# Curvature and Symmetry on Kähler manifolds

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?

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No way. There exist nontrivial, isometric $Z/k$-actions on $CP^1$, any torus and an infinite number of genus $\geq 2$ surfaces. Discrete symmetries do not have a close relationship to local properties of the metric. –  Johannes Ebert Mar 12 '12 at 8:26
In general of course it is hopeless, but I was wondering if there exists additional constraints to $X$ that would allow an implication like this. Maybe I formulated the question a bit awkwardly ... hmmm. –  The Common Crane Mar 12 '12 at 17:45

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 $X$.