Singular holomorphic functions for which all fibre-integrals are smooth

D. Barlet; H. Maire

Annales Polonici Mathematici (2000)

  • Volume: 74, Issue: 1, page 65-77
  • ISSN: 0066-2216

Abstract

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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) ≤ 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 ∞ .

How to cite

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Barlet, D., and Maire, H.. "Singular holomorphic functions for which all fibre-integrals are smooth." Annales Polonici Mathematici 74.1 (2000): 65-77. <http://eudml.org/doc/208377>.

@article{Barlet2000,
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     $s ↦ ∫_\{f=s\} ϱ ω^\{\prime \} ⋀ \bar\{ω^\{\prime \prime \}\}, ϱ ∈ C^\{∞\}_\{c\}(X), ω^\{\prime \}, ω^\{\prime \prime \} ∈ Ω_\{X\}^\{n\}$, 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 $ω^\{\prime \},ω^\{\prime \prime \} ∈ ⍹_\{X\}$, the Grothendieck sheaf, are $C^\{∞\}$.},
author = {Barlet, D., Maire, H.},
journal = {Annales Polonici Mathematici},
keywords = {singularities; fibre-integrals; Mellin transform; currents},
language = {eng},
number = {1},
pages = {65-77},
title = {Singular holomorphic functions for which all fibre-integrals are smooth},
url = {http://eudml.org/doc/208377},
volume = {74},
year = {2000},
}

TY - JOUR
AU - Barlet, D.
AU - Maire, H.
TI - Singular holomorphic functions for which all fibre-integrals are smooth
JO - Annales Polonici Mathematici
PY - 2000
VL - 74
IS - 1
SP - 65
EP - 77
AB - 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     $s ↦ ∫_{f=s} ϱ ω^{\prime } ⋀ \bar{ω^{\prime \prime }}, ϱ ∈ C^{∞}_{c}(X), ω^{\prime }, ω^{\prime \prime } ∈ Ω_{X}^{n}$, 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 $ω^{\prime },ω^{\prime \prime } ∈ ⍹_{X}$, the Grothendieck sheaf, are $C^{∞}$.
LA - eng
KW - singularities; fibre-integrals; Mellin transform; currents
UR - http://eudml.org/doc/208377
ER -

References

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  1. [A-G-Z-V] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps II, Birkhäuser, 1988. 
  2. [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. 
  3. [B 84] D. Barlet, Contribution effective de la monodromie aux développements asymptotiques, Ann. Sci. Ecole Norm. Sup. 17 (1984), 293-315. Zbl0542.32003
  4. [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. Zbl0574.32011
  5. [B-M 89] D. Barlet and H.-M. Maire, Asymptotic expansion of complex integrals via Mellin transform, J. Funct. Anal. 83 (1989), 233-257. Zbl0707.32003
  6. [B-M 99] D. Barlet and H.-M. Maire, Poles of the current | f | 2 λ over an isolated singularity, Internat. J. Math. (2000) (to appear). 
  7. [L] F. Loeser, Quelques conséquences locales de la théorie de Hodge, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 1, 75-92. Zbl0862.32020

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