A computation on the decomposition factors of D -modules over a hyperplane arrangement in space

Abstract

Let m be a positive integer,α_i:C^n⟶C^n, for i=1,2,…,m be linear forms and H_i={P∈C^n:α_i (P)=0} be the corresponding hyperplane for each i=1,2,…,m . The linear forms α_1,α_2,…,α_m define a hyperplane arrangement and X=C^n\V(α), where α=∏_(i=1)^m α_i and "V"(α)={P∈C^n:α(P)=0}. The coordinate ring O_X of X is the localization 〖C[x_1,…,x_n]〗_αand the ring O_X=〖C[x_1,…,x_n]〗_α is a holonomic A_n-module, where A_n is the n-th Weyl algebra, hence it has finite length. In this work, we will compute the number of decomposition factors of the A_3-module 〖C[x]〗_α, where α defines a central hyperplane arrangement in space, in terms of the no-broken circuits and describe the decomposition factors in terms of their supports.