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For a germ (X, 0) of normal complex space of dimension n + 1 with an isolated singularity at 0 and a germ f : (X, 0) → (C, 0) of holomorphic function with df(x) 6= 0 for x 6= 0, the fibre-integrals s 7→ \ f=s ̺ω ′ ∧ ω, ̺ ∈ C∞ c (X), ω ′ , ω ′′ ∈ Ω X , are C on C and have an asymptotic expansion at 0. Even when f is singular, it may happen that all these fibre-integrals are C. We study such maps and build a family of examples where also fibre-integrals for ω, ω ∈ ωX , the Grothendieck sheaf, are C . 0. Introduction. Let (X, 0) be a germ of normal complex space of dimension n+ 1 with an isolated singularity at 0 and let f : (X, 0) → (C, 0) be a germ of holomorphic function such that df(x) 6= 0 for x 6= 0. In a previous paper [B-M 99], we have explained how eigenvalues of the monodromy M of f acting on H(F ), where F is the Milnor fibre of f , contribute to create poles of the meromorphic extension of the current λ 7→ Γ (λ) T X |f | . For eigenvalues different from 1, our results generalize those of the first author [B 84] for smooth X. But for the eigenvalue 1 of M , poles of the above current appear at negative integers if, and only if, 1 is also an eigenvalue of the monodromy of f acting in the quotient H(F )/J , where J is the image of the mapH(X\{0}) → H(F ) induced by restriction. (See Example 3 for explicit computation of the image.) When this restriction is surjective (which implies M = 1 and is therefore a very strong hypothesis), it follows that λ 7→ T X |f | has only simple poles on the negative integers. 2000 Mathematics Subject Classification: 32C30, 32S30, 32S50, 58K15.

Singular holomorphic functions for which all fibre-integrals are smooth