Jongil Park

A classification of numerical Campedelli surfaces

video: comming soon.
abstract:
In order to classify complex surfaces of general type with \(p_g=0\) and \(K^2=2\) (Such surfaces are usually called numerical Campedelli surfaces), it seems to be natural to classify them first up to their topological types. It has been known by M. Reid and G. Xiao that the algebraic fundamental group \(\pi_1^{\text{alg}}\) of a numerical Campedelli surface is a finite group of order \(\leq 9\). Furthermore the topological fundamental groups \(\pi_1\) for any numerical Campedelli surfaces are also of order \(\leq 9\) in as far as they have been determined. Hence it is a natural conjecture that \(|\pi_1| \leq 9\) for all numerical Campedelli surfaces.
Conversely one may ask whether every group of order \(\leq 9\) occurs as the topological fundamental group or as the algebraic fundamental group of a numerical Campedelli surface. It has been proved that the dihedral groups \(D_3\) of order \(6\) or \(D_4\) of order \(8\) cannot be fundamental groups of numerical Campedelli surfaces. Furthermore, it has also been known that all other groups of order \(\leq 9\), except \(D_3\), \(D_4\), \(\mathbb{Z}/4\mathbb{Z}\), \(\mathbb{Z}/6\mathbb{Z}\), occur as the topological fundamental groups of numerical Campedelli surfaces.
Unlike the case of topological fundamental group, there is also a known numerical Campedelli surface with \(H_1=\mathbb{Z}/6\mathbb{Z}\) (in fact \(\pi_1^{\text{alg}}=\mathbb{Z}/6\mathbb{Z}\)). Therefore all abelian groups of order \(\leq 9\) except \(\mathbb{Z}/4\mathbb{Z}\) occur as the first homology groups (and algebraic fundamental groups) of numerical Campedelli surfaces. Nevertheless, the question on the existence of numerical Campedelli surfaces with a given topological type was completely open for \(\mathbb{Z}/4\mathbb{Z}\).
Recently Heesang Park, Dongsoo Shin and myself constructed a new minimal complex surface of general type with \(p_g=0\), \(K^2=2\) and \(H_1=\mathbb{Z}/4\mathbb{Z}\) (in fact \(\pi_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}\)) using a rational blow-down surgery and a \(\mathbb{Q}\)-Gorenstein smoothing theory, so that the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups are settled down. In this talk I'd like to review how to construct such a numerical Campedelli surface.