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 Aut(X) be the complex Lie group of holomorphic automorphisms of X. Examples show that dim Aut(X) can grow exponentially in n = dimX. In this note it is shown that dim Aut(X) ≤ n − 1 + (2n− 1 n− 1 ) when n ≥ 3. Thus, dim 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 = detTY , of a homogeneous projective rational manifold Y of dimension m: dimH0(Y,K∗ Y ) ≤ (2m+1 m ) .