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Singular holomorphic functions for which all fibre-integrals are smooth

D. Barlet;  H. Maire — 2000

Annales Polonici Mathematici

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 ' ⋀ 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 ∞ .

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