Revised Demailly's Affineness Criterion and Algebraization of Entire Grauert Tubes

TL;DR

This paper advances the understanding of Grauert tubes by proving a partial affineness result and generalizing Demailly's criterion for Stein manifolds, with potential broader implications.

Contribution

It introduces a generalized affineness criterion for Stein manifolds and applies it to partial algebraization of Grauert tubes, addressing a longstanding conjecture.

Findings

Complement of a codimension-one subset of a Grauert tube is affine

Generalized Demailly's criterion for Stein manifolds established

Partial algebraization of Grauert tubes achieved

Abstract

We provide a partial answer to Burns' 1982 conjecture on the affineness of entire Grauert tubes: the complement of a codimension-one subset of an entire Grauert tube is affine. This result is obtained by establishing a generalized version of Demailly's criterion for affineness of Stein manifolds, which may be of independent interest.