  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.
