Fake Enriques surfaces (Q1824006)

An Enriques surface is an elliptic surface with vanishing geometric genus \(p_ g=0\) and \(2K=0\), where K is the canonical bundle, and its fundamental group is \({\mathbb{Z}}/2{\mathbb{Z}}\). A homotopy Enriques surface is one which has the same oriented homotopy type as an Enriques surface. It is proved that the latter is precisely one of those elliptic surfaces with \(p_ g=0\), which have two multiple fibres of multiplicities 2p and 2q, with relatively prime odd integers p and q, which are respectively denoted by \(X(2p,2q)\). All of them are homeomorphically equivalent. For the \(C^{\infty}\)-structure, the author proves that there is an at most finite number of \(X(2p,2q)\) which are diffeomorphic to a fixed type \(X(2p_ 0,2q_ 0)\). This implies that there exist infinitely many \(C^{\infty}\)-structures on the underlying topological manifold of an Enriques surface. The last result is based on the computation of Donaldson's \(\Gamma\)-invariant for these surfaces.