On Complex Manifolds and Observable Schemes

TL;DR

This paper develops a method to construct Stein manifolds from certain algebras and extends this to complex manifolds using a scheme framework, with an eye towards applications in non-commutative geometry.

Contribution

It introduces a new construction of complex manifolds from Arens-Michael algebras and proposes a scheme-based approach for non-commutative geometry applications.

Findings

Constructs Stein manifolds from commutative Arens-Michael algebras.

Extends the construction to arbitrary complex manifolds via schemes.

Uses spectral functors to facilitate the non-commutative geometric framework.

Abstract

We work out the construction of a Stein manifold from a commutative Arens-Michael algebra, under assumptions that are mild enough for the process to be useful in practice. Then, we do the passage to arbitrary complex manifolds by proposing a suitable notion of scheme. We do this in the abstract language of spectral functors, in view of its potential usefulness in non-commutative geometry.