Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Sur certaines singularités d'hypersurfaces $ {\rm 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

Abstract: 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

$\displaystyle 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

$\displaystyle 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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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

American Mathematical Society