Suppose $X$ is a Kahler manifold that has an analytic isometry $S$, with $S^k = Id$ ($k \in \Bbb N$). In a situation like this(maybe with additional assumptions on $X$) can one say something about the positivity/negativity of the curvature of $X$? Particularly I would be interested in instances where the bisectional curvature might be positive/negative.
If this was already studied(it might be possible since I am relatively new to the field). Can someone please provide some references?