Uniformization of analytic spaces

Abstract

In this article, we give an explicit elementary proof of a version of resolution of singularities in characteristic zero. Local means that the centres of blowing up are chosen locally, so that a finite number of finite sequences of blowings-up may be required to cover a neighbourhood of a given point. (Hence we use the term rather than local desingularization, although uniformization in our sense is considerably stronger than the original idea of Zariski [16].) In the last two decades, several mathematicians (notably Abhyankar [1], Hironaka and Spivakovsky [12], and Youssin [14, 15]) have proposed simpler or more explicit versions of the inductive procedure in Hironaka's proof of his great theorem [8]. These approaches would seem to lead to some form of uniformization, though full details of none of them have yet appeared. The idea of seeking an explicit procedure to determine the centres of blowing up is that a sufficiently good choice should globalize automatically. (Our method, for example, gives global resolution of singularities of surfaces in any codimension.) Throughout this paper, K denotes either the field of real numbers R or the field of complex numbers C. We will work in the category of analytic spaces over K, although our techniques apply as well to algebraic spaces over a field