Complex hypersurfaces diffeomorphic to affine spaces (Q1340411)

The main result of this paper is the construction of a family of \((2m - 1)\)-dimensional complex hypersurfaces diffeomorphic to \(\mathbb{C}^{2m - 1}\). Let \(m > 1\), \(0 < a < d - 1\) be positive integers such that \((a,d) = (a,d - 1) = 1\). Then the authors prove the following result. The hypersurface \(X_{d,a}\) defined by \(\{f = 0\}\), where \(f = x^ a_ 0 x^{d - a}_ 1 + x_ 1 x^{d - 1}_ 2 + \cdots + x_{2m - 3} x^{d - 1}_{2m - 2} + x_{2m - 2} + x^ d_{2m - 1}\), has these properties: (i) \(X_{d,a}\) is smooth and diffeomorphic to \(\mathbb{C}^{2m - 1}\). (ii) For \(a = 1\), the map \(f : \mathbb{C}^{2m} \to \mathbb{C}\) is a topologically trivial fibration. (iii) For \(a > 1\), the hypersurface \(\{f = \lambda\}\) is not contractible if \(\lambda \in \mathbb{C} - \{0\}\). Further, for \(m = 2\) and \(a = 1\) the embedding of \(X_{d,1}\) in \(\mathbb{C}^ 4\) is equivalent to a linear one. These hypersurfaces are suitable Zariski-open subsets of certain projective hypersurfaces with isolated singularities and which are \(\mathbb{Z}\)-homology projective spaces. It will be interesting to know the logarithmic Kodaira dimensions of these hypersurfaces.