POLES OF THE CURRENT |f|^2λ OVER AN ISOLATED SINGULARITY

Let (X, 0) be the germ of a normal space of dimension n+1 with an isolated singularity at 0 and let f be a germ of holomorphic function with an isolated regularity at 0. We prove that the meromorphic extension of the current

has a pole of order k at λ=-m-r for m∈ℕ large enough and r∈[0, 1[ if, and only if, e^-2iπr is an eigenvalue with nilpotency order k of the monodromy of f acting on Hⁿ(F)/J, where F is the Milnor fibre of f and J is the image of the restriction map Hⁿ(X\{0})→Hⁿ(F).