Smooth affine varieties and complete intersections (Q1331739)

The author proves that if \(X\) is a smooth surface in the affine \(n\)-space \((n \geq 5)\), satisfying the following numerical condition: \(c^ 2_ 1(X) + c_ 2(X) = 0\), then \(X\) is a set-theoretic complete intersection. As previous results in the same direction [see \textit{N. Mohan Kumar}, Mém. Soc. Math. Fr., Nouv. Sér. 38, 135-143 (1989; Zbl 0715.14043); and \textit{S. Bloch}], the proof makes use of the Ferrand construction. Moreover, the following example is constructed: there exists a smooth affine fourfold \(X\) over \(\mathbb{C}\) such that \(\omega_ X\) is trivial and \(\Omega_ X\) is not stably trivial (hence \(X\) is not a complete intersection in an affine space). Previously known examples of this phenomenon have dimension at least five.