The Deligne-Mostow list and special families of surfaces

TL;DR

This paper investigates the existence of infinite families of surfaces with specific invariants of CM type, constructing numerous examples mainly of K3 surfaces and some of general type, using Deligne-Mostow's work.

Contribution

It constructs new families of surfaces with p_g=1 that form special subvarieties in moduli space, extending the understanding of CM type surfaces and their properties.

Findings

Existence of infinitely many surfaces with CM type invariants.

Construction of families of K3 surfaces and some of general type.

Proof that a very general K3 surface cannot be dominated by low-genus curves.

Abstract

We study whether there exist infinitely many surfaces with given discrete invariants for which the H^2 is of CM type. This is a surface analogue of a conjecture of Coleman about curves. We construct a large number of examples of families of surfaces with p_g = 1 that in the moduli space cut out a special subvariety; these provide a positive answer to our question. Most of these are families of K3 surfaces but we also obtain some families of surfaces of general type. As input for our construction we use the work of Deligne and Mostow. Finally we prove that a very general K3 surface cannot be dominated by a product of curves of small genus.