Proof of the metric Arnold's corank problem

In this article, we approach the Arnold corank problem, posed by Arnold in 1975, which asks whether the corank of holomorphic functions is an ambient topological invariant. Here, we obtain a complete positive answer to the metric Arnold corank problem, which asks whether the corank of holomorphic functions is an ambient bi-Lipschitz invariant. Consequently, we show that for complex hypersurfaces, the multiplicity equal to two is an ambient bi-Lipschitz invariant. We also prove that the Arnold corank problem holds true for holomorphic functions of three variables. Other topological invariants are also presented.

💡 Research Summary

The paper addresses a long‑standing question originally posed by V. Arnold in 1975: whether the corank of a holomorphic function germ—defined as the corank of its Hessian matrix at the origin—is an invariant of the ambient topological type of its zero set. The authors reformulate this question in a metric setting, asking whether the corank is preserved under ambient bi‑Lipschitz homeomorphisms (the “Metric Arnold corank problem”). Their main result (Theorem 3.1) gives an affirmative answer: if two reduced holomorphic germs f,g : (ℂⁿ,0)→(ℂ,0) have a bi‑Lipschitz homeomorphism φ sending V(f) onto V(g), then crk(f)=crk(g).

The proof proceeds in two stages. First, Lemma 3.2 treats the special case where both germs have multiplicity two at the origin. Using the Splitting Lemma, each function can be written as a sum of a non‑degenerate quadratic form in r (or s) variables plus higher‑order terms. The tangent cones S and T of V(f) and V(g) are then the quadratic cones {z₁²+…+z_r²=0} and {z₁²+…+z_s²=0}. Existing results on bi‑Lipschitz invariance of tangent cones (