Complements of hypersurfaces in projective spaces

We study the complement problem in projective spaces ${\mathbb{P}}^n$ over any algebraically closed field: If $H,H^{\mathrm{\prime }}\subseteq {\mathbb{P}}^n$ are irreducible hypersurfaces of degree $d$ such that the complements ${\mathbb{P}}^n\setminus H$, ${\mathbb{P}}^n\setminus H^{\mathrm{\prime }}$ are isomorphic, are the hypersurfaces $H$, $H^{\mathrm{\prime }}$ isomorphic? For $n=2$, the answer is positive if $d\le 7$ and there are counterexamples when $d=8$. In contrast we provide counterexamples for all $n,d\ge 3$ with $(n,d)\mathrm{\ne }(3,3)$. Moreover, we show that the complement problem has an affirmative answer for $d=2$ and give partial results in case $(n,d)=(3,3)$. In the course of the exposition, we prove that rational normal projective surfaces admitting a desingularisation by trees of smooth rational curves are piecewise isomorphic if and only if they coincide in the Grothendieck ring, answering affirmatively a question posed by Larsen and Lunts for such surfaces.