The metric Zariski multiplicity conjecture in dimension 4

Abstract

Recently, OpenAI announced a counterexample to the Zariski multiplicity conjecture in dimension 4. Here, we prove that the metric Zariski multiplicity conjecture is true in dimension 4. More precisely, let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be germs of reduced holomorphic functions, and let $φ\colon(\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ be a bi-Lipschitz homeomorphism such that $φ(V(f))=V(g)$. We prove that if every irreducible component of the tangent cone of $V(f)$ at the origin has a singular locus of dimension at most two, then $V(f)$ and $V(g)$ have the same multiplicity at the origin. In particular, the multiplicity of complex analytic hypersurface singularities in $\mathbb{C}^n$, $n\leq 4$, is an embedded bi-Lipschitz invariant; the cases $n=2$ and $n=3$ were known. This extends a previous result of the author for tangent cones whose components have singular loci of dimension at most one. We also prove that two irreducible homogeneous hypersurfaces whose singular loci have dimension at most two, and which are mapped onto each other by a homeomorphism of $\mathbb{C}^n$ fixing the origin, have the same degree.