# Multiplicities of Critical Points of Invariant Functions

Montaldi, James (1993) Multiplicities of Critical Points of Invariant Functions. Matematica Contempor ˆanea, 5. pp. 93-135.

## Abstract

The purpose of this expository article is to describe in an elementary and homogeneous manner, the relationship between the geometric and algebraic multiplicities of isolated critical points of holomorphic functions. In particular, I am interested in the setting where the function is invariant under some group action. The emphasis is on functions invariant under actions of finite groups as very little is known if the group is not finite. Most of the results described here are already explicitly in the literature; the only small extension is to functions that are not invariant, but equivariant under the action of a group G: a function f satisfying f(gx) = J(g).f(x) for some homomorphism J : G \to C∗. The results (in Section 7) on the multiplicity of critical points of homogeneous functions invariant under C∗ are also new.

Item Type: Article Algebraic multiplicity, geometric multiplicity, representations, G-complexes MSC 2010, the AMS's Mathematics Subject Classification > 14 Algebraic geometry Dr James Montaldi 21 Feb 2010 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/1415