Let $U\subset X$ be a dense open of a smooth projective variety $X$ over the complex numbers. I am looking for examples of the following phenomena:
Aut(X) is trivial, but Aut(U) is infinite.
Or, slightly easier probably:
Aut(X) is finite, but Aut(U) is infinite.
Of course, this means that $U$ should have many automorphisms that do not extend to $X$. It's not possible to find such examples in dimension one. What about dimension two? Maybe we can take a dense open of some generic K3 surface?
