Singular holomorphic functions for which all fibre-integrals are smooth

• Autorzy: D. Barlet , H.-M. Maire
• Języki publikacji: EN

Abstrakty

EN

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 $C^{∞}$.

Słowa kluczowe

EN

singularities; fibre-integrals; Mellin transform; currents

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 74
• Numer: 1
• Strony: 65-77

Daty

wydano

2000

otrzymano

1999-07-20

Twórcy

autor

D. Barlet

• Université H. Poincaré (Nancy I) et, Institut Universitaire de France, Institut E. Cartan UHP/CNRS/INRIA, UMR 7502, Boîte postale 239, F-54506 Vandœuvre-lès-Nancy, France

autor

H.-M. Maire

• Section de Mathématiques, Université de Genève, Case postale 240, CH-1211 Genève 24, Switzerland

Bibliografia

• [A-G-Z-V] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps II, Birkhäuser, 1988.
• [B 78] D. Barlet, Le faisceau $ω_X$ sur un espace analytique X de dimension pure, in: Lecture Notes in Math. 670, Springer, 1978, 187-204.
• [B 84] D. Barlet, Contribution effective de la monodromie aux développements asymptotiques, Ann. Sci. Ecole Norm. Sup. 17 (1984), 293-315.
• [B 85] D. Barlet, Forme hermitienne canonique sur la cohomologie de la fibre de Milnor d'une hypersurface à singularité isolée, Invent. Math. 81 (1985), 115-153.
• [B-M 89] D. Barlet and H.-M. Maire, Asymptotic expansion of complex integrals via Mellin transform, J. Funct. Anal. 83 (1989), 233-257.
• [B-M 99] D. Barlet and H.-M. Maire, Poles of the current $|f|^{2λ}$ over an isolated singularity, Internat. J. Math. (2000) (to appear).
• [L] F. Loeser, Quelques conséquences locales de la théorie de Hodge, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 1, 75-92.