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?



Is the automorphism group of a Calabi-Yau variety an arithmetic group
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](http://math.ucla.edu/~totaro/papers/public_html/algebraic.pdf). 


