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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Sur certaines singularités d’hypersurfaces $\textrm {II}$

Author: Daniel Barlet
Journal: J. Algebraic Geom. 17 (2008), 199-254
Published electronically: November 28, 2007
MathSciNet review: 2369085
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Abstract | References | Additional Information


The aim of the present article is to construct analytic invariants for a germ of a holomorphic function having a one-dimensional critical locus $S$. This is done for a large class of such germs containing for instance any quasi-homogeneous germ at the origin. More precisely, aside from the Brieskorn $(a,b)$-module at the origin and a (locally constant along $S^* : = S \setminus \{0\}$) sheaf $\mathcal {H}^n$ of $(a,b)$-modules associated with the transversal hypersurface singularities along each connected component of $S^*$, we construct also $(a,b)$-modules “with supports” $E_c$ and $E’_{c \cap S}$.

An interesting consequence of the local study along $S^*$ is the corollary showing that for a germ with an isolated singularity, the largest sub-$(a,b)$-module having a simple pole in its Brieskorn-$(a,b)$-module is independent of the choice of a reduced equation for the corresponding hypersurface germ.

We also give precise relations between these various $(a,b)$-modules via an exact commutative diagram. This is an $(a,b)$-linear version of the tangling phenomenon for consecutive strata we have previously studied in the “topological” setting for the localized Gauss-Manin system of $f$.

Finally we show that in our situation there exists a non-degenerate $(a,b)$-sesquilinear pairing \[ h : E \times E’_{c \cap S} \longrightarrow \vert \Xi ’ \vert ^2 \] where $\vert \Xi ’ \vert ^2$ is the space of formal asymptotic expansions at the origin for fiber integrals. This generalizes the canonical hermitian form defined in 1985 for the isolated singularity case (for the $(a,b)$-module version see the recent 2005 paper). Its topological analogue (for the eigenvalue $1$ of the monodromy) is the non-degenerate sesquilinear pairing \[ h : H^n_{c \cap S}(F, \mathbb {C})_{=1} \times H^n(F, \mathbb {C})_{=1} \to \mathbb {C} \] defined in an earlier paper for an arbitrary germ with a one-dimensional critical locus. Then we show this sesquilinear pairing is related to the non-degenerate sesquilinear pairing introduced on the sheaf $\mathcal {H}^n$ via the canonical Hermitian form of the transversal hypersurface singularities.

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  • D. Barlet, Développement asymptotique des fonctions obtenues par intégration sur les fibres, Invent. Math. 68 (1982), no. 1, 129–174 (French). MR 666639, DOI
  • D. Barlet, Forme hermitienne canonique sur la cohomologie de la fibre de Milnor d’une hypersurface à singularité isolée, Invent. Math. 81 (1985), no. 1, 115–153 (French). MR 796194, DOI
  • Daniel Barlet, Interaction de strates consécutives pour les cycles évanescents, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 4, 401–505 (French). MR 1123558
  • Daniel Barlet, Theory of $(a,b)$-modules. I, Complex analysis and geometry, Univ. Ser. Math., Plenum, New York, 1993, pp. 1–43. MR 1211877
  • D. Barlet, Théorie des $(a,b)$-modules. II. Extensions, Complex analysis and geometry (Trento, 1995) Pitman Res. Notes Math. Ser., vol. 366, Longman, Harlow, 1997, pp. 19–59 (French). MR 1477438
  • Daniel Barlet, La variation pour une hypersurface ayant une singularité isolée relativement à la valeur propre $1$, Séminaire de Singularités (Vandoeuvre-les-Nancy), Inst. Élie Cartan, vol. 15, Univ. Nancy I, Vandoeuvre-les-Nancy, 1997, pp. 5–29 (French). MR 1490315
  • Daniel Barlet, Interaction de strates consécutives. II, Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, 139–173 (French, with English summary). MR 2115970
  • Daniel Barlet, Modules de Brieskorn et formes hermitiennes pour une singularité isolée d’hypersurface, Singularités, Inst. Élie Cartan, vol. 18, Univ. Nancy, Nancy, 2005, pp. 19–46 (French, with English and French summaries). MR 2205835
  • Daniel Barlet, On the Brieskorn $(a,b)$-module of an isolated hypersurface singularity, C. R. Math. Acad. Sci. Paris 343 (2006), no. 11-12, 747–749 (English, with English and French summaries). MR 2284704, DOI
  • Daniel Barlet, Sur certaines singularités non isolées d’hypersurfaces. I, Bull. Soc. Math. France 134 (2006), no. 2, 173–200 (French, with English and French summaries). MR 2233704, DOI
  • Daniel Barlet, Interaction de strates consécutives. III. Le cas de la valeur propre 1, Manuscripta Math. 121 (2006), no. 2, 201–263 (French, with English summary). MR 2264022, DOI
  • Barlet, D. Sur les germes de fonctions holomorphes à lieu singulier de dimension 1 : le cas général, preprint Institut E. Cartan 2007/ $n^0$1.
  • Barlet, D. Finite determination for a regular $(a,b)$-module, preprint Institut E. Cartan 2007/ $n^0$15.
  • Barlet, D. and Saito, M. Brieskorn modules and Gauss-Manin systems for non isolated hypersurface singularities, preprint Institut E. Cartan 2004/ $n^0$54. A paraître à la London Math. Soc.
  • R. Belgrade, Dualité et spectres des $(a,b)$-modules, J. Algebra 245 (2001), no. 1, 193–224 (French, with English summary). MR 1868189, DOI
  • Egbert Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math. 2 (1970), 103–161 (German, with English summary). MR 267607, DOI
  • Kashiwara, M. On the maximally over determined systems of differential equations, Publ. R.I.M.S. 10 (1975), p. 563-579.
  • Malgrange, B. Systèmes différentiels à coefficients constants, Seminaire Bourbaki 15 (1962-1963) exposé 246.
  • Bernard Malgrange, Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup. (4) 7 (1974), 405–430 (1975) (French). MR 372243
  • Marcos Sebastiani, Preuve d’une conjecture de Brieskorn, Manuscripta Math. 2 (1970), 301–308 (French, with English summary). MR 267608, DOI
  • Mikio Sato, Takahiro Kawai, and Masaki Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Springer, Berlin, 1973, pp. 265–529. Lecture Notes in Math., Vol. 287. MR 0420735
  • Morihiko Saito, Period mapping via Brieskorn modules, Bull. Soc. Math. France 119 (1991), no. 2, 141–171 (English, with French summary). MR 1116843

Additional Information

Daniel Barlet
Affiliation: Institut Universitaire de France et Institut Elie Cartan UMR 7502, Nancy-Université, CNRS, INRIA, BP 239 - F - 54506 Vandoeuvre-lès-Nancy Cedex, France

Received by editor(s): October 18, 2005
Received by editor(s) in revised form: April 21, 2007
Published electronically: November 28, 2007
Dedicated: Á mon ami Masaki Kashiwara