Reflective modular varieties and their cusps

Abstract

We classify reflective automorphic products of singular weight under certain regularity assumptions. Using obstruction theory we show that there are exactly 11 such functions. They are naturally related to certain conjugacy classes in Conway's group $\text{Co}_0$. The corresponding modular varieties have a very rich geometry. We establish a bijection between their $1$-dimensional type-$0$ cusps and the root systems in Schellekens' list. We also describe a $1$-dimensional cusp along which the restriction of the automorphic product is given by the eta product of the corresponding class in $\text{Co}_0$. Finally we apply our results to give a complex-geometric proof of Schellekens' list.