Étale exoticity of the Pinchuk surface (Q2430036)

Let \(S\) be a complex surface. \(S\) is étale exotic if: 1. there is a diffeomorphism \(\phi :\mathbb{C}^{2}\rightarrow S\) which is realized by a birational map, 2. there is no regular étale map \(S\) into \(\mathbb{C}^{2}.\) The author gives a new proof of the fact that the surface \(S\subset \mathbb{C }^{3}\) parametrized by \[ X=V,\;\;Y=VU,\;\;Z=VU^{2}+U \] (its closure \(\overline{S}\) satisfies the equation \(XZ-Y(Y+1)=0\) and it is a Danielewski surface) is étale exotic. The first proof was given also by \textit{R. Peretz} [Ill. J. Math. 40, No. 2, 293--303 (1996; Zbl 0855.13015)]. This fact shows that the construction of a counterexample to the complex Jacobian conjecture in \(\mathbb{C}^{2}\) is not possible along the Pinchuk idea in real case.