A simply connected numerical Campedelli surface with an involution (Q368630)

Recently there has been growing interest in the description of involutions of minimal surfaces of general type with geometric genus 0, following results of \textit{A. Calabri} et al. [Trans. Am. Math. Soc. 359, No. 4, 1605--1632 (2007; Zbl 1124.14036)]. We recall that a numerical Campedelli (resp. numerical Godeaux) surface is a minimal surface of general type with geometric genus 0 and canonical degree 2 (resp. 1). The authors show the existence of a simply connected numerical Campedelli surface with an involution whose quotient is birational to a simply connected numerical Godeaux surface. It was indeed already known that one could have an involution on a numerical Campedelli surface with \(4\) isolated fixed points, thus yielding a numerical Godeaux surface at the quotient as in [\textit{D. Frapporti}, Collect. Math. 64, No. 3, 293--311 (2013; Zbl 1303.14045)]. Still, the simple connectedness here is an important property, since even the existence of simply connected numerical Godeaux and Campedelli surfaces has been a challenging problem for a long time, and we are still far from classifying all of them. The strategy combines double covering and \(\mathbb{Q}\)-Gorenstein smoothing techniques, and both surfaces are obtained by smoothing a (different) singular rational elliptic surface. To show the existence of the smoothing, the authors need to show that there is no local-to-global obstruction. This involves a new technique which generalizes a result of \textit{D. M. Burns jun.} and \textit{J. M. Wahl} [Invent. Math. 26, 67--88 (1974; Zbl 0288.14010)] describing the space of first-order deformations of a singular complex surface with only rational double points.