Equivalences between isolated hypersurface singularities

Let (9,+1 be the ring of germs of holomorphic functions (C ~+ 1, 0 ) ~ C. There are many important equivalence relations that have been defined on the elements of (9+ 1. ~ ' , ~s and ~f-equivalence are well known in function theory. Each of these equivalence relations can be defined in terms of a Lie group action on (9 +1For instance two functions are defined to be ~-equivalent if they are the same up to a holomorphic change of coordinates in the domain. In this case the Lie group acting on (9+ 1 is the group of all holomorphic change of coordinates preserving the origin. Simple complete characterizations of when two functions are ~'-, ~L,r or J~f-equivalent were given by Yau I-9] and by Mather and Yau [6]. .~_, ~t_, and ~-equivalence come from singularity theory. These equivalence relations are defined on the basis of algebra isomorphisms. For example, we can associate a C-algebra (9,+ JA(f), the Milnor algebra, to any fE(_9, + 1, where A(f) is the ideal in (9,+ 1 generated by the partial derivatives of f. We say that two functions are S-equivalent if their associated Milnor algebras are isomorphic. It is an interesting question to determine the relationships between these six equivalences. The goal of this paper is to study these relationships. For a holomorphic function f with a critical point at the origin, we determine when the equivalence classes o f f with respect to two different equivalence relations coincide. The purpose of this paper is two-fold. On the one hand, we give a necessary and sufficient condition for ~ e q u i v a l e n c e to coincide with ~-equivalence (cf. Theorem 5.1). This leads us to define the new notion of almost quasi-homogeneous functions. We suspect that the singularities defined by almost quasi-homogeneous functions may form a distinguished class of singularities which have some special properties shared by quasi-homogeneous ones. In Sect. 6, we discuss the relationship between .~and ~r Perhaps the most striking result here is Theorem 6.9, which provides us a lot of examples