MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
up vote 5 down vote accepted

You might be interested in the following paper,

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 $X$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.