Reduced Forms of a Singular Vector Field in C³ with Nilpotent Linear Part

Abstract

We study the reduction problem for a holomorphic singular vector field with nilpotent linear part at the origin, P = (y + a(X)) ∂_x + (z + b(X)) ∂_y + c(X)∂_z, where X = (x, y, z) ∈ C³. By introducing a notion of quasi-valuation, which was given in the previous paper [M-S] with A. Shirai, we characterize a class which can be reduced into simpler ones by formal change of coordinates which admit various variations in Theorems A and B.