Affine vs. Stein in rigid geometry

TL;DR

The paper addresses when affine and Stein notions align in non-Archimedean rigid geometry, showing a stronger interplay than in complex geometry. It develops a density framework whereby algebraic sections densely approximate analytic sections, hinges on an extension theorem for Cartier divisors via admissible blow-ups, and employs formal models to bridge schemes and Berkovich spaces. The resulting Remmert factorization with boundary and analytic-to-algebraic approximation yield that Stein spaces are quasi-affine, and under Noetherian or finite-type hypotheses, affine; in particular, for surfaces Stein ⇔ affine. These findings reveal a sharp rigidity in the rigid-analytic setting and provide new tools to translate analytic properties into algebraic geometry, with consequences for the affineness of algebraic groups and the behavior of holomorphic convexity. Overall, the work tightens the correspondence between algebraic and analytic geometry in the non-Archimedean world and extends classical complex-analytic intuitions into rigid geometry.

Abstract

We investigate the relationship between affine and Stein varieties in the context of rigid geometry. We show that the two concepts are much more closely related than in complex geometry, e.g. they are equivalent for surfaces. This rests on the density of algebraic functions in analytic functions. One key ingredient to prove such a density statement is an extension result for Cartier divisors.

Theorems & Definitions (183)

• Theorem A
• Corollary
• Corollary
• Theorem B
• Theorem C
• Example
• Theorem D
• Theorem E
• Theorem 1.1
• Theorem 1.2