Absence of the affine lines on the homology planes of general type (Q1210062)

Let \(X\) be a non-singular algebraic surface defined over the complex field \(\mathbb{C}\). We call \(X\) a homology plane (resp. \(\mathbb{Q}\)-homology plane) if the homology groups \(H_ i (X;\mathbb{Z})\) (resp. \(H_ i (X;\mathbb{Q}))\) vanish for all \(i>0\). A purpose of the present article is to show the following result. Let \(X\) be a \(\mathbb{Q}\)-homology plane of Kodaira dimension 2. Then there lies no curve \(C\) on \(X\) which is topologically isomorphic to the affine line \(\mathbb{A}^ 1\). The core of a proof is to show that \(X\) and \(X-C\) are respectively embedded as Zariski open sets into almost minimal pairs and that the inequality of Miyaoka-Yau type, after a relevant modification, can be applied to derive a contradiction if one assumes the existence of a curve topologically isomorphic to \(\mathbb{A}^ 1\).