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?


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