On the Milnor fibres of initial forms of topologically equivalent holomorphic functions

TL;DR

The paper addresses whether the Milnor fibre of the initial form $f_*$ of a holomorphic germ is invariant under topological equivalence, with a focus on plane curves. It proves a positive answer in dimension $2$ and links this to Zariski multiplicity conjectures, including right-equivalence and modulo $2$ variants, while examining weaker/alternative equivalences such as bi-Lipschitz and blow-spherical. The methods combine irreducible decompositions, bi-Lipschitz classifications for curves in the plane, and blow-spherical techniques, along with deformation arguments comparing $f(x)$ to $f(x)+t^k$ to relate Milnor fibres. The results reveal deep connections between Milnor-fibre topology and multiplicity invariants, offering conditional implications that reduce the Zariski problem to Milnor-fibre invariance in specific settings and outlining a path toward broader equivalence results.

Abstract

Budur, Fernandes de Bobadilla, Le and Nguyen (2022) conjectured that if two germs of holomorphic functions are topologically equivalent, then the Milnor fibres of their initial forms are homotopy equivalent. In this note, we give affirmative answers to this conjecture in the case of plane curves. We show also that a positive answer to this conjecture implies in a positive answer to the famous Zariski multiplicity conjecture both in the case of right equivalence or in the case of hypersurfaces with isolated singularities.

Theorems & Definitions (38)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Conjecture 4
• Conjecture 5
• Conjecture 6
• Proposition \ref{prop:conj_initial_implies_zariski}
• Proposition \ref{prop:conj_initial_implies_zariski_isolated}
• Proposition \ref{prop:conj_initial_dim_two}
• Definition 1.1