Exercise 13, § 6

Let H be a closed subgroup of G; put $X = G/H$ and $n=$ dim X. Show that the following conditions are equivalent:

(i) There exists a 2-form $\omega$ on X such that $d\omega = 0$ and such that the alternating form $\omega_x$ on $T_x(X)$ is separating for all $x\in X$;

(ii) $n$ is even, and there exists an element $\omega$ of $H^2(G/H)$ such that $\omega^{n/2}\not= 0$; (iii) H is the centralizer of a torus in G;

(iv) H is of maximum rank, and there exists a G-invariant complex structure on X;

(v) there exists a complex structure $j$ and a 2-form $\omega$ on X such that $d\omega = 0$, $\omega_x(ju, jv) =\omega_x(u, v)$ and $\omega_x(u, ju)>0$ for all $x$ in X and all non-zero $u, v$ in $T_x(X) (^*$ in other words, a Kähler structure on $X_*)$;

(vi) there exists a complex structure $j$ and a 2-form $\omega$ on X satisfying the conditions in (v) and invariant under $G (^*$ that is, a G-invariant Kähler structure on $X_*)$.

(To prove that (ii) $=\Rightarrow$ (iii), put $S = C(H)_0$ and $Z = Z_G(S)$; by using Exerc. 12 $d$), show that the canonical map $H^2(G/Z)\rightarrow H^2(G/H)$ is surjective, and deduce that Z = H. The equivalence of (iii) and (iv) follows from Exerc. 8 of §4. To prove that (iii) $=\Rightarrow$ (vi), construct an H-invariant separating positive hermitian form on $\mathfrak{g}/L(H)$ and consider its imaginary part; use Exerc. $11a).)$