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

Abstract

We prove that the smooth affine hypersurfaces \[ X_{m,n}=\{ζ^mz_0^n+ (1+z_1ζ)z_2+ζz_3^2=1\}\subset\mathbb{C}^5,\;\; m,n\geq 2, \] have the density property. For every odd integer $d\geq 3$, 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.