2024/11/14 by Chenglong Yu, Zhiwei Zheng, Yu, Chenglong +1 · 1 citation
Mathematics · #14D05 #22E40 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Group Theory (math.GR) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2411.09682
openalex publication_date 2024/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper gives the commensurability classification of Deligne--Mostow ball quotients and shows that the 104 Deligne--Mostow lattices form 38 commensurability classes. First, we find commensurability relations among Deligne--Mostow monodromy groups, which are not necessarily discrete. This generalizes previous work by Sauter and Deligne--Mostow in dimension two. In this part, we consider certain projective surfaces with two fibrations over the projective line, which induce two sets of Deligne--Mostow data. Correspondences between moduli spaces provide a geometric realization of commensurability relations. Secondly, we obtain commensurability invariants from conformal classes of Hermitian forms and toroidal boundary divisors. This completes the commensurability classification of Deligne--Mostow lattices and provides an alternative approach to the results of Kappes--Möller and McMullen on non-arithmetic Deligne--Mostow lattices.