The complex geometry of a hypothetical complex structure on $S^6$

TL;DR

This survey explores the theoretical aspects and implications of a hypothetical complex structure on the six-sphere, discussing algebraic dimension results and biholomorphism groups to understand its complex geometry.

Contribution

It provides an overview of key results and examples related to the complex geometry of a hypothetical complex six-sphere, highlighting recent research developments.

Findings

Discussion of algebraic dimension constraints

Examples illustrating complex structures on $S^6$

Analysis of biholomorphism groups

Abstract

This article is a survey about or introduction to certain aspects of the complex geometry of a hypothetical complex structure on the six-sphere. We discuss a result of Peternell--Campana--Demailly on the algebraic dimension of a hypothetical complex six-sphere and give some examples. We also give an overview over an application of Huckleberry--Kebekus--Peternell on the group of biholomorphisms of such a complex six-sphere.