Let $G$ be a compact Lie group and $a\in\mathfrak{g}^\*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O\_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also projective variety. How can we compute the Kodaira dimension of co-adjoint orbit as projective variety? 



> 
> Motivation: The Kodaira dimension of co-adjoint orbits are important,
>  because we can classify these type of projective varieties by Kodaira
>  dimension which is birationally invariant.
> 
> 
> 


In fact I am looking for 


$$\kappa(\mathcal O\_a)=\limsup\_{m\to \infty}\frac{\log\text{dim}H^0(\mathcal O\_a, K\_{\mathcal O\_a}^{\otimes m})}{\log m}$$



Kodaira dimension of co-adjoint orbit
Since your manifold $X$ has positive Ricci curvature, the line bundles $K\_X^m$ are all ${ negative}$ for $m\geq 1$, i.e. they admit a smooth Hermitian metric with negative curvature. By Kodaira Vanishing, we conclude that $H^0(X,K\_X^m)=0$ for all $m\geq 1$, i.e. the Kodaira dimension is $-\infty$.


