BIMSA > Algebraic and Complex Geometry Seminar Commensurability among Deligne-Mostow Monodromy Groups
Commensurability among Deligne-Mostow Monodromy Groups
Organizer
Da Li Shen
Speaker
Zhiwei Zheng
Time
Friday, December 6, 2024 3:00 PM - 4:00 PM
Venue
A3-2-301
Online
Zoom 559 700 6085 (BIMSA)
Abstract
This talk is based on a joint work with Chenglong Yu (YMSC, Tsinghua University). We give the commensurability classification of Deligne–Mostow ball quotients and show that the 104 Deligne–Mostow lattices form 38 commensurability classes. Firstly, we find commensurability relations among Deligne–Mostow monodromy groups, which are not necessarily discrete. This recovers and 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. The correspondences of moduli spaces provide the 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 also reproves Kappes–M\"oller and McMullen’s results on non-arithmetic Deligne–Mostow lattices.