Does every complex flag manifold have a natural Kähler structure? If so, what is it?

Every flag manifold $M=G^{\mathbb{C}}/P=G/C(S)$ where $P$ is a parabolik subgroup and $C(S)=P\cap G$ is the centralizer of a torus $S\subset G$, admits a finite number of invariant Kahler structures. In particular the complex presentation $G^{\mathbb{C}}/P$ gives rise to an finite number of invariant complex structures (i.e. integrable almost complex structures commuting with the isotropy representation of $M$). Any such complex structure is determined by an invariant ordering $R_{M}^{+}$ on the set of complementary roots $R_{M}=R\backslash R_{K}$ of $M$ and explicitly is given by $$ J_{o}E_{\pm \alpha}=\pm i E_{\pm\alpha}, \quad a\in R_{M}^{+} $$ where $E_{\alpha}$ are root vectors with respect a Weyl basis of $\frak{g}^{\mathbb{C}}$. On the other hand, the real presentation $G/C(S)$ makes $M$ a (homogeneous) Kahler manifold, as a (co)adjoint orbit of an element $w\in\frak{g}$ in the Lie algebra $\frak{g}$ of the compact connected (semi)simple Lie group. Flag manifolds exhaust all compact homgeneous Kahler manifolds corresponding to a compact connected semisimple Lie group. To be more specific, $M$ admits a finite number of Kahler structures which are parametrized by the wellknown $\frak{t}$chambers (connected components of the set of regular elements of $\frak{t}$) where $$ {\frak{t}} =( H\in{\frak{h}} : (H, \Pi_{0})=0 ) $$ is a real form of the center ${\frak{z}}$ of the isotropy subgroup $K=C(S)$. Here $\frak{h}$ is the Cartan subalgebra corresponding to a maximal torus $T$ of $G$ which contains $S$, and $\Pi_{0}\subset\Pi$ is the subgroup of simple roots which define (the semisimple part of) the complexification $\frak{k}^{\mathbb{C}}$ (note that $K=C(S)=P\cap G$ is areductive Lie group). We have $$ {\frak{z}}^{\mathbb{C}}={\frak{t}}\oplus i {\frak{t}}, \ \ \ {\frak{k}}^{\mathbb{C}}={\frak{z}}^{\mathbb{C}}\oplus{\frak{k}}_{ss}^{\mathbb{C}} $$ where ${\frak{z}}^{\mathbb{C}}$ is the complexification of the center ${\frak{z}}$ and ${\frak{k}}_{ss}^{\mathbb{C}}$ is the semisimple part of the reductive complex Lie subalgebra ${\frak{k}}^{\mathbb{C}}$ In particular, there exists a natural 11 corespondence between elements from ${\frak{t}}$ and closed invariant 2forms on $M$. Symplectic 2forms (nondegenerate) correspond to regular elements $t$ of ${\frak{t}}$. Note that the corresponding symplectic form corresponding to a regular elemnt $t_{0}$ is the KirillovKostantSouriau 2form in the (co)adjoint orbit $Ad(G)t_{0}$, that is $$ \omega_{t_{0}}(X, Y)=B(t_{0}, [X, Y]), \ \ X, Y\in T_{t_{0}}M. $$ For more details see: D. Alekseevsky: Flag manifolds (11. Yugoslav Geometrical seminar, Divcibare, 1017 October 1993, 335. This article is a very good review on the geometry of flag manifolds. 


Yes. Use Plucker embedding to embed it into $CP^n$ then restrict FubiniStudy metric. 


The question has already been answered by Bugs Bunny, but I thought I'd point out that there is a nice paper by H.C. Wang from the 1950s that discusses the complex structure of homogeneous manifolds in some detail. One of the results proved there is that a compact, simply connected complex homogeneous manifold (such as a complex flag manifold) is Kähler if and only if it has nonzero (ordinary) Euler characteristic. That complex flag manifolds have nonzero Euler characteristics follows, for example, from the Bruhat decomposition. 

