A bound for the dimension of the automorphism group of a homogeneous compact complex manifold

Let $X$ be a homogeneous compact complex manifold, and let $\operatorname{Aut}(X)$ be the complex Lie group of holomorphic automorphisms of $X$. Examples show that $\dim \operatorname{Aut} (X)$ can grow exponentially in $n = \dim X$. In this note it is shown that

when $n \ge 3$. Thus, $\dim \operatorname{Aut} (X)$ is at most exponential in $n$. The proof relies on an upper bound for the dimension of the space of sections of the anticanonical bundle, $K_Y^* = \det T_Y$, of a homogeneous projective rational manifold $Y$ of dimension $m$: $\dim H^0(Y,K_Y^*) \le \binom{2m+1}{m}$.