Let $X$ be a compact Kaehler manifold. Is the group of holomorphic isometries of $X$ compact?

Question

Let $X$ be a compact Kaehler manifold. Then it was shown by Bochner and Montgomery that, with respect to the compact open topology, $Aut_{\mathbb{C}}(X)$, the group of complex automorphisms of $X$, can be endowed with a structure of complex Lie group. Is it true that the group of holomorphic isometries $Iso_{\mathbb{C}}(X)$ is a compact subgroup of $Aut_{\mathbb{C}}X$?

Answer 1

A holomorphic map is an isometry iff it preserves the geodesic distance (the latter implying, as $X$ is compact – that’s not entirely trivial – bijectivity).

Now, the set of geodesic distance-preserving maps $X \rightarrow X$ is compact by Ascoli, and the set of holomorphic maps $X \rightarrow X$ is closed in $C(X,X)$ (for the compact open topology).

So their intersection is compact, and it is $Iso_{\mathbb{C}}(X)$.