Hi there,


Suppose $X$ is a Kähler manifold that has an analytic isometry $S$, with $S^k = \operatorname{Id}$ ($k \in \Bbb N$). 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? 



Curvature and Symmetry on Kähler manifolds
You might be interested in the following paper, <http://arxiv.org/pdf/1011.1464v1.pdf>


Abstract: We show that the number of birational automorphism
of a variety of general type $X$ is bounded by $c · vol(X, K\_X)$, where
$c$ is a constant which only depends on the dimension of $X$.


