Exercise 10, § 4

We retain the notations of the preceding exercise.

a) Assume that dim Z = 1. Then there are exactly two complex structures on $G/H$ invariant under G; there exists an automorphism of G that leaves H stable and interchanges these two structures (use Exerc. 8).

b) Determine the complex structures on $\mathbf{P}_n(\mathbf{C})$ invariant under $\mathbf{S}\mathbf{U}(n+1,\mathbf{C})$. c) Assume that dim Z = 0 and Card(Z) $\not= 2$. Show that there exist almost complex structures on $G/H$ invariant under G (if $z$ is an element of C(H) that is not central in G, Int $z$ induces an automorphism of $G/H$ of odd order (Exerc. 9); use Exerc. $8b$)$)$. Show that there is no complex structure on $G/H$ invariant under G (use Exerc. $8d$)$)$.

d) There is no complex structure on $\mathbf{S}_6$ invariant under $\mathbf{S}\mathbf{O}(7,\mathbf{R})$ (use Exerc. 7 of §3).