Let $M$ be a compact Kähler manifold. If $\phi:M\to M$ is an orientation-preserving isometric involution does it have to be either holomorphic or anti-holomorphic?



Orientation-preserving isometric involution on compact Kähler manifold
No. Let $M$ be the product of three copies of $\mathbb{C}/(\mathbb{Z}[i])$ (i.e., the square torus). Give it the obvious product metric. Now consider the map
$$
\phi\bigl([z\_1],[z\_2],[z\_3]\bigr) = \bigl([z\_1],[\,\overline{z\_2}\,],[\,\overline{z\_3}\,]\bigr).
$$
This is an orientation-preserving isometry that is neither holomorphic nor antiholomorphic.


