Milnor number of an invariant singularity: generalization of Chulkov's inequality

TL;DR

This paper extends Chulkov's Milnor-number bound from cyclic prime-order actions to arbitrary finite abelian group actions on germs with isolated singularities and vanishing 2-jet. By embedding the problem into a realified setting via $f \oplus f$ and employing Roberts' inequality for real group actions, it derives the bound $\mu(f) \ge l(\tau) - 1$, where $l(\tau)$ is the shortest non-fixed orbit length. The argument hinges on properties of the Jacobian algebra, equivariant Milnor numbers, and invariant Morse deformations, together with a decomposition that connects $\nu(f \oplus f)$ to $\mu(f)$. The paper also confirms the bound's tightness with explicit low- and high-dimensional examples, including a construction yielding $\mu = 2^n$ with orbit lengths demonstrating the sharpness of the estimate.

Abstract

S.P. Chulkov has proven that the Milnor number of a function germ $f$ with zero 2-jet invariant with respect to a nontrivial action of $\mathbb{Z}_p$ is at least $p-1$ for prime $p$. In this paper we prove a generalization of this inequality for actions of arbitrary abelian groups using M. Roberts' results on invariant morsifications for invariants of real group actions.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Theorem 3.1
• proof
• Theorem 3.2