Grauert's Approximation Theorem in any Characteristic and Applications

TL;DR

This paper extends Grauert's approximation theorem from complex numbers to arbitrary real valued fields of any characteristic, enabling new applications in deformation theory of singularities.

Contribution

It generalizes Grauert's division and approximation theorems to broader fields, facilitating the study of singularities over any real valued field.

Findings

Established a convergent semiuniversal deformation for isolated singularities over any real valued field.

Proved a splitting lemma for hypersurface singularities in arbitrary characteristic.

Extended Grauert's approximation theorem beyond complex analytic settings.

Abstract

In his seminal Inventiones paper from 1972 Grauert proved the existence of a semiuniversal deformation of an arbitrary complex analytic isolated singularity. For the proof he invented an approximation theorem for solving a system of "nested" analytic equations, which is now called Grauert's approximation theorem. To prove this, Grauert introduced standard bases for ideals in power series rings and proved a generalized Weiertrass division theorem. All this was done for convergent power series over the complex numbers. The purpose of this article is to extend Grauert's division and approximation theorem to convergent power series over arbitrary real valued fields of any characteristic. As an application, which was actually the motivation for this article, we derive the existence of a convergent semiuniversal deformation for an isolated singularity and a splitting lemma for not necessarily isolated hypersurface singularities over any real valued field.