Forgetful maps between Deligne-Mostow ball quotients

TL;DR

This paper investigates forgetful maps between Deligne-Mostow moduli spaces, classifies those extending to orbifold maps, and explores their geometric implications including fibrations and retractions in complex hyperbolic geometry.

Contribution

It provides a classification of extendable forgetful maps between Deligne-Mostow spaces and identifies their geometric significance in hyperbolic orbifolds.

Findings

Identification of cases where forgetful maps extend to orbifold maps

Connection to Livné fibrations and Mostow/Toledo maps

Existence of retractions onto totally geodesic submanifolds

Abstract

We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also include a retraction of a 3-dimensional ball quotient onto one of its 1-dimensional totally geodesic complex submanifolds.