It is well-known that the dimension of the isometric group of $n$-dimensional compact Riemannian manifolds is no larger than $\frac{1}{2}n(n+1)$, which is attained precisely by $S^n$ and $\mathbb{R}P^n$.

For K"{a}hler manifold, to my limited knowledge, a similar result is as follows: the dimension of the automorphic group of compact homogeneous K"{a}hler manifolds is no larger than $n(n+2)$, with equality only for $\mathbb{C}P^n$.

My question is: how about the situation for general compact K"{a}hler manifold? is the above result still true or any counterexample? And how about the situation for Fano K"{a}hler-Einstein manifolds?

It is well-known that the dimension of the isometry group of an $n$-dimensional compact Riemannian manifold is no larger than $\frac{1}{2}n(n+1)$, which is attained precisely by $S^n$ and $\mathbb{R}P^n$.

For Kähler manifolds, to my limited knowledge, a similar result is as follows: the dimension of the automorphism group of compact homogeneous Kähler manifolds is no larger than $n(n+2)$, with equality only for $\mathbb{C}P^n$.

My question is: how about the situation for general compact Kähler manifolds? Is the above result still true or are there any counterexamples? And how about the situation for Fano Kähler-Einstein manifolds?