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Abstract

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) ≤ 0 for x ≤ 0, the fibre-integrals     $s ↦ ∫_{f=s} ϱ ω' ⋀ \bar{ω''}, ϱ ∈ C^{∞}_{c}(X), ω', ω'' ∈ Ω_{X}^{n}$, 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 $ω',ω'' ∈ ⍹_{X}$, the Grothendieck sheaf, are $C^{∞}$.