# Is the automorphism group of a Calabi-Yau variety an arithmetic group

Let $X$ be a smooth projective variety over the complex numbers with trivial canonical bundle. Suppose that $X$ is Calabi-Yau.

Is the automorphism group of $X$ an arithmetic group?

What if $X$ is a K3 surface?

• pub.math.leidenuniv.nl/~luijkrmvan/K3Banff/abstracts.html
– user21574
Feb 16 '16 at 14:21
• I don't understand what information the previous comment is providing. But anyway, the answer for $K3$ surfaces is no. A counterexample, where the group is not even commensurable with an arithmetic group, was given by Totaro in Example 6.3 of this paper: math.ucla.edu/~totaro/papers/public_html/algebraic.pdf Feb 16 '16 at 14:41
• @potentiallydense Many thanks for your comment. That answers my question. Could you post your comment as an answer? Feb 16 '16 at 15:05

## 1 Answer

The answer for $K3$ surfaces is no. A counterexample, where the group is not even commensurable with an arithmetic group, was given by Totaro in Example 6.3 of this paper.