Complex Spaces and Nonstandard Schemes

TL;DR

This paper uses nonstandard mathematics to reinterpret complex spaces as standard parts of algebraic nonstandard schemes, establishing a new categorical framework and exploring implications for prime ideals, differential forms, and hyperbolicity.

Contribution

It introduces a novel category of algebraic nonstandard schemes that bridges algebraic schemes and complex spaces, providing new insights into their structure and properties.

Findings

Complex spaces are the standard parts of algebraic nonstandard schemes.

Maximal and minimal prime ideals in Stein algebras correspond to nonstandard points.

New applications to differential forms, meromorphic functions, and hyperbolicity are demonstrated.

Abstract

We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the "standard part" of certain algebraic nonstandard schemes. We construct a category of such schemes, sitting in between usual algebraic schemes (over the complex numbers) and that of complex spaces. We clarify the structure of prime ideals in a Stein algebra, coming from nonstandard points and show in particular that ANY maximal and minimal prime ideal in a Stein algebra is the vanishing ideal of a nonstandard point. Other applications of our point of view are given for differential forms (a la Leibniz), generic points (as nonstandard ones), meromorphic functions, hyperbolicity. The essential tools taken from nonstandard mathematics and adapted for our purposes, are summarized in the appendix.