Periods of an arrangement of six lines and Campedelli surfaces

TL;DR

This paper constructs a period map for Campedelli surfaces, linking their moduli to a quotient of Enriques surfaces, and explores monodromy properties using classical methods.

Contribution

It introduces a new period map for Campedelli surfaces via a covering approach, connecting their moduli space to an arithmetic quotient.

Findings

The period map is an isomorphism with a Baily-Borel compactification.

Properties of monodromy for Campedelli surfaces are established.

The approach uses classical techniques without modern machinery.

Abstract

We define a period map for classical Campedelli surfaces, using a covering trick as in the case of Enriques surfaces: the period map is shown to come from a family of Enriques surfaces, obtained as quotients of the Campedelli surface by an involution. The period map realises an isomorphism between a projective variety obtained by invariant theory, and the Baily-Borel compactification of an arithmetic quotient, in the same fashion as in the work of Matsumoto, Sasaki and Yoshida. The result is proved from scratch using traditional methods. As another consequence we determine properties of the monodromy of Campedelli surfaces with a choice of double cover.