Calabi-Yau Varieties via Cyclic Covers, and Complex Hyperbolic Structures for their Moduli Spaces

TL;DR

This work classifies Calabi–Yau varieties arising as degree-$d$ cyclic covers of products of projective spaces whose period maps factor through complex hyperbolic balls, and it formalizes when these ball-type structures occur. By developing the VHS framework for the $m{chi}$-eigenspace of middle cohomology, establishing an infinitesimal Torelli theorem, and performing explicit Betti-number computations, the authors identify ball-type partitions for $Z=(P^1)^n$ and $Z=P^3$, including a Fermat-type half-twist construction that lifts ball-type from lower to higher dimension. They prove arithmeticity of the resulting monodromy groups and connect their CY ball quotients to Deligne–Mostow lattices, showing vast commensurability relations and organizing a broad network of lattices arising from these CY constructions. The paper also investigates refinements and completeness of the CY families, providing criteria for when a simultaneous crepant resolution yields a complete, smooth moduli family, and extending the Deligne–Mostow relations to higher dimensions with a comprehensive table of cases.

Abstract

In this paper we mainly study Calabi-Yau varieties that arise as triple covers of products of projective lines branched along simple normal crossing divisors. For some of those families of Calabi-Yau varieties, the period maps factor through arithmetic quotients of complex hyperbolic balls. We give a classification of such examples. One of the families was previously studied by Voisin, Borcea and Rohde. For these ball-type cases, we will show arithmeticity of the monodromy groups. These ball quotients are all commensurable to ball quotients in Deligne-Mostow theory. As a byproduct, we prove some commensurability relations among arithmetic groups in Deligne-Mostow theory.

Figures (1)

• Figure 1: $(3,1)+(0,2)$

Theorems & Definitions (30)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['theorem: main']}