# Large isometry groups of Kaehler manifolds

Let $$M$$ be a closed simply-connected Kaehler manifold that is not isomorphic to a product of lower-dimensional Kaehler manifolds. Pick an orientation for $$\mathbb{C}$$; this endows $$M$$ with an orientation.

Assume that $$M$$ admits no self-diffeomorphisms (other than the identity) that preserve both metric and complex structure. This implies that the group of orientation-preserving isometries of $$M$$ is finite (this follows from the fact that every Killing vector field is holomorphic and the fact that isometry group is a compact Lie group in compact-open topology).

The question is: can this group have arbitrarily large order? What if we fix dimension of the manifolds?