A complex surface of general type with \(p_{g}=0\), \(K^{2}=2\) and \(H_{1}=\mathbb{Z}/4\mathbb{Z}\) (Q2849027)

The main result of this paper is the existence of a minimal surface of general type with \(p_g=0\) and \(K^2=2\) (these are known as \textit{numerical Campedelli surfaces}) whose algebraic fundamental group is cyclic of order \(4\).NEWLINENEWLINEThis result closes a long story: by a Theorem of Miles Reid (unpublished) integrated by some results by other authors (who excluded a couple of cases) the algebraic fundamental group of a numerical Campedelli surface is either the group of the quaternions (of order \(8\)) or abelian of order at most \(9\). Now we can say that all the groups above occur: indeed when this paper appeared the cyclic group of order \(4\) was the only one for which the existence question was not settled.NEWLINENEWLINEThe argument is similar to the one used by the same authors and Y. Lee in other occasions to settle similar existence question. In this case, the authors consider an elliptic Enriques surface previously constructed by Kondo and show that it is a birational to a surface with three singular points of class \(T\), singular points which admit a local \({\mathbb Q}\)-Gorenstein smoothing. Then they prove that this singular surface admits a global \({\mathbb Q}\)-Gorenstein smoothing. The more delicated part of this paper is the computation of the first homology group (more complicated than the previously mentioned similar construction where it had always cardinality \(\leq 2\)). This implies by the above mentioned result of M. Reid that also the algebraic fundamental is cyclic of order \(4\).NEWLINENEWLINEIt is worth mentioning that a few months later \textit{D. Frapporti} [Collect. Math. 64, No. 3, 293--311 (2013; Zbl 1303.14045)] constructed by completely different methods a concrete example of a numerical Campedelli surface with topological fundamental group cyclic of order \(4\). This implies that also the algebraic fundamental group of the Frapporti surface is cyclic of order \(4\).