Smooth contractible hypersurfaces in \(\mathbb{C}^ n\) and exotic algebraic structures on \(\mathbb{C}^ 3\) (Q1323460)

We present new examples of exotic algebraic structures on \(\mathbb{R}^ 6\), i.e., examples of smooth complex affine algebraic manifolds that are diffeomorphic to \(\mathbb{R}^ 6\), but not isomorphic to \(\mathbb{C}^ 3\). These exotic algebraic structures are contractible hypersurfaces in \(\mathbb{C}^ 4\), each of them admits a regular \(\mathbb{C}^*\)-action with one fixed point and they have Kodaira logarithmic dimension 2. We also present a general method of constructing contractible hypersurfaces in \(\mathbb{C}^ n\) which are not equivalent to a hyperplane up to a polynomial coordinate substitution.