Automorphism groups of the complements of hypersurfaces

It is a well-known fact that every automorphism of a smooth hypersurface of degree $d$ in $\mathbb{P}^n, n\ge 2,$ comes from the automorphism group of $\mathbb{P}^n$ unless $(n,d)=(2,3),(3,4)$. In my talk, I reinvestigate this phenomenon inside out, i.e., the problem when the automorphism group of the complement of the hypersurface in $\mathbb{P}^n$ coincides with the subgroup of the automorphismgroup of $\mathbb{P}^n$ that keeps the hypersurface fixed. This talk is base on a joint work with Ivan Cheltsov and Adrien Dubouloz.