Abstract:
Let R be the polynomial ring C[x_1,...,x_n] in n variables with complex coefficients. Let f be a non constant polynomial in R. We denote by Syz_R(f_1,...,f_n,f) the R-module of syzygies among the partial derivatives f_i of f w.r.t. x_i. To this syzygy module we associate two (different but related) modules over the Weyl algebra D:=D(R) of linear differential operators on the polynomial ring R. We will show how, in some interesting cases, how these D-modules can help to compute the (singular) cohomology of the complement in C^n of the hypersurface defined by f=0.