Fast vanishing cycles on perturbations of (complex) weighted-homogeneous germs

2023/11/22 by Dmitry Kerner, Rodrigo E. Mendes, Kerner, Dmitry +1 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2311.13423

openalex publication_date 2023/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X_o be a complex weighted-homogeneous complete intersection germ, (possibly non-reduced). Let X be a perturbation of X_o by ``higher-order-terms". We give sufficient criteria to detect fast cycles on X, via the weights of X_o. This is an easy obstruction to be non-metrically conical. A simple application of our results gives, e.g. * Suppose the germs X_o,X,X∩ V(x₁) are ICIS. If X is IMC then the n lowest weights of X_o coincide. * Let the surface germ X=V(f)⊂ (C³,o) be Newton-non-degenerate and IMC. Then for each of the faces of the Newton diagram the two lowest weights coincide. As an auxiliary result we prove (under certain assumptions, for \k=\R,\C): the weighted-homogeneous foliation of the pair X_o \sset (\k^N,o) deforms to a foliation of the pair X \sset (\k^N,o). In particular, the deformation by higher order terms is ambient-trivializable by a semialgebraic Lipschitz homeomorphism.

Cited by

• Lipschitz Geometry of Mixed Polynomials

Related

• Detecting fast vanishing loops in complex-analytic germs (and detecting germs that are inner metrically conical)
• Deforming the weighted-homogeneous foliation, and trivializing families of semi-weighted homogeneous ICIS
• Fast loops on semi-weighted homogeneous hypersurface singularities
• Decomposition in bunches of the critical locus of a quasi-ordinary map
• Weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersections