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
blowdown 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.
