Question

Hi there,

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?

Answer 1

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 $X$.