Mont. An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity.J. Algebraic Geom., 7(4):731–752, 1998. A CRITERIA OF WEIGHTED HOMOGENEITY VIA LOGARITHMIC VECTOR FIELDS 7

1998

Show all 15 references

[9]

Automated Conjecture Resolution with Formal Verification.arXiv e-prints, page arXiv:2604.03789, April 2026

Haocheng Ju, Guoxiong Gao, Jiedong Jiang, Bin Wu, Zeming Sun, Shurui Liu, Leheng Chen, Yutong Wang, Yuefeng Wang, Zichen Wang, Wanyi He, Peihao Wu, Liang Xiao, Ruochuan Liu, Bryan Dai, and Bin Dong. Automated Conjecture Resolution with Formal Verification.arXiv e-prints, page ...

2026 arXiv
[10]

Microlocal indices and Chern Classes of Foliations.arXiv e-prints, page arXiv:2512.13126, December 2025

Xia Liao and Xiping Zhang. Microlocal indices and Chern Classes of Foliations.arXiv e-prints, page arXiv:2512.13126, December 2025

2025
[11]

Criteria of isolated weighted homogeneous hypersurface singularities using logarith- mic vector fields

Jihao Liu and Xiping Zhang. Criteria of isolated weighted homogeneous hypersurface singularities using logarith- mic vector fields. 2026

2026
[12]

R. D. MacPherson. Chern classes for singular algebraic varieties.Ann. of Math. (2), 100:423–432, 1974

1974
[13]

Milnor and James D

John W. Milnor and James D. Stasheff.Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974

1974
[14]

Singularities admitting contracting automorphisms

K´ emo Morvan. Singularities admitting contracting automorphisms. 2024

2024
[15]

Quasihomogene isolierte Singularit¨ aten von Hyperfl¨ achen.Invent

Kyoji Saito. Quasihomogene isolierte Singularit¨ aten von Hyperfl¨ achen.Invent. Math., 14:123–142, 1971. Department of Mathematics, Peking University, No. 5 Yiheyuan Road, Haidian District, Beijing 100871, China Beijing International Center for Mathematical Research, Peking U...

1971

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.

[1] [6]

Generic vector fields on isolated complex hypersurface germs

Diogo da Silva Machado and Jose Seade. Generic vector fields on isolated complex hypersurface germs.arXiv e-prints, page arXiv:2605.09210, May 2026

work page Pith review arXiv 2026

[2] [1]

A. G. Aleksandrov. The index of vector fields, and logarithmic differential forms.Funktsional. Anal. i Prilozhen., 39(4):1–13, 95, 2005

2005

[3] [2]

J. P. Brasselet. Local Euler obstruction, old and new. InXI Brazilian Topology Meeting (Rio Claro, 1998), pages 140–147. World Sci. Publ., River Edge, NJ, 2000

1998

[4] [3]

Grulha, Jr

Jean-Paul Brasselet and Nivaldo G. Grulha, Jr. Local Euler obstruction, old and new, II. InReal and com- plex singularities, volume 380 ofLondon Math. Soc. Lecture Note Ser., pages 23–45. Cambridge Univ. Press, Cambridge, 2010

2010

[5] [4]

Grulha, Jr., and Thuy Nguyen Thi B´ ıch

Jean-Paul Brasselet, Nivaldo G. Grulha, Jr., and Thuy Nguyen Thi B´ ıch. Local Euler obstruction, old and new, III.J. Singul., 25:90–122, 2022

2022

[6] [5]

Springer-Verlag, Berlin, 2009

Jean-Paul Brasselet, Jos´ e Seade, and Tatsuo Suwa.Vector fields on singular varieties, volume 1987 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 2009

1987

[7] [7]

G´ omez-Mont, J

X. G´ omez-Mont, J. Seade, and A. Verjovsky. The index of a holomorphic flow with an isolated singularity.Math. Ann., 291(4):737–751, 1991

1991

[8] [8]

An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity.J

Xavier G´ omez-Mont. An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity.J. Algebraic Geom., 7(4):731–752, 1998. A CRITERIA OF WEIGHTED HOMOGENEITY VIA LOGARITHMIC VECTOR FIELDS 7

1998