The complex geometry of two exceptional flag manifolds

TL;DR

This paper investigates the complex geometry of two five-dimensional Kähler manifolds homogeneous under the exceptional Lie group G2, proving a rigidity result for one and exploring complex structures and Chern numbers.

Contribution

It proves the complex structure rigidity for the second G2-homogeneous manifold and relates Kähler structures to the existence of complex structures on the 6-sphere.

Findings

Rigidity of complex structure on one G2-homogeneous manifold

Existence of complex structures on the 6-sphere related to Kähler assumptions

Computation of Chern numbers for G2-invariant almost complex structures

Abstract

We discuss the complex geometry of two complex five-dimensional K\"ahler manifolds which are homogeneous under the exceptional Lie group $G_2$. For one of these manifolds rigidity of the complex structure among all K\"ahlerian complex structures was proved by Brieskorn, for the other one we prove it here. We relate the K\"ahler 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 $G_2$-invariant almost complex structures on these manifolds.