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

Abstract

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 [Formula: see text] 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).