The complex geometry of two exceptional flag manifolds

Abstract

We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group $$G_2$$ G 2 . For one of these manifolds, rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn; for the other one, we prove it here. We relate the Kähler assumption in Brieskorn’s theorem to the question of existence of a complex structure on the six-dimensional sphere, and we compute the Chern numbers of all $$G_2$$ G 2 -invariant almost complex structures on these manifolds.