A new class of smooth algebraic varieties with the density property and a counterexample to the Tóth-Varolin Conjecture

We prove that the smooth affine hypersurfaces $$X_{m,n}=\{{\zeta }^m{z}_0^n+(1+z_1\zeta )z_2+\zeta z_3^2=1\}\subset {\mathbb{C}}^5,m,n\ge 2,$$ have the density property. For every odd integer $d\ge 3$, we identify $X_{d-1,d}$ with a Kaliman modification along a Brieskorn variety of type $(d,2,2,2)$. We prove that the fourfolds $X_{d-1,d}$ are diffeomorphic to ${\mathbb{C}}^4$ but not biholomorphic to ${\mathbb{C}}^4$. These, therefore, give counterexamples to the Tóth-Varolin conjecture.