MathDB

Weighted homogeneity via logarithmic vector fields

Less than 1 year old · traced to

Let (W,0)⊂Cn+1(W,0)\subset\mathbb{C}^{n+1} be an open domain, let D⊂WD\subset W be a reduced hypersurface germ with an isolated singularity at 00, and let f∈OW,0f\in\mathcal{O}_{W,0} define DD. A holomorphic vector field is tangent to DD when it preserves the hypersurface, and its singularity at 00 is non-degenerate when its linear part at 00 is non-singular. The link spheres of DD are the intersections of DD with sufficiently small spheres centered at 00.

Machado–Seade conjecture. The following statements are equivalent: (D,0)(D,0) is weighted homogeneous; there exists a holomorphic vector field tangent to DD with an isolated singularity at 00 that is transverse to the link spheres of DD; and there exists a holomorphic vector field tangent to DD with a non-degenerate isolated singularity at 00.

The conjecture gives a criterion for detecting weighted homogeneity of isolated hypersurface singularities through logarithmic vector fields. The supplied abstract states that the authors prove it affirmatively, so the conjecture is solved.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Weighted homogeneity via logarithmic vector fields

    Do the logarithmic-vector-field conditions proposed by da Silva Machado and Seade characterize weighted homogeneous isolated hypersurface singularities?

References

Primary source

Jihao Liu and Xiping Zhang, “A Criteria of Weighted Homogeneity via Logarithmic Vector Fields”, arXiv:2606.29886 (2026).

geometry-topologylogarithmic-vector-fieldssingularity-theoryweighted-homogeneous-singularities
Posted

Progress summary

Refreshed
Claimed solved

Two 2026 papers claim to prove that the proposed vector-field test exactly identifies the relevant weighted singularities, subject to dimension and coordinate qualifications.

The question, posed by da Silva Machado and Seade, asks whether their logarithmic-vector-field conditions characterize weighted homogeneous isolated hypersurface singularities. The claimed characterization requires a biholomorphic coordinate change and, in one formulation, restrictions on dimension and irreducibility.

Known results

  • The weighted Euler vector field gives the implication from weighted homogeneity to the logarithmic-vector-field conditions.
  • Earlier work reportedly proved the transversality converse for odd nn and the nondegeneracy converse for even nn; the source supplies no names or dates.
  • For isolated hypersurface singularities, weighted homogeneity is equivalent to [1mμ=τ[1m\mu=\tau.[0m

2026 claimed proofs

Two arXiv papers state that the conjecture is true. Under n≥2n\geq 2, or n=1n=1 with irreducible DD, they claim equivalence among weighted homogeneity after coordinate change, link-transverse logarithmic fields, and tangent fields with invertible linear part. One paper also exhibits a coordinate system where transversality fails, showing that the coordinate qualification is essential. The proofs remain unverified by independent corroboration.

Current status (as of August 2026): A complete characterization is claimed under the stated hypotheses, but its proof is not yet independently verified; the coordinate-dependent formulation and excluded cases remain the precise scope.

Sources

Solutions 0

No solutions have been posted yet.