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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?

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  • $\begingroup$ More generally, if $G\subset\mathrm{GL}(n,\mathbb{R})$ is Lie subgroup of finite type, i.e., if the sheaf of vector fields whose flows preserve the translation-invariant $G$-structure $B_0$ on $\mathbb{R}^n$ has finite dimensional stalks of dimension $d$, then the group of automorphisms of any $G$-structure $B$ on an $n$-manifold is a Lie group of dimension at most $d$. This was essentially known to É. Cartan. It includes all the pseudo-Riemannian geometries and their sub-geometries such as almost Hermitian; in particular, it includes the cases you mention above. $\endgroup$
    – Robert Bryant
    Commented Mar 22, 2018 at 12:03

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For the first question, see the review of Tanno's 1969 paper:

enter image description here

I am not sure I understand the second question.

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