Weighted homogeneity from transverse logarithmic vector fields

For a reduced isolated complex hypersurface germ of dimension at least two, or an irreducible curve germ, is weighted homogeneity after coordinate change equivalent to a logarithmic holomorphic vector field transverse to every sufficiently small link in those same coordinates?

Why it matters and what counts as progress

Why it matters: The equivalence would characterize an algebraic grading through local differential geometry.

What counts as progress: Both implications under the exact germ hypotheses, including the invertible-linear-part characterization, or a germ separating the proposed conditions.

How the problem developed

9 May 2026 · Conjecture [1]
Diogo da Silva Machado and José Seade Proposed a link between transverse logarithmic vector fields and weighted homogeneity, with partial cases for isolated hypersurface singularities.
29 Jun 2026 · Reported claim [2]
Jihao Liu and Xiping Zhang Reported an equivalence after a suitable coordinate change for reduced isolated hypersurface germs of dimension at least two and for irreducible curve germs.

AI contribution

Criteria of isolated weighted homogeneous hypersurface singularities using Logarithmic vector fields

Jihao Liu, Xiping Zhang

For a reduced isolated complex hypersurface germ of dimension at least two, or an irreducible curve germ, weighted homogeneity after coordinate change is equivalent to a logarithmic holomorphic vector field transverse in the real-Euclidean sense to every sufficiently small link in those same coordinates. An invertible ambient linear part is a third equivalent condition.

AI involvementAI-autonomous

ImportanceNotable

Technical verificationPartial

Human understandingFull